Carlo Gasbarri

Angelo Lopez

Thomas Dedieu

Aiuto Da

Robert Laterveer

Christian Lehn

Mingmin Shen

Charles Vial

Claire Voisin

Tutti

Zia Maria

Zio Nino

Zia Denise

Zio Lino E I Miei Amici Julio

la sua guida, espressi durante tutti e tre gli anni di dottorato.

rtie IF Dans la première partie, on étudie le cône des classes pseudoeectives des variétés k-Fano. Toutes nos variétés sont algébriques, irréductibles, réduites et dénies sur le corps des nombres complexes. X est une variété de dimension n, l'anneau R indique un des anneaux Z, Q ou R. On note M ⊗ Z R avec M R pour chaque Z-module M . Définition 0.0.1. Un k-cycle sur X est une somme nie Z-linéaire a i V i de sousvariétés de dimension k. Le groupe de tous les k-cycles est notée Z k (X).

On peut dénir sur Z k (X) plusieurs relations d'équivalence.

• Équivalence rationnelle (Denition 2.1.3), Rat k (X) est le groupe de tous les k-cycles rationnellement équivalents à zéro. • Équivalence algébrique (Denition 2.1.6), Alg k (X) est le groupe de tous les kcycles algébriquement équivalents à zéro. Le groupe des k-cycles modulo l'équivalence rationnelle sur X est le groupe de Chow

A k (X) = Z k (X) /Rat k (X).

• Équivalence homologique (Denition 2.1.7), Hom k (X) est le groupe de tous les k-cycles homologiquement équivalents à zéro. • Équivalence numérique (Denition 2.1.12), Num k (X) est le groupe de tous les kcycles numériquement équivalents à zéro. Le groupe des k-cycles modulo l'équivalence numérique sur X est le groupe quotient

N k (X) = Z k (X) /Num k (X).
Les relations d'équivalence dénies sur Z k (X) satisfont la chaîne d'inclusions suivante

Rat k (X) ⊆ Alg k (X) ⊆ Hom k (X) ⊆ Num k (X) ⊆ Z k (X),
impliquant l'existence du diagramme suivant.

(0.0.1)

A k (X) / / / / Z k (X) /Alg k (X) / / / / Z k (X) /Hom k (X) _ π k / / / / N k (X) H 2k (X, Z)
Soit E un bré vectoriel sur X. On peut démontrer qu'il y a un unique ensemble d'applications sur les groupes de Chow

c i (E) • _ : A k (X) → A k-i (X),
et sont appelées classes de Chern. Les principales propriétés des classes de Chern sont énoncées dans Theorem 2.1.8. Chaque polynôme en classes de Chern à coecients dans R dénit une action sur la somme directe ⊕ k A k (X) R . En particulier, si P est un polynôme homogène de degré d, c i (E) est de degré i et α ∈ A k (X) R , alors

P • α ∈ A k-d (X) R .
Les caractères de Chern sont des exemples intéressants de polynômes homogènes à coecients rationnels. On peut les calculer en utilisant la formule suivante (cf. [pulWV,Example 15.1.2(b)]), où c i = c i (E).

ch k (E) = 1 k! det         c 1 2c 2 3c 3 . . . kc k 1 c 1 2c 2 . . . . . . 1 . . . . . . 3c 3 . . . c 1 2c 2 1 c 1        
Le polynôme ch k (E) est homogène de degré k. Les premièrs caractères de Chern de E sont

ch 1 (E) = c 1 (E) ch 2 (E) = 1 2 (c 2 1 (E) -2c 2 (E)) ch 3 (E) = 1 6 (c 3 1 (E) -4c 1 (E)c 2 (E) + 3c 3 (E))
Si X est lisse et T X est le bré tangent, les classes de Chern et les caractères de Chern de T X sont respectivement notés c k (X) et ch k (X). On dénit le cône des classes eectives et pseudoeectives comme suit.

Définition 0.0.2. Un k-cycle r i V i ∈ Z k (X) R sur X est eectif si r i ≥ 0 pour chaque i. Une classe α ∈ N k (X) R est eective si α = [ r i V i ] pour chaque k-cycle eectif r i V i . On note Eff k (X), le cône des classes de k-cycles eectifs en N k (X) R , et par Eff k (X) son adhérence. Les classes pseudoeectives sont les classes de Eff k (X).

On sait que si X est lisse, il existe une forme bilinéaire symétrique non-dégénérée :

_ • _ : N k (X) R ⊗ N n-k (X) R → R.
Définition 0.0.3. Soit X une variété lisse. Une classe α ∈ N k (X) R est positive si α • β > 0 pour chaque β ∈ Eff n-k (X)\{0}, elle est nef si α • β ≥ 0 pour chaque

β ∈ Eff n-k (X).
On peut à présent dénir l'objet de la première partie de cette thèse. Définition 0.0.4. Une variété est k-Fano si ch s (X) est positif pour chaque 1 ≤ s ≤ k, elle est k-Fano faible pour k > 1 si X est (k -1)-Fano et ch k (X) est nef.

Cette dénition généralise la dénition de variété de Fano ; en fait les variétés de Fano sont exactement les 1-Fano par le Théorème de Kleiman (Theorem 2.1.20). Les variétés 2-Fano ont été introduites par de Jong et Starr dans [dtHTD dtHU], où ils ont démontré que par le point général de chaque variété 2-Fano avec pseudo-indice au moins égal à 3, il passe une surface rationnelle. Cela généralise un résultat classique, c'est-à-dire que chaque variété de Fano est uniréglée (par le point général il passe une courbe rationnelle). Ensuite, Araujo et Castravet ont démontré dans [egIP,Theorem 1.5(3)] que (avec des hypothèses supplémentaires) le point général d'une variété 3-Fano est traversé par une variété rationnelle de dimension trois. De plus, elles ont classié les variétés 2-Fano avec indice au moins égal à n -2 ([egIQ]).

Nous sommes intéressés par les cônes des classes pseudoeectives des variétés k-Fano. En fait nous avons la suivante.

Conjecture. Soit X une variété k-Fano. Alors le cône Eff k (X) est polyhédral.

Pour le 1-Fano nous savons que la conjecture est vraie. En fait, on a le corollaire suivant du Théorème du Cône de Mori (Theorem 2.1.25).

Corollaire 0.0.5. Soit X une variété de Fano. Alors le cône Eff 1 (X) est polyhédral. Il est donc naturel de se demander si ce résultat aussi peut se généralises aux k-Fano, de même que l'existence de sous-variétés rationnelles.

En général, si X est de Fano nous sommes incapables de dire si Eff k (X) est polyhédral dans les cas k = 0, 1, n. Un exemple par Tschinkel (énoncé dans [hivII, Example 6.10]) montre qu'une variété de Fano peut avoir Eff 2 (X) qui n'est pas polyhédral (Proposition 2.5.4).

La stratégie adoptée dans la plupart de la thèse est d'utiliser le lemme suivant (Lemma 2.1.29).

Lemme 0.0.6. Soit X une variété projective. Alors (1) Si rkA k (X) = 1 ou b 2k (X) = 1, alors Eff k (X) est un rayon.

(2) Si rkA k (X) = 2 ou b 2k (X) = 2, alors Eff k (X) est un rayon ou il est engendré par deux rayons.

La démonstration est une application du diagramme (0.0.1). En fait, le cône Eff k (X) est un cône dans un espace vectoriel de dimension au plus 2, c'est donc polyhédral.

Nous calculons donc les nombres de Betti pour certaines variétés k-Fano. Enn, nous utilisons la classication de Araujo et Castravet [egIQ, Theorem 3] (Theorem 2.5.2) pour démontrer le suivant. Théorème 0.0.7. Soit X une variété k-Fano avec indice i X ≥ n -2. Alors le cône Eff k (X) est polyhédral pour k = 2, 3.

Avant de démontrer ce théorème, on considère certains cas particuliers. Proposition 0.0.8. Soit X une intersection complète lisse dans un espace projectif à poids. Alors b 2k (X) = 1 pour chaque k = n 2 , et donc le cône Eff k (X) est polyhédral. Si X est dans un espace projectif, k-Fano faible et 1 ≤ s ≤ k, alors b 2s (X) ≤ 2, et donc le cône Eff k (X) est polyhédral.

La conjecture est donc vraie pour les intersections complètes dans un espace projectif. On considère maintenant les intersections complètes dans trois familles des variétés homogènes : G(r, s), OG(r, s) e SG(r, s). Soit G un groupe algébrique réductible sur C, un sous-groupe P de G est dit parabolique si G/P est une variété projective. Une variété homogène est une variété pour laquelle il existe une action transitive d'un groupe algébrique. Pour les variétés homogènes, il est bien connu que tous les cônes de classes pseudoeectives sontpolyhédrales. On peut démontrer qu'une variété homogène est de Fano si et seulement si elle est un produit G 1 /P 1 × ... × G k /P k , ou P i est un sous-groupe parabolique du groupe simple G i . Pour les Grassmanniennes on a le résultat suivante. Proposition 0.0.9. Soit X une variété 2-Fano faible, intersection complète dans G(r, s). Alors, b 4 (X) ≤ 2. En particulier le cône Eff 2 (X) est polyhédral.

Pour les Grassmanniennes orthogonales on démontre que : Proposition 0.0.10. Soit s, r des entiers positifs tels que 2 ≤ r ≤ s 2 , e s 2r = 1, 2 si s est pair. Si s = 2r (respectivement, s = 2r), X est une 2-Fano faible, intersection complète dans une composante connexe de OG(r, s) sous le plongement de Plücker (respectivement, demi-spin), X est très général si X ⊆ OG(2, 7), alors le cône Eff 2 (X) est polyhédral.

An de prouver cela, nous utilisons à la fois [pWT, Theorem 2] et Lemme 0.0.11. Soit X une composante connexe de OG(r, s),

2 ≤ r ≤ m = s 2 . Alors b 4 (X) =      1 si r = m 3 si 1 ≤ m -r ≤ 2, s pair 2 autrement
Pour les intersections complètes dans les Grassmanniennes symplectiques, nous avons le résultat suivant. Proposition 0.0.12. Soit X une 2-Fano faible, intersection complète dans SG(r, s).

Alors, b 4 (X) ≤ 2. En particulier le cône Eff 2 (X) est polyhédral.

Ces propositions ont été obtenues en utilisant divers résultats dans [egIQ], dans laquel sont données plusieurs conditions sur une variété X pour être 2-Fano faible. Nous répondons également à [egIQ, Question 39 and 40] ; en fait, nous démontrons le résultat suivant Théorème 0.0.13. Soit Y = G(2, 5) o G(2, 6), et soit X une intersection complète de type (1, 1) en Y avec le plongement de Plücker. Alors X n'est pas 2-Fano faible.

Pour prouver ce théorème, nous avons utilisé [deIS, Corollary 5.1] (impliquant que si X est général alors ce n'est pas 2-Fano), et [yttIS,Proposition 3]. Ce théorème complète la classication des variétés 2-Fano faibles avec indice i X ≥ n -2 dans [egIQ, Theorem 4]. Dans la démonstration, nous avons utilisé le suivant. Lemme 0.0.14. Soit X 11 une intersection complète de type (1, 1) en G(2, 5) avec le plongement de Plücker. Alors b 4 (X) = 2.

Pour démontrer ce lemme, nous avons calculé le diamant de Hodge de X 11 . Puisque le théorème de Sommese (Theorem 2.4.2) ne sut pas pour déterminer le diamant entier, nous avons également utilisé [noVT].

rtie PF Dans la seconde partie, nous étudions le lieu d'indétermination d'une application rationnelle introduite dans [oiIT,Proposition 4.8]. Le lieu d'indétermination est le plus grand sous-ensemble fermé constitué des points où l'application n'est pas dénie. Nous pouvons facilement vérier que si

X f h / / W Y g > >
est un diagramme commutatif avec f et h des applications rationnelles et g un morphisme, alors on a Ind(h) ⊆ Ind(f ). De plus, par un résultat de Hironaka [rirTR, I.Question (E) p.140], pour chaque application rationnelle f : X Y entre variétés lisses, il y a une résolution des indéterminations. C'est-à-dire, un diagramme commutatif

X π f X f / / Y
ou X est lisse et π est une application rationnelle et un isomorphisme en dehors de Ind(f ).

Définition 0.0.15. Soit X une variété de Kähler compacte. On dit que X est une variété hyperkählérienne si X est simplement connexe et H 0 (X, ∧ 2 Ω X ) est engendré par une 2-forme non dégénérée en tout point de X.

Par un résultat classique, la première classe de Chern d'une variété avec métrique de Ricci plate est nulle. De plus, pour la décomposition de Bogomolov [fogURD feVQ], ces variétés ont un revêtement étale ni donné par le produit des tores complexes, des variétés de Calabi-Yau et des variétés hyperkählériennes. En dimension 2, les seuls exemples des variétés hyperkählériennes sont les surfaces K3. Beauville a démontré dans [feVQ, Théorèmes 3 and 4] que pour chaque n ≥ 0, le schéma de Hilbert S [n] paramétrant les sous-schémas de longueur n d'une surface K3 S, et la variété de Kummer généralisée K n (T ) associée à un tore complexe T de dimension 2, sont des variétés hyperkählériennes. Une variété hyperkählérienne obtenue à partir d'une déformation de S [n] (respectivement, d'une variante de Kummer généralisée) est dite de type K3 [n] (respectivement, de type K n (T )). Ces variétés sont particulièrement intéressantes car elles permettent de construire des variétés hyperkählériennes complexes de toute dimension paire. Par la suite O'Grady dans [y9qHQD y9qWW] a construit deux nouveaux exemples en dimension 6 et 10 de variétés hyperkählériennes qui ne sont pas des déformations de types connus. Dans la deuxième partie de la thèse, nous utilisons deux exemples particuliers de variétés hyperkählériennes de type K3 [n] . Le premier exemple est celui fourni par Beauville et Donagi.

Théorème 0.0.16 ([fhVS]). Soit Y ⊆ P 5 une hypersurface cubique lisse de dimension 4. Soit F la variété de Fano de Y . Alors

(1) La variété F est hyperkählérienne de type K3 [2] .

(2) Si Y est pfaenne, alors F est isomorphe à S [2] pour une surface K3 S.

Remarque 0.0.17. Cette variété a été utilisée pour donner un exemple de classe nef mais pas pseudoeective. En fait, en général, il n'est pas vrai que (cf. [evIU, Corollary] ou [vehIS,Corollary 6.3]). Il y a un diagramme commutatif

Nef k (X) ⊆ Eff k (X) pour 2 ≤ k ≤ dim X -2.
F 3 (Y ) φ / / G(4, 6) Z g 9 9 où F 3 (Y ) → G(4, 6
) est le morphisme qui envoie chaque cubique rationnelle sur l'espace P 3 qui la contient. Le morphisme g est ni de degré 72 sur un ouvert de Z .

Voisin a construit une application rationnelle de degré 6 dans [oiIT, Proposition 4.8]

ψ : F × F Z.
De manière synthétique, l'application ψ envoie une paire de droites non concourantes (l, l ) ∈ F × F sur la classe d'une courbe rationnelle normale de degré 3 contenue dans le système linéaire |L -L -K S l,l | de la surface cubique S l,l := L, L ∩ Y . Cela nous donne un diviseur uniréglé sur Z. En fait, si on a une résolution des indéterminations L'application ψ est dénie en tout point de l'ouvert rte IF Nella prima parte studiamo il cono delle classi pseudoeettive delle varietà k-Fano. Indichiamo con X una varietà (cioè, uno schema algebrico complesso ridotto e irriducibile) di dimensione n. L'anello R indica uno qualunque tra Z, Q oppure R.

F × F π ψ " " F × F ψ / / Z
U ADE := (l, l ) ∈ F × F |(l, l ) / ∈ I, S l,l := L, L ∩ Y a des singularités ADE et la composition σ • ψ : F × F Z coïncide
Indichiamo M ⊗ Z R con M R per ogni Z-modulo M . Definizione 0.0.1. Un k-ciclo su X è una somma nita Z-lineare a i V i di sottovarietà di dimensione k. Il gruppo di tutti i k-cicli è indicato con Z k (X).
Possiamo denire su Z k (X) varie relazioni di equivalenza.

• Equivalenza razionale (Denition 2.

1.3), Rat k (X) è il gruppo di k-cicli razionalmente equivalenti a zero. • Equivalenza algebrica (Denition 2.1.6), Alg k (X) è il gruppo di k-cicli algebri- camente equivalenti a zero. Il gruppo dei k-cicli modulo equivalenza razionale su X è il gruppo di Chow A k (X) = Z k (X) /Rat k (X).
• Equivalenza omologica (Denition 2.1.7), Hom k (X) è il gruppo di k-cicli omologicamente equivalenti a zero. • Equivalenza numerica (Denition 2.1.12), Num k (X) è il gruppo di k-cicli numericamente equivalenti a zero. Il gruppo dei k-cicli modulo equivalenza numerica su X è il gruppo quoziente

N k (X) = Z k (X) /Num k (X).
Le relazioni di equivalenza denite su Z k (X) soddisfano la seguente catena di inclusioni

Rat k (X) ⊆ Alg k (X) ⊆ Hom k (X) ⊆ Num k (X) ⊆ Z k (X),
che implica l'esistenza del seguente diagramma.

(0.0.2)

A k (X) / / / / Z k (X) /Alg k (X) / / / / Z k (X) /Hom k (X) _ π k / / / / N k (X) H 2k (X, Z) Sia E un brato vettoriale su X. Sui gruppi di Chow si dimostra l'esistenza di un unico insieme di applicazioni c i (E) • _ : A k (X) → A k-i (X), dette classi di Chern. Le principali proprietà delle classi di Chern sono enunciate in Theorem 2.1.8. Ogni polinomio nelle classi di Chern a coecienti in R agisce sulla somma diretta ⊕ k A k (X) R . In particolare, se P è un polinomio omogeneo di grado d, dove c i (E) ha grado i e α ∈ A k (X) R , allora P • α ∈ A k-d (X) R .
I caratteri di Chern sono tra i più interessanti esempi di polinomi omogenei a coecienti razionali. Si possono calcolare utilizzando la seguente formula chiusa (vedi [pulWV,Example 15.1.2(b)

]), in cui c i = c i (E). ch k (E) = 1 k! det         c 1 2c 2 3c 3 . . . kc k 1 c 1 2c 2 . . . . . . 1 . . . . . . 3c 3 . . . c 1 2c 2 1 c 1         Il polinomio ch k (E) è omogeneo di grado k. I primi caratteri di Chern di E sono ch 1 (E) = c 1 (E) ch 2 (E) = 1 2 (c 2 1 (E) -2c 2 (E)) ch 3 (E) = 1 6 (c 3 1 (E) -4c 1 (E)c 2 (E) + 3c 3 (E))
Se X è liscia e T X è il brato tangente, le classi di Chern e i caratteri di Chern di T X sono indicati con, rispettivamente, c k (X) e ch k (X). Deniamo adesso il cono delle classi eettive e pseudoeettive.

Definizione 0.0.

2. Un k-ciclo r i V i ∈ Z k (X) R su X è eettivo se r i ≥ 0 per ogni i. Una classe α ∈ N k (X) R è eettiva se α = [ r i V i ] per qualche k-ciclo eettivo r i V i . Indichiamo con Eff k (X) il cono generato dalle classi di k-cicli eettivi in N k (X) R , e con Eff k (X) la sua chiusura. Le classi pseudoeettive sono le classi di Eff k (X).
Sappiamo che se X è liscia, abbiamo una forma bilineare simmetria non degenere:

_ • _ : N k (X) R ⊗ N n-k (X) R → R. Definizione 0.0.3. Sia X una varietà liscia. Un ciclo α ∈ N k (X) R è positivo se α • β > 0 per ogni β ∈ Eff n-k (X)\{0}, ed è nef se α • β ≥ 0 per ogni β ∈ Eff n-k (X).
Deniamo adesso l'oggetto centrale della prima parte della tesi.

Definizione 0.0.4. Sia X una varietà liscia. Allora X si dice k-Fano se ch s (X) è positivo per ogni

1 ≤ s ≤ k, ed è k-Fano debole per k > 1 se X è (k -1)-Fano e ch k (X) è nef.
Questa denizione generalizza la denizione di varietà Fano, infatti le varietà Fano sono esattamente le 1-Fano per il criterio di Kleiman (Theorem 2.1.20). Le 2-Fano sono state introdotte da de Jong e Starr in [dtHTD dtHU], dove hanno dimostrato che in una varietà 2-Fano con pseudo-indice almeno 3, passa una supercie razionale nel punto generale. In questo modo si migliora un risultato classico, e cioè che una varietà Fano è rigata (cioè passa una curva razionale per il punto generale). Successivamente, Araujo e Castravet hanno dimostrato in [egIP, Theorem 1.5(3)] che (sotto opportune ipotesi) in una 3-Fano debole passa una 3-varietà razionale per il punto generale. Inoltre hanno classicato le varietà 2-Fano con indice almeno n -2 ([egIQ]).

Noi saremo interessati ai coni di classi pseudoeettive delle varietà k-Fano. Infatti abbiamo la seguente.

Congettura. Sia X una varietà k-Fano. Allora Eff k (X) è poliedrale.

Nel caso delle 1-Fano, è già noto che la congettura è vera. Infatti si ha la seguente applicazione del Teorema del Cono di Mori (Theorem 2.1.25).

Corollario 0.0.5. Sia X una varietà Fano. Allora Eff 1 (X) è poliedrale. Quindi è lecito chiedersi se anche questo corollario può essere generalizzato per le k-Fano, proprio come è stata generalizzata l'esistenza di sottovarietà razionali.

In generale, se X è Fano non possiamo dire nulla sulla poliedralità di Eff k (X) nel caso k = 0, 1, n. Infatti un esempio di Tschinkel (enunciato in [hivII, Example 6.10]) dimostra che una Fano può avere Eff 2 (X) non poliedrale (si veda Proposition 2.5.4).

La strategia adottata in gran parte della tesi è quella di utilizzare il seguente lemma (Lemma 2.1.29).

Lemma 0.0.6. Sia X una varietà proiettiva. Allora

(1) Se rkA k (X) = 1 oppure b 2k (X) = 1, allora Eff k (X) è una semiretta. (2) Se rkA k (X) = 2 oppure b 2k (X) = 2, allora Eff k (X) è una semiretta oppure è generato da due raggi estremali.
La dimostrazione segue dal diagramma (0.0.2). Infatti si ha che Eff k (X) è un cono in uno spazio vettoriale di dimensione al più 2, quindi è poliedrale.

Dunque calcoliamo i numeri di Betti per più varietà k-Fano possibile. Inne utilizziamo la classicazione di Araujo e Castravet [egIQ, Theorem 3] (vedi Theorem 2.5.2) per dimostrare il seguente. Teorema 0.0.7. Sia X una varietà k-Fano

con indice i X ≤ n -2. Allora Eff k (X) è poliedrale per k = 2, 3.
Prima di dimostrare il teorema, analizziamo alcuni casi particolari. Proposizione 0.0.8. Sia X una intersezione completa liscia in uno spazio proiettivo pesato. Allora b 2k (X) = 1 per ogni k = n 2 . In particolare Eff k (X) è poliedrale. Se X è in uno spazio proiettivo, k-Fano debole e 1 ≤ s ≤ k, allora b 2s (X) ≤ 2. In particolare Eff s (X) è poliedrale.

Dunque, la congettura è vera nel caso delle intersezioni complete in uno spazio proiettivo. Adesso analizzeremo le intersezioni complete in tre famiglie di varietà omogenee: G(r, s), OG(r, s) e SG(r, s). Dato un gruppo algebrico riduttivo G su C, i sottogruppi P tali che G/P è una varietà proiettiva vengono chiamati parabolici. Una varietà omogenea è una varietà per cui esiste un gruppo algebrico che agisce transitivamente. Per le varietà omogenee è già noto che tutti i coni di classi pseudoeettive sono poliedrali. Si dimostra che una varietà omogenea è Fano se e solo è un prodotto G 1 /P 1 × ... × G k /P k , con P i sottogruppo parabolico del gruppo semplice G i . Per le Grassmanniane usuali abbiamo la seguente. Proposizione 0.0.9. Sia X una 2-Fano debole intersezione completa in G(r, s).

Allora, b 4 (X) ≤ 2. In particolare Eff 2 (X) è poliedrale.

Per le Grassmanniane ortogonali dimostriamo che. Proposizione 0.0.10. Siano s, r interi positivi tali che 2 ≤ r ≤ s 2 , e s 2 -r = 1, 2 se s è pari. Sia s = 2r (rispettivamente, s = 2r), X una 2-Fano debole intersezione completa in una componente connessa in OG(r, s) con la mappa di Plücker (rispettivamente, half-spinor), X molto generale se X ⊆ OG(2, 7). Allora Eff 2 (X) è poliedrale.

Per questa proposizione utilizziamo sia [pWT, Theorem 2], che il seguente. Lemma 0.0.11. Sia X una componente connessa di OG(r, s),

2 ≤ r ≤ m = s 2 . Allora b 4 (X) =      1 r = m 3 1 ≤ m -r ≤ 2, s pari 2 altrimenti.
Per le intersezioni complete nelle Grassmanniane simplettiche abbiamo il seguente risultato.

Proposizione 0.0.12. Sia X una 2-Fano debole intersezione completa in SG(r, s). Per dimostrare questo lemma abbiamo calcolato il diamante di Hodge di X 11 . Poiché il Teorema di Sommese (Theorem 2.4.2) non è suciente per determinare tutto il diamante, abbiamo utilizzato anche [noVT].

rte PF Nella seconda parte studiamo il luogo di indeterminazione della mappa razionale introdotta in [oiIT,Proposition 4.8]. Data una mappa razionale, il luogo delle indeterminazioni è il sottoschema chiuso composto dai punti in cui la mappa non è denita. Facilmente possiamo vericare che dato un diagramma commutativo

X f h / / W Y g > >
con f e h mappe razioni e g un morsmo, abbiamo che Ind(h) ⊆ Ind(f ). Inoltre per il risultato di Hironaka [rirTR, I.Question (E) p.140], per ogni f : X Y mappa razionale tra varietà lisce, esiste una risoluzione delle indeterminazioni. Cioè esiste un diagramma commutativo

X π f X f / / Y
dove X è liscia e π è una mappa birazionale che è un isomorsmo fuori da Ind(f ).

Definizione 0.0.15. Sia X una varietà compatta di Kähler. Diciamo che X è hyperKähler, se è semplicemente connessa e lo spazio delle 2 forme olomorfe globali H 0 (X, ∧ 2 Ω X ) è generato da una forma simplettica ovunque non degenere.

Per un risultato classico, la prima classe di Chern di una varietà con usso di Ricci piatto è nulla. Inoltre per la decomposizione di Bogomolov [fogURD feVQ] queste varietà hanno un ricoprimento étale nito dato dal prodotto di un toro complesso, di una varietà Calabi-Yau e di una varietà hyperKähler. In dimensione 2 gli unici esempi di varietà hyperKähler sono le superci K3. Beauville ha dimostrato in [feVQ, Théorèmes 3 and 4] che per ogni n ≥ 0, sia lo schema di Hilbert dei punti S [n] dove S è una supercie K3, sia la varietà di Kummer generalizzata K n A associata ad una supercie Abeliana A, sono varietà hyperKähler. Una varietà hyperKähler che si ottiene da una deformazione di S [n] (rispettivamente, di una varietà Kummer generalizzata) è detta di tipo K3 [n] (rispettivamente, di tipo K n A). Queste varietà sono particolarmente interessanti perché permettono di costruire varietà hyperKähler di ogni dimensione complessa pari. Successivamente O'Grady in [y9qHQD y9qWW] ha costruito due nuovi esempi in dimensione 6 e 10 di varietà hyperKähler che non sono deformazioni di tipo conosciuto. Nella seconda parte della tesi usiamo due particolari esempi di varietà hyperKähler di tipo K3 [n] . Il primo esempio è quello fornito da Beauville e Donagi. Teorema 0.0.16 ([fhVS]). Sia Y ⊆ P 5 una ipersupercie cubica liscia di dimensione 4. Sia F la sua varietà di Fano. Allora

(1) La varietà F è hyperKähler di tipo K3 [2] .

(2) Se Y è Pfaana, allora F è isomorfa a S [2] per qualche supercie K3 S.

Osservazione 0.0.17. Questa varietà è stata utilizzata per dare un esempio di classe nef ma non pseudoeettiva. Infatti, in generale non è vero che Nef k (X) ⊆ Eff k (X) per 2 ≤ k ≤ dim X -2. Il primo esempio di questo fenomeno è stato dato in [hivII]. Ottem ha mostrato che se la cubica liscia Y è molto generale, allora Eff 2 (F ) Nef 2 (F ). Inoltre, la seconda classe di Chern c 2 (F ) è nef ma non è eettiva [yttIS, Theorem 1].

L'altro esempio è stato dato più recentemente da C. Lehn, M. Lehn, Sorger e van Straten in [vvvIU]. Essi hanno osservato che se F 3 (Y ) è una compatticazione dello spazio delle cubiche razionali in Y , dove Y è una cubica liscia di dimensione 4 che non contiene un piano, allora esiste una brazione in P 2 φ : F 3 (Y ) → Z , dove Z è una varietà liscia. Inoltre esiste un divisore di Z che può essere contratto, la relativa contrazione σ : Z → Z produce la varietà hyperKähler Z, che è di tipo K3 [4] (si veda [evIU, Corollary] oppure [vehIS,Corollary 6.3]). Esiste un diagramma commutativo (4,6) è la mappa che associa ad ogni cubica razionale l'unico spazio P 3 che la contiene. Il morsmo g è nito di grado 72 su un aperto di Z .

F 3 (Y ) φ / / G(4, 6) Z g dove F 3 (Y ) → G
Voisin ha costruito in [oiIT, Proposition 4.8] una mappa razionale di grado 6

ψ : F × F Z.
In breve, la mappa ψ manda coppie di rette non incidenti (l, l ) ∈ F × F nella classe di una curva razionale normale di grado 3 contenuta nel sistema lineare |L -L -K S l,l | della supercie cubica S l,l := L, L ∩ Y . Questo ci fornisce un divisore rigato su Z.

Infatti data una risoluzione delle indeterminazioni

F × F π ψ " " F × F ψ / / Z
l'immagine tramite ψ del divisore eccezionale di π è un divisore rigato di Z. Il nostro obiettivo è calcolare il luogo delle indeterminazioni di ψ. Prima dimostreremo la seguente.

Proposizione 0.0.18. Esiste una mappa razionale ψ : F × F Z dominante, di grado 6 tale che Ind(ψ ) = I.

La mappa ψ è denita in tutti i punti dell'aperto

U ADE := (l, l ) ∈ F × F |(l, l ) / ∈ I, S l,l := L, L ∩ Y ha singolarità ADE CHAPTER 1

Introduction

This thesis deals with two dierent problems, studied respectively in Chapter 2 and Chapter 3.

IFIF sntrodution to ghpter P

The study of cones of curves or divisors on smooth complex projective varieties X is a classical subject in Algebraic Geometry and is still an active research topic. However, little is known when we pass to higher dimensions. For example it is a classical result that the cone of nef divisors is contained in the cone of pseudoeective divisors, but in general Nef k (X) ⊆ Eff k (X) is not true. These phenomena can appear only if dim X ≥ 4 and very few examples are known. In particular [hivII] gives two examples of such varieties. Furthermore [yttIS] proves that if X is the variety of lines of a very general cubic fourfold in P 5 , then the cone of pseudoeective 2-cycles on X is strictly contained in the cone of nef 2-cycles.

The central subject of the rst part will be the k-Fano varieties.

Definition

1.1.1. A smooth Fano variety X is k-Fano if the s th Chern character ch s (X) is positive (see Denition 2.3.1) for 1 ≤ s ≤ k, and weak k-Fano for k > 1 if X is (k -1)-Fano and ch k (X) is nef.
There is a large interest in studying varieties with positive Chern characters. For example varieties with positive ch 1 (X) are Fano, hence uniruled, that is there is a rational curve through a general point. Fano varieties with positive second Chern character were introduced by J. de Jong and J. Starr in [dtHTD dtHU]. They proved a (higher dimensional) analogue of this result: weak 2-Fano varieties of pseudo-index at least 3 have a rational surface through a general point. Furthermore if X is weak 3-Fano then there is a rational threefold through a general point of X (under some hypothesis on the polarized minimal family of rational curves through a general point of X, [egIP, Theorem 1.5(3)]).

Another problem concerns how the geometry of the cones of pseudoeective k-cycles depends on the positivity of the Chern characters ch s (X). Mori's Cone Theorem resolves this problem for k = 1: the positivity of ch 1 (X) implies the polyhedrality of the cone of pseudoeective 1-cycles and the extremal rays are spanned by classes of rational curves. By Kleiman's Theorem, a variety with positive ch 1 (X) is just a Fano variety, that is with c 1 (X) ample, but this is not enough, in general, for the polyhedrality of cones of pseudoeective k-cycles for k > 1: Tschinkel showed a Fano variety where Eff 2 (X) has innitely many extremal rays (see Example 2.3.8). Therefore more positivity is needed in order to obtain polyhedrality of cones of pseudoeective k-cycles for k > 1.

In this paper we investigate a possible way of generalizing Mori's result:

Conjecture 1.1.2. If X is k-Fano, then Eff k (X) is a polyhedral cone.
The computing of the fourth Betti number is enough to show the polyhedrality of some of the cones of 2-cycles for a large class of varieties: complete intersections in weighted projective spaces, rational homogeneous varieties and most complete intersections in them, etc. This allows us to test the conjecture for many 2-Fano varieties, and in particular we prove that it holds for del Pezzo and Mukai varieties. Using the classication of Araujo-Castravet, we also prove the following.

Theorem 1.1.3 (= Theorem 2.5.3). Let X be a n-dimensional 2-Fano variety with i X ≥ n -2. Then Eff 2 (X) is polyhedral. Also, Eff 3 (X) is polyhedral with the possible exception of the complete intersection of type (2, 2) in P 8 .

In particular, Conjecture 1.1.2 is true for any n-dimensional k-Fano variety with i X ≥ n -2 and k = 2, 3.

Let X be a complete intersection in G(2, 5) or G(2, 6) with two hyperplanes under the Plücker embedding. Araujo and Castravet proved that X is not 2-Fano, but questioned if it is weak 2-Fano [egIQ,Proposition 32 and Questions 39,41]. In [deIS, Corollary 5.1] it is proved that a general such X is not 2-Fano by showing that there exists an eective surface S such that [i(S)] = σ ∨ 1,1 , where i is the inclusion. In this circumstance we can prove that all the smooth complete intersections of this type are not weak 2-Fano, and this completes the classication given in [egIQ, Theorem 3 and4

]. Theorem 1.1.4 (= Theorem 2.4.9). Let Y = G(2, 5) or G(2, 6), let X be a smooth complete intersection of type (1, 1) in Y under the Plücker embedding. Then X is not weak 2-Fano.
These ideas can be improved in three very promising directions: to generalize Tschinkel's example to higher dimensions, to prove the conjecture for some Fano 4-folds of index 1, and to use minimal families of rational curves to prove the conjecture for other 2-Fano's.

IFPF sntrodution to ghpter Q

It is a classical result that a manifold with a Ricci at metric has trivial rst Chern class, and by Bogomolov's decomposition [fogURD feVQ] such manifolds have a nite étale cover given by the product of a Torus, Calabi-Yau varieties and hyperKähler varieties. HyperKähler manifolds are interesting in their own and they are the subject of an intensive research. The rst examples are K3 surfaces, and Beauville proved in [feVQ, Théorèmes 3 and 4] that for any n ≥ 0, the Hilbert schemes of points X [n] , where X is a K3 surface or the generalized Kummer varieties K n A associated to an Abelian surface A, are hyperKähler varieties. Any hyperKähler variety that is a deformation of X [n] where X is a K3 surface (respectively, a generalized Kummer variety) is called K3 [n] -type (respectively, K n A-type). These examples are particularly interesting because they permit to construct hyperKähler varieties of any even complex dimension. Later O'Grady in [y9qHQD y9qWW] constructed two new examples in dimension 6 and 10 of hyperKähler varieties that are not deformation of known types. There are few explicit complete families of hyperKähler of K3 [n] -type. Beauville and Donagi proved in [fhVS, Proposition 1] that the variety of lines F (Y ) of a smooth cubic fourfold Y ⊆ P 5 is an hyperKähler variety of K3 [2] -type. Another example was given much more recently by C. Lehn, M. Lehn, Sorger and van Straten in [vvvIU]. They observed that if F 3 (Y ) is a compactication of the space of rational cubic curves in Y , then F 3 (Y ) is a P 2 -bration based on a smooth variety Z (Y ). Moreover there is a divisor of Z (Y ) that can be contracted and this contraction produces a hyperKähler variety Z(Y ). This variety is of K3 [4] -type by [evIU, Corollary] or [vehIS,Corollary 6.3].

On the other hand the study of k-cycles on smooth complex projective varieties is a classical subject and it is very interesting on hyperKähler manifolds with respect to several regards. Indeed, as we said before, in general it is not true that

Nef k (X) ⊆ Eff k (X) for 2 ≤ k ≤ dim X -2.
The rst example of such phenomenon was given in [hivII], but another example is a hyperKähler variety. Indeed, Ottem has shown that if the cubic Y is very general, then Eff 2 (F (Y )) Nef 2 (F (Y )), and the second Chern

class c 2 (F (Y )) is nef but it is not eective [yttIS, Theorem 1].
It is known, due to Mumford's Theorem [wumTV, Theorem], that for a projective hy-perKähler variety X of dimension 2n the kernel of the cycle map cl : A 2n (X) → H 4n (X) is innite-dimensional (see [oiHQ, III.10] for more details). Nevertheless Beauville conjectured in [feHU] that Conjecture 1.2.1 (Beauville). Let X be a projective hyperKähler manifold. Then the cycle class map is injective on the subalgebra of A * (X) generated by divisors.

See [oiIT] for an introduction to these topics. On the other hand Shen and Vial in [IT] used a codimension 2 algebraic cycle to give evidence to the existence of a certain decomposition for the Chow ring of F (Y ) for a very general cubic fourfold Y .

Voisin constructed in [oiIT, Proposition 4.8] a degree 6 rational map

ψ : F (Y ) × F (Y ) Z(Y ).
Roughly speaking, the map ψ sends pairs of non-incident lines (l, l ) ∈ F (Y ) × F (Y ) to the (class of the) degree 3 rational normal curve in the linear system |L -L -K S l,l | of the cubic surface S l,l := L, L ∩ Y . The Chapter 3 is devoted to the study of the indeterminacy locus of this map. In particular we prove the following.

Theorem 1.2.2 (= Theorem 3.4.4). The indeterminacy locus of the Voisin map ψ

is the variety I of the intersecting lines in Y .

We hope that the explicit description of Ind(ψ) will contribute to the study of c 2 (Z(Y )), the study of algebraic cycles on Z(Y ) and to other aspects of the geometry of Z(Y ). We hope to return to these topics in a future work.

IFQF xottions

A variety is a reduced and irreducible algebraic scheme over C. Unless otherwise stated, X is a variety of dimension n.

CHAPTER 2

Betti numbers and pseudoeective cone in 2-Fano Varieties No way of thinking or doing, however ancient, can be trusted without proof.

Henry David Thoreau, Walden.

PFIF qenerl fts out yles

In this chapter the ring R is either

Z, Q or R, while the notation M R means M ⊗ Z R for any Z-module M . Definition 2.1.1. A k-cycle on X is a nite Z-linear formal sum a i V i of subva- rieties of dimension k. The group of all k-cycles is indicated by Z k (X).
In this paragraph we introduce three well-known equivalence relations on Z k (X). Let C(X) be the eld of rational functions on X. For any codimension 1 subvariety Y ⊆ X, since the ring O X,Y is a discrete valuation ring, there exists a group homomorphism f ) where u is a unit in O X,Y and t is the generator of the maximal ideal. In other words, ord Y (f ) is the order of vanishing of f along Y .

ord Y : C(X) * -→ Z f → ord Y (f ) where ord Y (f ) is the integer such that f = u • t ord Y (
Definition 2.1.2. Let r be a rational function on a subvariety Y ⊆ X of dimension k + 1. The k-cycle on X dened by r is

[div(r)] := W ⊆Y ord W (r)W
where the sum is taken over all the codimension 1 subvarieties of Y .

Definition 2.1.3. A k-cycle α is rationally equivalent to zero, if there are a nite number of (k + 1)-dimensional subvarieties Y i of X, and

r i ∈ C(Y i ) * , such that α = [div(r i )].
The subgroup of all the k-cycles rationally equivalent to zero is indicated by Rat k (X),

the group of k-cycles modulo rational equivalence on X is the Chow group

A k (X) = Z k (X) /Rat k (X).
Remark 2.1.4. When X is smooth and projective variety, by [pulWV, Chapter 8] the direct sum A * (X) := ⊕ k A k (X) has a ring structure, graded by codimension. The ring A * (X) is called Chow ring and its product operation is called intersection product.

Remark 2.1.5. There is a remarkable morphism of group

deg : A 0 (X) R → R
that maps any Chow class of a point to 1.

Let H i (X, R) and H i (X, R) be the singular homology and cohomology groups of X for 1 ≤ i ≤ 2n and coecients in R. By [pulWV,p.372] there exists a cycle map cl : Z k (X) → H 2k (X, Z) that is a homomorphism of groups and maps each subvariety of X to its homology class.

Definition 2.1.6. A k-cycle α is algebraically equivalent to zero, if there is a non-singular variety B, a cycle α ∈ Z k+dim B (X × B), and points b 1 , b 2 ∈ B such that α is the dierence of the specializations of α at b 1 and b 2 , that is α = α b 1 -α b 2 .
The subgroup of all the k-cycles algebraically equivalent to zero is indicated by Alg k (X).

Definition 2.1.7. A k-cycle α is homologically equivalent to zero, if cl(α) = 0.
The subgroup of all the k-cycles homologically equivalent to zero is indicated by

Hom k (X).
Now we want to dene the numerical equivalence. First, let us do a digression on vector bundles.

Let E be a vector bundle on an algebraic scheme X. By [pulWV, p.47], there exists a unique set of Chern class operations

c i (E) • _ : A k (X) → A k-i (X),
with the following properties.

Theorem 2.1.8 ([pulWV, Theorem 3.2]). The Chern classes of an algebraic scheme X satisfy the following.

(1) (Vanishing) For all bundles E on X, all i > rk(E),

c i (E) = 0.
(2) (Commutativity) For all bundles E, F on X, integers i,j, and classes α on X,

c i (E) • (c j (F ) • α) = c j (F ) • (c i (E) • α). (3) (Projection formula) Let E be a vector bundle on X, f : X → X a proper morphism. Then f * (c i (f * E) • α) = c i (E) • f * (α)
for all classes α on X , all i.

(4) (Pull-back) Let E be a vector bundle on X, f : X → X a at morphism. Then

c i (f * E) • f * α = f * (c i (E) • α)
for all classes α on X, all i.

(5) (Whitney sum) For any exact sequence

0 → E → E → E → 0 of vector bundles on X, then c k (E) = i+j=k c i (E )c j (E ). (6) (Normalization) If E is a line bundle on a variety X, D a Cartier divisor on X with O X (D) ∼ = E, then c 1 (E) • [X] = [D].
In the contest of point 6 (Normalization), we denote by

c 1 (E) the divisor c 1 (E) • [X], if no confusion is possible.
Remark 2.1.9 ([pulWV, Remark 3.2.2]). The commutative law implies that any polynomial in the Chern classes with coecient in R operates on the direct sum ⊕ k A k (X) R . In particular, if P is any homogeneous polynomial of weight d, where c i (E) has weight i, and α

∈ A k (X) R , then P • α ∈ A k-d (X) R .
c i = c i (E) ch k (E) = 1 k! det         c 1 2c 2 3c 3 . . . kc k 1 c 1 2c 2 . . . . . . 1 . . . . . . 3c 3 . . . c 1 2c 2 1 c 1        
The polynomial ch k (E) is homogeneous of weight k. The rst Chern characters of E are

ch 1 (E) = c 1 (E) ch 2 (E) = 1 2 (c 2 1 (E) -2c 2 (E)) ch 3 (E) = 1 6 (c 3 1 (E) -4c 1 (E)c 2 (E) + 3c 3 (E))
Remark 2.1.11. If X is a smooth variety and T X is its tangent bundle, the Chern classes and the Chern characters of T X are usually indicated, respectively, by c k (X) and

ch k (X). Definition 2.1.12. A k-cycle α is numerically equivalent to zero, if deg (P • [α]) =
0 for any weight k homogeneous polynomial P in Chern classes of vector bundles on X. The subgroup of all the k-cycles numerically equivalent to zero is indicated by Num k (X), the group of k-cycles modulo numerical equivalence on X is the factor group

N k (X) = Z k (X) /Num k (X).
Remark 2.1.13. There is a chain of inclusions [pulWV, p.374]

Rat k (X) ⊆ Alg k (X) ⊆ Hom k (X) ⊆ Num k (X) ⊆ Z k (X)
that gives rise to a diagram (2.1.1)

A k (X) / / / / Z k (X) /Alg k (X) / / / / Z k (X) /Hom k (X) _ π k / / / / N k (X) H 2k (X, Z)
We set (2.1.2)

π k,R : Z k (X) /Hom k (X) ⊗ R N k (X) R the tensor product of π k and id R . Definition 2.1.14. A k-cycle r i V i ∈ Z k (X) R on X is eective if r i ≥ 0 for all i. A class α ∈ N k (X) R is eective if α = [ r i V i ] for some eective k-cycle r i V i .
2.1.0.1. Cones of cycles. A cone is a subset of a real vector space that is stable under multiplication by positive scalars.

Definition 2.1.15. The eective cone Eff k (X) is the cone generated by the eective classes in N k (X) R . The closure Eff k (X) of the eective cone is the pseudoeective cone, its classes are called pseudoeective classes.

It is useful to consider the abstract dual notions of N k (X).

Definition 2.1.16. Let X be a projective variety. The numerical dual groups are

N k (X) =
homogeneous Chern polynomials P of weight k Chern polynomials P such that deg (P

• [α]) = 0 for all α ∈ N k (X)
The numerical class groups N k (X) are also denoted by N k (X) ∨ . The dual class group N 1 (X) is called Néron-Severi group and it has the following equivalent denition. Definition 2.1.17 ([vzHR,Denition 1.1.15]). Let X be a complete algebraic scheme over C. A Cartier divisor D on X is numerically equivalent to zero if

D • α := deg (c 1 (O X (D)) • α) = 0
for all 1-cycles classes α. The subgroup of divisors numerically equivalent to zero is indicated by Num(X), the group of divisors modulo numerical equivalence on X is the Néron-Severi group

N 1 (X) = Div(X) /Num(X).
Remark 2.1.18. There exists a natural map

ϕ : N k (X) Q → N n-k (X) Q by setting ϕ([P ]) = P • [X].
By linearity of the intersection product, N k (X) is torsion free. When X is smooth and projective, ϕ is an isomorphism [pulWV, Example 19.1.5], and gives a perfect pairing

N k (X) Q ⊗ N n-k (X) Q → Q. Definition 2.1.19. Let X be a projective variety. An element β ∈ N k (X) is nef if β • α ≥ 0 for any eective α ∈ Eff k (X). The nef cone Nef k (X) is the cone generated by classes of nef classes in N k (X) R .
When X is smooth and projective, we dene Nef n-k (X) to be the image of Nef k (X) under the isomorphism described in Remark 2.1.18. Two very important theorems by Kleiman explain the relation between ample and nef divisors.

Theorem 2.1.20 (Kleiman's criterion for amplitude). Let X be a projective variety, and let D be an R-divisor on X. Then D is ample if and only if D • α > 0 for every α ∈ Eff 1 (X)\{0}. Equivalently, choose any norm • on N 1 (X) R , and denote by S the unit sphere of classes in N 1 (X) R of length 1. Then D is ample if and only if D • α > 0 for every α ∈ Eff 1 (X)\{0} ∩ S.

Proof. See [vzHR,Theorem 1.4.29].

Theorem 2.1.21 (Kleiman's Theorem). Let X be a projective variety, let Amp(X) ⊆ N 1 (X) R be the cone generated by the ample divisors. Then Nef 1 (X) is the closure of Amp(X), and Amp(X) is the interior of Nef 1 (X).

Proof. See [vzHR,Theorem 1.4.23]. Let X be a smooth projective variety. Since every ample divisor D has a multiple with a section, the (n -1)-cycle D • [X] is eective. From the relation Nef 1 (X) = Amp(X), using the isomorphism ϕ, it follows that

Nef n-1 (X) ⊆ Eff n-1 (X).
This relation easily implies Nef 1 (X) ⊆ Eff 1 (X). Anyway, in higher codimension the picture is more complicated, indeed: Theorem 2. 1.22 ([hivII,Theorem B]). Let E be an elliptic curve having complex multiplication, and set

X = E × ... × E (n times). Then for k = 0, 1, n -1, n, we have Eff k (X) = Nef k (X), while in any other case Eff k (X) Nef k (X).
Theorem 2.1.23 ([hivII, Theorem B]). Let A be a very general principally polarized abelian surface, and let X = A × A. Then Eff 2 (X) Nef 2 (X).

Theorem 2.1.24 ([yttIS, Theorem 1]). Let Y ⊆ P 5 be a very general cubic fourfold and let F be its variety of lines. Then Eff 2 (F ) Nef 2 (F ). In fact, c 2 (F ) is positive on every surface, but has no eective multiple.

I do not know any other example of variety X such that Nef k (X) Eff k (X) for k = 1, n -1. Clearly such variety must have dimension at least four.

In this rst part we will be mainly interested on the shape of the cones Eff k (X). The cone Eff 1 (X) is called Mori's Cone, and it is indicated also by N E(X). The following is the celebrated Mori's Cone Theorem and it gives us a lot of information on the shape of Eff 1 (X).

Theorem 2.1.25 ([worVP]). Let X be a smooth projective variety, and suppose that K X fails to be nef. Given any divisor D on X, let

D ≥0 = {α ∈ N 1 (X)/D • α ≥ 0} Eff 1 (X) D≥0 = Eff 1 (X) ∩ D ≥0
(1) There are countably many rational curves

C i ⊆ X, with 0 ≤ -(C i • K X ) ≤ n + 1 which together with Eff 1 (X) K X ≥0 generate Eff 1 (X), i.e Eff 1 (X) = Eff 1 (X) K X ≥0 + i R + • [C i ].
(2) Fix an ample divisor H. Then given any > 0, there are only nitely many of these curves -say C 1 , ..., C t -whose classes lie in the region -(

K X + H) ≥0 . Therefore Eff 1 (X) = Eff 1 (X) (K X + H) ≥0 + t i=1 R + • [C i ].
Proof. See [vzHR, Theorem 1.5.33].

Remark 2.1.26. People are primarily interested in the cone Eff 1 (X) for its use in the Mori Minimal Model Program. Mori's idea is that each extremal ray in the K X -negative part of the cone denes a map X → X 1 that contract the locus of the ray. The goal is repeating this process up to getting a variety X m with K Xm nef. Anyway, there are several problems to handle with: the image of it could be too singular, or the process could not be nite. If Mori's Cone is polyhedral (at least in the K X -negative part), then there is a nite number of contractions. It follows that the structure of the variety X can be fully understood.

In this thesis we study the polyhedrality of Eff k (X) for k = 1, 2, ..., n.

Definition 2.1.27. A Fano variety is a smooth projective variety such that the anti-canonical divisor is ample.

There are cases where the polyhedrality of some pseudoeective cone is particularly simple to determinate. The following corollary show that Mori's Cone of Fano varieties is polyhedral.

Corollary 2.1.28. Let X be a Fano variety. Then Eff 1 (X) is polyhedral.

Proof. Let H = -K X , and = 1 2 . Since -K X is ample, using Theorem 2.1.20, we have

Eff 1 (X) (K X + H) ≥0 = Eff 1 (X) 1 2 K X ≥0 = {0} .
Now apply statement 2 of the Cone Theorem.

Lemma 2.1.29. Let X be a projective variety. Then

(1) If either rkA k (X) = 1 or b 2k (X) = 1, then Eff k (X) is a half-line. (2) If either rkA k (X) = 2 or b 2k (X) = 2, then Eff k (X) is either a half-line or it is
spanned by two extremal rays.

Proof. In the rst case, by diagram (2.1.1), we have a surjection Z N k (X) and, as N k (X) is torsion-free, it must be N k (X) ∼ = Z. In the second case, again by diagram (2.1.1), there is a surjection Z 2 N k (X) and then either A classical tool in the study of cohomology of varieties is the following.

N k (X) ∼ = Z or N k (X) ∼ = Z 2 . Since Eff k (X) generates N k (X) R , it
Theorem 2.1.31 (Kodaira Vanishing Theorem). Let X be a smooth projective variety, and let A be an ample divisor on X. Then H i (X, K X + A) = 0 for i > 0.

Proof. See [vzHR,Theorem 4.2.1].

PFPF qrssmnnins nd huert yles PFPFIF tionl homogeneous vrietiesF Let us give a brief introduction to the rational homogeneous varieties. This is a family of varieties including the projective space, the Grassmann variety and many other classical objects. Let GL n (k) be the group of invertible square matrices with coecients in a eld k. Let G be a reductive linear algebraic group dened over k = C, T a maximal torus of G. Let g be the Lie algebra of G. It is well known that there exists a root space decomposition of

g g = t⊕ α∈Φ g α ,
where Φ is the root system of g with respect to T , and g α denotes the root subspace of α. Choose a subset Φ + ⊆ Φ of positive roots, and let ∆ ⊆ Φ + be a base of the root system. The Lie subalgebra Definition 2.2.5 ([roHU,p.354]). A subgroup P with the property that G/P is projective (i.e., compact) is called a parabolic subgroup.

Remark 2.2.6. In general, a parabolic subgroup needs to be a normal subgroup. Then G/P has not a structure of group.

Theorem 2.2.7 ([roHU,p.355]). For an algebraic subgroup H ⊆ G the following two conditions are equivalent:

(1) H is maximal connected solvable (a Borel subgroup).

(2) H is minimal parabolic.

As a corollary, we get that there exists a bijective correspondence between subsets of ∆ and parabolic subgroups of G containing B. Indeed, let ∆ ⊆ ∆ be a subset of simple roots. Then ∆ denes a Lie subalgebra

p ∆ := b⊕ α∈Φ + ∩Φ ∆ g -α ,
where Φ ∆ is the root subsystem generated by the simple roots not in Definition 2.2.9. A homogeneous rational variety is a homogeneous variety that it is rational.

Since the action of G on G/P by left multiplication is transitive, G/P is a homogeneous variety. There are only nitely many rational homogeneous varieties (up to isomorphism) of xed dimension.

Theorem 2.2.10 ([fTP]). Let X be a projective homogeneous variety. Then

(1) X is isomorphic to a direct product A × R, where A is an abelian variety and R is a homogeneous rational variety.

(2) If X is rational, then X is isomorphic to a product G 1 /P 1 × ... × G k /P k , where for all i ≤ k, G i is simple and P i ⊆ G i is a parabolic subgroup.

Furthermore, the rational part of a homogeneous variety is Fano [frSV]. Denote by S the corresponding set of reections in the Weyl group W of Φ. Then the pair (W, S) is a Coxeter system, i.e. it satises the following. Definition 2.2.11 ([fouTV, Chapitre IV, Dénition 3]). We say that (W, S) is a

Coxeter system if it satises the following condition: for all s, s ∈ S, let m(s, s ) be the order of ss , let I be the set of pairs (s, s ) such that m(s, s ) is nite. The generator set S and the relations (ss ) m(s,s ) = 1 for (s, s ) in I are a presentation of the group W .

Let l : W → N 0 be the length function relative to the system S of generators of W . Furthermore, we x a subset Θ of S and denote by W Θ the subgroup of W generated by Θ, and by P a parabolic subgroup of G associated to Θ [roHU, Theorem p.362]. Let w 0 (respectively, w θ ) be the unique element of maximal length of W (respectively, W Θ ). A simple calculation shows that dim G/P = l(w 0 ) -l(w θ ). The element w 0 and w θ are characterized by the property [fouTV, Chapitre IV, Exercise 22]

l(ww 0 ) = l(w 0 ) -l(w), ∀w ∈ W (2.2.1) l(ww θ ) = l(w θ ) -l(w), ∀w ∈ W Θ (2.2.2) that imply immediately w 2 0 = 1 and w 2 θ = 1. It follows that, for every w ∈ W l(w 0 w) = l((w 0 w) -1 ) = l(w -1 w -1 0 ) = l(w -1 w 0 ) = l(w 0 ) -l(w -1 ) = l(w 0 ) -l(w).
Furthermore, set W Θ = {w ∈ W/l(ws) = l(w) + 1 ∀s ∈ Θ}. We have, for every (w, w) ∈ W Θ × W Θ , (2.2.3) l(w w) = l(w) + l( w).

Proposition 2.2.12. Let X be a smooth projective n-dimensional variety and let G be an ane group which acts transitively on X. Suppose that, for every k = 1, ..., n -1, there exists a nite family of subvarieties

{Ω a } a∈I k of dimension k such that (1) {[Ω a ] /a ∈ I k } = H 2k (X, Z) or A k (X), and (
2) ∀a ∈ I k , ∃b ∈ I n-k such that Ω c • Ω b = δ a,c ∀c ∈ I k . Then Nef k (X) = Eff k (X) = Eff k (X)
is polyhedral and simplicial.

Proof. We will suppose that the classes of the subvarieties {Ω a } a∈I k generate H 2k (X, Z), the case A k (X) being similar. Let ω a be the class of

Ω a in N k (X). Let γ ∈ Nef k (X). By (2.1.2) there is a class β ∈ Z k (X) /Hom k (X) ⊗ R ⊆ H 2k (X, R) such that π k,R (β) = γ. By (1) we have that β = a∈I k γ a [Ω a ] and then γ = γ a π k ([Ω a ]) = γ a ω a . Let a ∈ I k and let b ∈ I n-k be as in (2). Then γ • ω b = γ a ≥ 0 because γ is nef and ω b is eective. Therefore γ ∈ Eff k (X), then Nef k (X) ⊆ Eff k (X). Furthermore, from ω c • ω a = δ a,c
it follows that the system {ω a } a∈I k is linearly independent. Let A a subvariety of X of dimension k, and let B be a subvariety of X of codimension k. By Kleiman's Theorem [uleUR] there is an element g ∈ G such that gA is rationally equivalent to A and generically transverse to B.

Then A • B = (gA) • B = #((gA) ∩ B) ≥ 0, so Eff k (X) ⊆ Nef k (X). It is clear that Nef k (X) is generated by {ω a /a ∈ I k }. Since Nef k (X)
is closed and, as seen above, generated by the ω a , we get that Nef k (X) = Eff k (X) is polyhedral.

Proposition 2.2.13. Let X be a rational homogeneous variety. Then Nef k

(X) = Eff k (X) = Eff k (X) is polyhedral.
Proof. The description of the Chow ring of any rational homogeneous variety given in [uöWI, Corollary(1.5)] is

A * (X) = w∈W Θ Z[X w ],
where X w is the closure of the set BwP/P , with dimension l(w) [uöWI,Proposition(1.3)]. Let I k = {w ∈ W Θ /l(w) = k}. Given w ∈ W Θ we claim that w 0 ww θ ∈ I dim X-k . Indeed for all s ∈ Θ, using (2.2.1) and (2.2.3), we have

l(w 0 ww θ s) = l(w 0 ) -l(ww θ s) = l(w 0 ) -l(w) -l(w θ s) = l(w 0 ) -l(w) -l(w θ ) + l(s) = l(w 0 ) -l(ww θ ) + 1 = l(w 0 ww θ ) + 1.
Similarly we can prove that l(w 0 ww θ ) = l(w 0 ) -l(w θ ) -l(w). Now given w ∈ I k we have, by [uöWI, Proposition(1.4)], that (2) of Proposition 2.2.12 is satised.

The pseudoeective cone is also polyhedral in the case when the action of G on X has nitely many orbits, see [pwWS, Corollary p.2] and [viIS].

PFPFPF qrssmnn vrietyF Consider the group SL s (C) := GLs(C) /C * , the root space of its Lie algebra is of type A s-1 . Let {α 1 , ..., α s-1 } be the simple roots. If we choose any root α r , it can be proved that the parabolic subgroup P αr is the group of block matrices

P αr = A B 0 C ∈ SL s (C)/A ∈ GL r (C), C ∈ GL s-r (C) .
Definition 2.2.14. The Grassmann variety is the rational homogeneous variety G(r, s) := SL s (C)/P αr .

The Grassmann variety can also be dened as the variety that parametrizes rdimensional vector subspaces in a C s . Let us see for example the case G(1, 2), where we expect to recover the usual projective line.

Example 2.2.15. The elements of SL 2 (C)/P α 1 are cosets

x y z t P α 1 where a generic element is

x y z t a b 0 c = ax bx + cy az bz + ct ,
then it is obvious that

x y z t P α 1 = x y z t P α 1 ⇐⇒ [x : z] = [x : z ] ∈ P 1 .
Since G(1, s + 1) ∼ = P s , and by duality it can proved that G(r, s) ∼ = G(s -r, s), we will usually take integers r and s such that 2 ≤ r ≤ s 2 . Note that the dimension of G(r, s) is r(s -r). We can dene some tautological bundle over G(r, s). The rst one is C s × G(r, s), that is the trivial vector bundle of rank s over G(r, s). We have also the universal subbundle S → G(r, s), that is the subbundle of C s × G(r, s) whose ber of at each point [W ] ∈ G(r, s) is the vector space W ⊆ C s . Finally there is an exact sequence (2.2.4)

0 → S → C s × G(r, s) → Q → 0
where Q is called universal quotient bundle. It can be checked that the cotangent bundle of G(r, s) is

Ω G(r,s) = S ⊗ Q ∨ .
If we apply Whitney sum (Theorem 2.1.8) and [rrUU, A.3.(C4)] to the exact sequence (2.2.4), we have (2.2.5)

-K G(r,s) = -c 1 (det Ω G(r,s) ) = -c 1 (Ω G(r,s) ) = s c 1 (det S ∨ ).
The line bundle det S ∨ is the Plücker line bundle, it denes an embedding called Plücker embedding:

G(r, s) → P(∧ r C s ).
The Plücker embedding simply sends a k-plane [W ] = w 1 , ..., w r to the multivector w 1 ∧ ... ∧ w r . Further details are in [qrUV,p.209]. Then det S ∨ is very ample, so by (2.2.5) we get that G(r, s) is Fano. Let us understand now the closed subvarieties of G(r, s). Given any ag of linear subspaces

F • 0 ⊆ F 1 ⊆ F 2 ⊆ ... ⊆ F s ⊆ C s
and a sequence of integers λ = {λ i } such that 0 ≤ λ r ≤ ... ≤ λ 1 and |λ| = λ i , the space Ω (λ) (F • ) is dened as the closure of the locus

{[W ] ∈ G(r, s)/ dim(W ∩ F s-r+i-λ i ) = i for 1 ≤ i ≤ r} .
The following is known as Ehresmann's Theorem [ihrQR].

Theorem 2.2.16. The integral homology of the Grassmannian G(r, s) has no torsion and is freely generated as a group by the cycles Ω (λ) (F • ) in real codimension 2|λ|, where λ = {λ i } ranges over all nonincreasing sequences of integers between 0 and s -r. In particular, all cohomology in G(r, s) is algebraic.

Proof. See [qrUV,Proposition pp. 195,196].

Remark 2.2.17. Since the cohomology of G(r, s) is algebraic, its Hodge Diamond is vertical. That is,

h p,q (G(r, s)) = 0 ⇐⇒ p = q [qrUV, p,163]. We set σ (λ) = Ω (λ) (F • ) ∈ H 2|λ| (X, Z)
. By Ehresmann's Theorem, two cycles σ (λ) and σ (λ ) are dierent if and only if λ = λ , i.e. the cycles σ (λ) does not depend on the ag. The direct sum

H * (G(r, s), Z) = 0≤k≤r(s-r) H 2k (G(r, s), Z)
has a structure of ring. This means that a product σ

(λ) • σ (λ ) is a cycle µ γ µ σ (µ) ∈ H 2(|λ|+|λ |) (G(r, s), Z),
where µ ranges over all nonincreasing sequences of r integers between 0 and s -r such that |µ| = |λ| + |λ |, and γ µ is an integer. The set of rules used to determinate γ µ is called Schubert Calculus, and it is the best tool for answering many combinatorial questions coming from geometry. Definition 2.2.18. The cycles σ (λ) are called Schubert cycles. In general, the computations made to determinate each γ µ are very complicated, but the following theorem can be useful at least in some particular cases.

Theorem 2.2.19 (Pieri's Formula). Let λ = (a, 0, ..., 0) and λ be two nonincreasing sequences of r integers between 0 and s -r. Then

σ (λ) • σ (λ ) = λ i ≤ µ i ≤ λ i-1 |µ| = a + |λ | σ (µ).
Remark 2.2.20. Even if Pieri's formula works in a very particular situation, it is enough to compute all the product rule of the ring H * (G(r, s), Z). This is because the Schubert cycles σ (λ) where λ = (a, 0, ..., 0) generate the ring H * (G(r, s), Z) [qrUV,p.205]. We will denote such cycles simply by σ a .

Lemma 2.2.21. The rst even Betti numbers of G(r, s

) are: b 0 (G(r, s)) = 1, b 2 (G(r, s)) = 1, and b 4 (G(r, s)) = 2.
Proof. It is enough to count the number of sequences λ with the prescribed value of |λ|. Remember that each sequence is nonincreasing, i.e. 0 ≤ λ r ≤ ... ≤ λ 1 . We have λ i = 0 ⇐⇒ λ = (0, ..., 0)

λ i = 1 ⇐⇒ λ = (1, 0, ..., 0),
and

λ i = 2 ⇐⇒      λ = (1, 1, 0, ..., 0) or λ = (2, 0, ..., 0).
This lemma implies that H 4 (G(r, s), Z) is generated by the Schubert cycles σ 2 and σ 1,1 . Then the ring H r(s-r)-4 (G(r, s), Z) is generated by two Schubert cycles, indicated by σ ∨ 2 and σ ∨ 1,1 , such that

σ 2 • σ ∨ 2 = σ 1 • σ ∨ 1 = 1 σ 1 • σ ∨ 2 = σ 2 • σ ∨ 1 = 0.
From now on, we will use implicitly the natural identication

Z ∼ = H 2r(s-r) (G(r, s), Z), then [uII, p. 39] χ(T G(2,s) (m)) = k (m+2,m+1,1, ..., 1 , s-3 0) χ(∧ 2 T G(2,s) (m)) = k (m+3,m+1,1, ..., 1 , s-4 0,0) + k (m+3,m+3,2, ..., 2 , s-3 0).
In our case, the computations give us

χ(T G(2,5) (m)) = 1 24 (m + 1)(m + 2)(m + 3)(m + 4) 2 (m + 6) χ(∧ 2 T G(2,5) (m)) = 1 24 (m + 2)(m + 4)(m + 6)(m + 7)(m + 3) 2 + 1 16 (m + 1)(m + 4) 2 (m + 6)(m + 7)(m + 3).
By Serre duality

χ(Ω α G(2,s) (-m)) = χ(∧ α T G(2,s) (m -s)),
then we get (2.2.6), (2.2.7) and (2.2.8). PFPFQF yrthogonl qrssmnn vrietyF We are now going to dene two types of rational homogeneous varieties: orthogonal Grassmannians OG(r, s) and symplectic Grassmannians SG(r, s). By [ihrQR] the (co)homology of OG(r, s) and SG(r, s) is algebraic and freely generated by the closure of certain loci dened by ags. In particular, we have denitions of Schubert cycles and Schubert calculus analogous to those of G(r, s).

Consider the group SO s (C) of orthogonal matrices, where s is an odd (respectively, even) number and m = [ s 2 ]. The root space of its algebra Lie is of type B m (respectively, D m ). Let {α 1 , ..., α m } be the simple roots. Definition 2.2.26. The orthogonal Grassmann variety is the rational homogeneous variety OG(r, s) := SO s (C)/P αr .

The orthogonal Grassmann variety can also be dened as the variety that parametrizes r-dimensional vector subspaces of C s , that are isotropic with respect to a non-degenerate symmetric bilinear form ω on C s . The scheme OG(r, 2m) has two isomorphic connected components if r = m or m -1. In these two cases, we will denote by OG + (r, 2m) a connected component of OG(r, 2m).

Remark 2.2.27. The variety OG(r, s) can be embedded in G(r, s) as the zero locus of a section of the bundle Sym 2 S ∨ . Indeed, since any symmetric bilinear form ω on C s is an element of Sym 2 (C s ) ∨ , any such ω is a global section of Sym 2 (C s × G(r, s)) ∨ . By (2.2.4), we have the following surjective map of vector bundles (2.2.9)

Sym 2 (C s × G(r, s)) ∨ Sym 2 S ∨
which can be described point-wise by

(ω, [V ]) → (ω |V , [V ]).
Thus, the composition of ω with (2.2.9) is a global section ω of Sym 2 S ∨ , with zerolocus given by the set of isotropic subspaces of C s with respect to ω. If we take ω non-degenerate, the zero locus of ω has the expected dimension r(2s-3r-1)

2

.

We now dene Schubert cycles in orthogonal Grassmannian. Let us recall the useful notation in [gosII]. Given a connected component X ⊆ OG(r, s), we will write s = 2m + 1 -with ∈ {0, 1} and 2 ≤ r ≤ m. Let t be an integer such that 0 ≤ t ≤ r, and t ≡ m (mod 2) if 2r = s. Given a sequence of integers λ = (λ 1 , ..., λ t ) of length t such that m -≥ λ 1 > ... > λ t > -.

Let λ = ( λt+1 , ..., λm ) be the unique sequence of length m -t such that

• m -1 ≥ λt+1 > ... > λm ≥ 0,
• λj + λ i = m -for every i = 1, .., t and j = t + 1, ..., m. The Schubert varieties in X are parametrized by pairs (λ, µ), where µ is any subsequence of λ of length r -t. Given an isotropic ag of vector subspaces

F • 0 ⊆ F 1 ⊆ F 2 ⊆ ... ⊆ F m ⊆ F ⊥ m-1 ⊆ F ⊥ m-2 ⊆ ... ⊆ F ⊥ 1 ⊆ C s , Ω (λ,µ) (F • ) is dened as the closure of the locus {[W ] ∈ X/ dim(W ∩ F m+1--λ i ) = i for 1 ≤ i ≤ t; dim(W ∩ F ⊥ µ j ) = j for t < j ≤ r .
Let us dene another sequence λ of length t in this way:

• λ = λ if either = 0 or = 1 and t ≡ m (mod 2); otherwise • λ = λ ∪ {b} where b = min{a ∈ N/0 ≤ a ≤ m -1, a / ∈ λ, a + µ j = m -1 ∀j = t + 1, ..., k}.
Let λ be the unique sequence associated to λ as above. Then the pair (λ, µ) is a subsequence of (λ , λ ). Suppose (λ, µ) = (λ i 1 , ..., λ it , λ i t+1 , ..., λ ir ) and let the discrepancy of λ and µ be the non-negative number (2.2.10)

dis(λ, µ) = r j=1 (m -r + j -i j ).

Then the codimension of a Schubert cycle

Ω (λ,µ) (F • ) is (see [gosII, p.2448]) codim(Ω (λ,µ) (F • )) = t i=1 λ i + dis(λ, µ). Let Ω (λ,µ) (F • ) be of codimension k and set σ (λ,µ) = Ω (λ,µ) (F • ) ∈ H 2k (X, Z).
The set of all σ (λ,µ) of codimension k is a basis of H 2k (X, Z) (as we said in the beginning of this subsection). Let us compute some Betti number of OG(r.s).

Lemma 2.2.28. Let X be a connected component of OG(r, s),

2 ≤ r ≤ m = s 2 , we have b 4 (X) =      1 r = m 3 1 ≤ m -r ≤ 2, s even 2 otherwise.
Proof. We have to count the number of sequences (λ, µ) such that

t i=1 λ i + dis(λ, µ) = 2.
For 1 ≤ j ≤ r let c j = m -r + j -i j . It can easily be seen that

m -r ≥ c 1 ≥ c 2 ≥ .... ≥ c r ≥ 0
and we can write

dis(λ, µ) = r i=1 c j .
We are in one of the following cases:

(1) t i=1 λ i = 0 and dis(λ, µ) = 2, or (2) t i=1 λ i = 1 and dis(λ, µ) = 1, or (3) t i=1 λ i = 2 and dis(λ, µ) = 0. Let s be odd. Then Case (1) t must be 0. If m -r ≥ 1 then c 1 = c 2 = 1, and, if m -r > 1,

we have also the possibility c 1 = 2. These cases correspond to

(λ, µ) = (∅, (r, r -1, r -3, ..., )) (∅, (r + 1, r -2, r -3, ..., )).

Case (2) Only one possibility if

m -r = 1, that is λ = (1) and c 2 = 1.
This case corresponds to (λ, µ) = ((1), (r -2, r -3, ..., )). No other possibilities if m -r = 1. Case (3) It must be λ = (2), then i 1 = 1 and since c j = 0 ∀j ≥ 1, c 1 = m -r + 1 -1 = 0 implies m = r. This is the case (λ, µ) = ((2), (m -1, m -3, ...)). Let s be even. If s = 2r, then the discrepancy is 0 because c j ≤ m -r ∀j ≥ 1, then it is possible only the case 3, that is

(λ, µ) = ((2), (m -1, m -2, m -4, . . .)) m odd ((2, 0), (m -2, m -4, . . .)) m even.
Suppose m > r. Let m be even, then Case (1) It must be λ = ∅, then λ = λ = ∅ and λ = (m -1, m -2, m -3, m -4, ...).

If m -r ≥ 1 then c 1 = c 2 = 1, and, if m -r ≥ 2, we have also the possibility c 1 = 2. These cases corresponds to

(λ, µ) = (∅, (r, r -1, r -3, ..., )) (∅, (r + 1, r -2, r -3, ..., )).
Case (2) It must be λ = (1, 0), then we can have λ = (0) or λ = (1). Suppose λ = (0), λ = (m -2, m-3, ...), and we have to choose a µ such that b = 1 in order to have λ = λ∪{1} which implies λ = (m-3, m-4, ...). This can happen only if m -2 / ∈ µ, that is, it is enough to choose µ as a subsequence of (m-3, m-4, ...). This case implies that i 1 = 2, then c 1 = m-r+1-2 = m-r-1, then it must be m -r = 2. Since c j = 0 ∀j ≥ 2, that corresponds to the case (λ, µ) = ((0), (m -4, m -5, ..., )).

Suppose λ = (1), λ = (m -1, m-3, ...), and we have to choose a µ such that b = 0 in order to have λ = λ∪{0} which implies λ = (m-3, m-4, ...). This can happen only if m -1 / ∈ µ, that is, it is enough to choose µ as a subsequence of (m-3, m-4, ...). This case implies that i 1 = 1, then c 1 = m-r +1-1 = m-r, then it must be m -r = 1. Since c j = 0 ∀j ≥ 2, that corresponds to the case (λ, µ) = (( 1), (m -3, m -4, m -5, ..., )).

Case (3) It must be λ = (2, 0), then we can have λ = (0) or λ = (2).

If λ = (2), then c 1 = m -r, then the discrepancy is not 0. So λ = (0), λ = (m -2, m -3, ...), c j = 0 ∀j ≥ 1, and we have to choose a µ such that b = 2 in order to have λ = λ∪{2} which implies λ = (m-2, m-4, ...). This can happen only if m -2 ∈ µ and m -3 / ∈ µ. That is, the sequence

((0), µ) = ((0), ( λ i 1 , ..., λ ir ))
seen as a subsequence of ((2, 0), (m -2, m -4, ...)) = (λ , λ ) must satisfy i 1 = 2.

The condition

c j = 0 implies i j = m -r + j, then i 1 = m -r + 1 = 2 implies m -r = 1. Then, if m -r = 1, we have the sequence (λ, µ) = ((0), (m -2, m -4, ...)).
Let m be odd, then Case (1) It must be λ = (0), then we can have λ = λ = (0) or λ = ∅.

Suppose λ = λ = (0), this implies λ = (m -2, m -3, m -4, ...) and c 1 = m -r. Then -if m -r ≥ 3, then this case in not possible since the rst summand of the discrepancy (which it must be 2) is m -r, -if m -r = 2, then c j = 0 for j ≥ 2, that is i j = m -r + j for j ≥ 2, then (λ, µ) = ((0), ( λm-r+2 , λm-r+3 , ..., )) = ((0), (r -2, r -3, ...)),

-if m -r = 1, then c j = 0 for j ≥ 3 and c 2 = 1, that is (λ, µ) = ((0), ( λm-r+1 , λm-r+3 , ..., )) = ((0), (r -1, r -3, ...)).

Suppose λ = ∅, λ = (m -1, m -2, ...), and we have to choose a µ such that b = 0 in order to have λ = λ ∪ {0} which implies λ = (m -2, m -3, m -4, ...).

This can happen only if

m -1 / ∈ µ, that is, it is enough to choose µ as a subsequence of (m -2, m -3, m -4, ...). If m -r ≥ 1 we have c 1 = c 2 = 1, that corresponds to the case (λ, µ) = (∅, (r, r -1, r -3, ..., )).
But, in order to make m -1 / ∈ µ, we must have r = m -1, then this case only happen if m -r ≥ 2. If m -r ≥ 2, we have also the possibility c 1 = 2, that corresponds to the case (λ, µ) = (∅, (r + 1, r -2, r -3, ..., )).

But, in order to make m -1 / ∈ µ, r + 1 = m -1, then this case only happen if m -r ≥ 3.

Case (2) It must be λ = (1), then we can have λ = λ = (1) or λ = ∅. Suppose λ = λ = (1), then λ = (m -1, m -3, m -4, ...), c 1 = m -r, and c j = 0 for j ≥ 2. So, if m -r = 1, we have the sequence (λ, µ) = ((1), ( λm-r+2 , λm-r+3 , ..., )) = (( 1), (m -3, m -4, ...)).

Suppose λ = ∅, λ = (m -1, m -2, ...), c 1 = 1, c j = 0 ∀j ≥ 2, and we have to choose a µ such that b = 1 in order to have λ = λ ∪ {1} which implies

λ = (m -1, m -3, m -4, ...).

This can happen only if

m -1 ∈ µ and m -2 / ∈ µ.
That is, the sequence

(∅, µ) = (∅, ( λ i 1 , ..., λ ir ))
seen as a subsequence of

((1), (m -1, m -3, m -4, ...)) = (λ , λ ) must satisfy i 1 = 2. The condition c 1 = 1 implies 1 = m -r + 1 -i 1 , then 1 = m -r + 1 -2 that is m -r = 2, while the condition c j = 0 ∀j ≥ 2 implies i j = m -r + j.
Then, if m -r = 2, we have the sequence

(λ, µ) = ((∅), (m -1, m -4, m -5, ...)).
Case (3) It must be λ = (2), then we can have

λ = λ = (1) or λ = ∅. If λ = (2), then c 1 = m -r, then the discrepancy is not 0. So λ = ∅, λ = (m -1, m -2, ...), c j = 0 ∀j ≥ 1,

This can happen only if

m -1, m -2 ∈ µ and m -3 / ∈ µ.
That is, the sequence

(∅, µ) = (∅, ( λ i 1 , ..., λ ir ))
seen as a subsequence of

((2), (m -1, m -2, m -4, ...)) = (λ , λ ) must satisfy i 1 = 2 and i 2 = 3. The condition c j = 0 implies i j = m -r + j, then i 1 = m -r + 1 = 2 and i 2 = m -r + 2 = 3 imply m -r = 1. Then, if m -r = 1, we have the sequence (λ, µ) = ((∅), (m -1, m -2, m -4, ...)).
Lemma 2.2.29. b 6 (OG + (r, 2r)) = 2.

Proof. We have to calculate the number of Schubert cycles of dimension 6, that is the number of sequences r -1 ≥ λ 1 > ... > λ t ≥ 0 such that t i=1 λ i = 3, t ≡ r (mod 2).

We get

• If r is odd, λ = (3) and λ = (2, 1, 0); • If r is even, λ = (3, 0) and λ = (2, 1).

PFPFRF ympleti qrssmnn vrietyF Consider the group Sp s (C) of symplectic matrices, and m = s 2 . The root space of its Lie algebra is of type C m . Let {α 1 , ..., α m } be the simple roots.

Definition 2.2.30. The symplectic Grassmann variety is the rational homogeneous variety SG(r, s) := Sp s (C)/P αr .

The Symplectic Grassmann variety can also be dened as the variety that parametrizes r-dimensional vector subspaces of C s , that are isotropic with respect to a non-degenerate skew-symmetric bilinear form σ on C s . Remark 2.2.31. The variety SG(r, s) can be embedded in G(r, s) as the zero locus of a section of the bundle ∧ 2 S ∨ in the same way as Remark 2.2.27.

We now dene Schubert cycles in symplectic Grassmannians SG(r, s) with 2 ≤ r ≤ m = s 2 . Let us recall the useful notation in [gosIV]. Let t be an integer such that 0 ≤ t ≤ r. Given a sequence of integers λ = (λ 1 , ..., λ t ) of length t such that

m ≥ λ 1 > ... > λ t > 0.
Let λ = ( λt+1 , ..., λm ) be the unique sequence of length m -t such that

• m -1 ≥ λt+1 > ... > λm ≥ 0,
• λj + λ i = m for every i = 1, .., t and j = t + 1, ..., m. The Schubert varieties in SG(r, s) are parametrized by pairs (λ, µ), where µ is any subsequence of λ of length r -t. Given an isotropic ag of vector subspaces

F • 0 ⊆ F 1 ⊆ F 2 ⊆ ... ⊆ F m ⊆ F ⊥ m-1 ⊆ F ⊥ m-2 ⊆ ... ⊆ F ⊥ 1 ⊆ C s Ω (λ,µ) (F • )
is dened as the closure of the locus

{[W ] ∈ SG(r, s)/ dim(W ∩ F m+1-λ i ) = i for 1 ≤ i ≤ t; dim(W ∩ F ⊥ µ j ) = j for t < j ≤ r}.
Suppose (λ, µ) = (λ 1 , ..., λ t , λi t+1 , ..., λir ), the codimension of

Ω (λ,µ) (F • ) is (see [gosIV, p. 57]) codim(Ω (λ,µ) (F • )) = t i=1 λ i + dis(λ, µ)
Where the discrepancy dis(λ, µ) is dened in (2.2.10).

The set all ω (λ,µ) = Ω (λ,µ) (F • ) of codimension k is a basis of H 2k (SG(r, s), Z) (as we said in the beginning of the previous subsection). The following lemma can be proved in the same way as in the case of OG(r, 2m + 1).

Lemma 2.2.32. Let 2 ≤ r ≤ m = s 2 , then b 4 (SG(r, s)) = 2 m -r ≥ 1 1 r = m.

PFQF righer pno vrieties

We now deal with the central object of this chapter, the higher Fano manifolds. Let us dene positive cycles and recall the denition of nef cycle.

Definition 2.3.1. Let X be a smooth variety. A class α ∈ N k (X) R is positive if α • β > 0 for every β ∈ Eff n-k (X)\{0}, and it is nef if α • β ≥ 0 for every β ∈ Eff n-k (X).
The cone generated by nef classes of k-cycles is Nef k (X).

Definition 2.3.2. A smooth Fano variety X is k-Fano if the s th Chern character ch s (X) is positive for 1 ≤ s ≤ k, and weak k-Fano for k > 1 if X is (k -1)-Fano and ch k (X) is nef. Proposition 2.3.3. Let X be a smooth variety . Then X is Fano if and only if X is 1-Fano. In particular, if X is 1-Fano then Eff 1 (X) is polyhedral.
Proof. Using properties of Chern classes [rrUU, A.3.( C4)], we have

-K X = c 1 (det Ω ∨ X ) = c 1 (Ω ∨ X ) = c 1 (X).
Then the following claim

K X is ample ⇐⇒ c 1 (X) is positive,
it follows tautologically by Theorem 2.1.20. Then Eff 1 (X) is polyhedral by Corollary 2.1.28. It is known that Fano varieties are uniruled [heHI,Theorem 3.4]. That is, there exist a rational curve through a general point. This result was generalized by de Jong and Starr in 2006. Indeed, they proved that if X is weak 2-Fano with pseudo-index is at least 3, then a general point of X is contained in a rational surface [dtHU,Proposition 1.3].

Following [uolWT, p. 109], for a projective scheme X and a general point x ∈ X, there exists a scheme RatCurves n (X, x) parametrizing rational curves through x. Let H x be a minimal family of rational curves through x. For example H x can be an irreducible component of RatCurves n (X, x) parametrizing rational curves on X, having minimal degree with respect to some xed ample line bundle on X. By denition of RatCurves n (X, x), H x is normal. Due to [ueHP,Theorem 3.4], there exists a nite morphism τ x : H x → P(T x X ∨ ), sending a smooth curve to its tangent direction at x. Definition 2.3.4. Let X be a smooth projective uniruled variety and let x ∈ X be general. A polarized minimal family of rational curves through x ∈ X is a pair

(H x , L x ), where L x := τ * x O(1).
Araujo and Castravet gave a further generalization of de Jong and Starr's result.

Theorem 2.3.5 ([egIP, Theorem 1.5]). Let X be a smooth projective Fano variety, and let (H x , L x ) be a polarized minimal family of rational curves through a general point

x ∈ X, d = dim H x ≥ 1. If X is 2-Fano and (H x , L x ) (P d , O(2)), (P 1 , O(3))
, then there is a generically injective morphism g : (P 2 , p) → (X, x) mapping lines through p to curves parametrized by H x . Moreover if X is weak 3-Fano and d ≥ 2.

(1) There is a rational 3-fold through x, except possibly if (H x , L x ) ∼ = (P 2 , O(2)) and τ x (H x ) is singular.

(2) Let (W h , M h ) be a polarized minimal family of rational curves through a general point h ∈ H x . Suppose that (H

x , L x ) (P d , O(2)) and (W h , M h ) (P k , O(2)), (P 1 , O(3)).
Then there is a generically injective morphism h : (P 3 , q) → (X, x) mapping lines through q to curves parametrized by H x .

Then, also weak 3-Fano are uniruled except possibly in one case. We think that Proposition 2.3.3 can be generalized to k-Fano, in the same way the uniruled property as been generalized to k-Fano. That is, we think that it is true the following.

Conjecture 2.3.6. If X is k-Fano, then Eff k (X) is a polyhedral cone.

Remark 2.3.7. The conjecture is interesting only for Fano varieties of dimension at least 4. Indeed, if X is Fano, then it is also a Mori Dream Space [fgrwIH, Corollary 1.3.2]. Hence Eff n-1 (X) is polyhedral [ruHH,Proposition 1.11(2)]. We deduce that Eff 2 (X) is polyhedral if X is a Fano threefold (see [gsHT] for an introduction to Mori Dream Spaces).

In general, if X is Fano we cannot say that the other cones Eff k (X) are polyhedral for k > 1.

Example 2.3.8 (Tschinkel). This example rst appeared in [hivII, Example 6.10], and shows that to be Fano is not enough to have polyhedrality for Eff 2 (X). Let f : X b → P 4 be the blow-up of P 4 along the surface

Y b described in [gutHH]. That is, Y b is a smooth quartic in P 3 such that Eff 1 (Y b ) = Nef 1 (Y b ) is a round cone. Then X b is Fano. But Eff 2 (X b ) is not polyhedral since every curve C ∈ Eff 1 (Y b ) denes an extremal surface [π -1 (C)] ∈ Eff 2 (X b ), where f |E = π : E → Y b
is the restriction to the exceptional divisor. Anyway, X b is not weak 2-Fano. We work out some of the details in Proposition 2.5.4.

PFRF gomplete sntersetions

Definition 2.4.1. Let X be a projective variety. We say that a vector bundle E is (very) ample over X if the Serre line bundle O P(E) (1) is a (very) ample line bundle on the projective bundle P(E).

A complete intersection is the zero-locus of a section of a direct sum O(a 1 )⊕...⊕O(a r ), where a i > 0 and O(1) is a very ample line bundle. In this section we want to study the polyhedrality of the cones of pseudoeective cycles on complete intersections. Since the direct sum of ample line bundles is ample [vzHR,Proposition 6.1.13(i)], we have that the following theorem can be applied to any complete intersection.

Theorem 2.4.2 (Sommese's Theorem). Let X be a non-singular variety, let E be an ample vector bundle over X of rank e, and let Z be the zero-locus of a section of E. Then

H i (X, Z; Z) = 0 for i ≤ n -e.
In particular the restriction map H i (X, Z) → H i (Z, Z) is an isomorphism for i < n -e, and injective when i = n -e.

Proof. See [vzHR,Theorem 7.1.1].

PFRFIF gomplete intersetions in weighted projetive spesF Let w 0 , w 1 , ..., w n be a set of strictly positive integers and let w = (w 0 , w 1 , ..., w n ) be a vector. Consider the action associated to w

C * × C n+1 → C n+1 λ (x 0 , x 1 , ..., x n ) → (λ w 0 x 0 , λ w 1 x 1 , ..., λ wn x n )
We dene the quotient

P(w) := C n+1 \{0} /C *
The space P(w) can be seen as the projective variety ProjC[x 0 , ..., x n ] where deg(x i ) = w i (see [himWP,p. 230]). Sommese's Theorem cannot be applied to it, since P(w) is the quotient of a smooth quasi projective variety by a nite group. Then it has orbifold singularities [himWP, B10]. But we can use the following.

Theorem 2.4.3 (Lefschetz's Theorem over Q). Let V be a weighted complete intersection in P(w), then the restriction map

H i (P(w), Q) → H i (V, Q)
is an isomorphism for i < dim V , and injective when i = dim V .

Proof. See [himWP,B22].

Proposition 2.4.4. Let X be a n-dimensional smooth complete intersection in a

weighted projective space. If k = n 2 then b 2k (X) = 1. In particular Eff k (X) is polyhedral. Proof. Recall [himWP, B13] that dim H 2i (P(w), Q) = 1 for every 0 ≤ i ≤ dim P(w). By Theorem 2.4.3 we have that H 2k (X, Q) ∼ = H 2k (P(w), Q) for 2k < n, then b 2k (X) = 1 for k < n 2 . But b 2n-2k (X) = b 2k (X), then it follows that, for k = n 2 , b 2k (X) = 1 and by Lemma 2.1.29 that Eff k (X) is a half-line.
Furthermore, if X is a k-Fano complete intersection in a projective space, then we can solve Conjecture 2.3.6, even for weak Fano.

Theorem 2.4.5. Let X be a n-dimensional weak k-Fano complete intersection in a projective space. If 1 ≤ s ≤ k, then b 2s (X) ≤ 2. In particular Eff s (X) is polyhedral.

Proof. Let X be of type (d 1 , ..., d c ) in P n+c , with d i ≥ 2 for 1 ≤ i ≤ c. By Proposition 2.4.4, we can suppose n even and s = n 2 . We know from [egIQ, 3.3.1] that [eiUP,p.20] and the theorem follows by Lemma 2.1.29.

ch n 2 (X) is nef if and only if d n 2 1 + ... + d n 2 c ≤ n + c + 1. Since n ≥ 4, it follows easily that c = 1. On the other hand d n 2 1 ≤ n + 2 is possible only for d 1 = 2, that is X is an n-dimensional quadric. But b n (X) = 2

PFRFPF gomplete intersetions in qrssmnn vrietiesF

In this subsection we will compute the number b 4 (X), where X is a complete intersection in a Grassmannian variety. We rst consider the smooth complete intersection of type (1, 1) in G(2, 5), in light of Proposition 2.4.8. Lemma 2.4.6. Let X be a smooth complete intersection of type (1, 1) in a Grassmann variety G(2, 5) under the Plücker embedding. Then b 4 (X) = 2.

Proof. Assume that X is of type (d 1 , ..., d c ). If n > 4, by Theorem 2.4.2, we have b 4 (X) = b 4 (G(r, s)) ≤ 2 and we can apply Lemma 2.1.29. If n = 4, using [egIQ, Proposition 31], we have the following conditions: c = r(s -r) -4 and c i=1 d i ≤ s -1.

It is easy to see that this leads to the following cases 3,6) (1,1,1,1,1) (1, 2)

G(r, s) Type G(r, s) Type G(2, 7) (1, 1, 1, 1, 1, 1) G(2, 5) (1, 1) G(
G(2, 6) (1, 1, 1, 1) (1, 3) (1, 1, 1, 2) (2, 2)
None of them is weak 2-Fano by [egIQ, Proposition 31 and 32(iv)], and Theorem 2.4.9.

After Lemma 2.2.21, we introduced the following notation: the group

H 4 (G(r, s), Z) is generated by {σ 2 , σ 1,1 }, while H r(s-r)-4 (G(r, s), Z) is generated by a basis {σ ∨ 2 , σ ∨ 1,1 } dual to {σ 2 , σ 1,1 }. Proposition 2.4.8 (from [deIS, Corollary 5.1]). Let Y = G(2, 5) or G(2, 6
), let X be a generl complete intersection of type (1, 1) in Y under the Plücker embedding.

Then X is not weak 2-Fano. In particular, there exist a surface in S ⊆ X such that [i(S)] = σ ∨ 1,1 where i : X → Y is the inclusion.

Now we can prove the following.

Theorem 2.4.9. Let Y = G(2, 5) or G(2, 6), let X be a smooth complete intersection of type (1, 1) in Y under the Plücker embedding. Then X is not weak 2-Fano.

Proof. Let O Y (1) be the Plücker line bundle and let

U ⊆ P(H 0 (Y, O Y (1))) × P(H 0 (Y, O Y (1)))
be the open set parametrizing the smooth complete intersections in Y of bidegree (1, 1).

For t ∈ U, we denote by X t the corresponding variety. Let X := {(x, t) ∈ Y ×U : x ∈ X t } and consider the family

X pr 2 pr 1 / / Y U Suppose Y = G(2, 5). Let i : X t → Y be the inclusion, the map i * : H 4 (Y, Z) → H 4 (X t ,
[i(S t )] = σ ∨ 1,1 . Then there exist a t , b t ∈ Z such that S t = a t σ 2|Xt + b t σ 1,1|Xt . From Example 2.2.22, we know that (σ 2|Xt ) 2 = (σ 2 2 ) • σ 2 1 = 2 (σ 1,1|Xt ) 2 = (σ 2 1,1 ) • σ 2 1 = 1 σ 2|Xt • σ 1,1|Xt = (σ 2 • σ 1,1 ) • σ 2 1 = 1. Using the condition [i(S t )] = σ ∨ 1,1 = σ 2,2 , we have 0 = σ 2,2 • σ 2 = S t • σ 2|Xt = 2a t + b t 1 = σ 2,2 • σ 1,1 = S t • σ 1,1|Xt = a t + b t then a t = -1 and b t = 2. Let S := pr * 1 (-σ 2 + 2σ 1,1 ), then the surface S |Xt is such that [S t ] = [S |Xt ],
and since we see that it is eective for a general t, hence it is eective for all 1 t. Let t ∈ U, then X t is not weak 2-Fano since using [egIQ, Proposition 32]

ch 2 (X t ) • S |Xt = 1 2 (σ 2|Xt -σ 1,1|Xt ) • (-σ 2|Xt + 2σ 1,1|Xt ) = - 1 2 . Suppose Y = G(2, 6). By Theorem 2.4.2 we have that H 4 (Y, Z) ∼ = H 4 (X t , Z), then b 8 (X t ) = b 4 (X t ) = 2. Now consider i * : H 8 (Y, Z) → H 8 (X t , Z)
, where i : X t → Y is the inclusion. Using again Example 2.2.22, we have

σ 4|Xt • σ 2|Xt = (σ 4 • σ 2 ) • σ 2 1 = 1 σ 2,2|Xt • σ 2|Xt = (σ 2,2 • σ 2 ) • σ 2 1 = 1 σ 4|Xt • σ 1,1|Xt = (σ 4 • σ 1,1 ) • σ 2 1 = 0 σ 2,2|Xt • σ 1,1|Xt = (σ 2,2 • σ 1,1 ) • σ 2 1 = 1.
Hence σ 4|Xt and σ 2,2|Xt are a basis of H 8 (X t , Z), since that group is torsion free (see Remark 2.4.10). Then

[S t ] = a t σ 4|Xt +b t σ 2,2|Xt
, where as before S t is the surface described in [deIS, Corollary 5.1] for general t ∈ U. Using the condition [i(S t )] = σ ∨ 1,1 = σ 3,3 , we have

0 = σ 3,3 • σ 2 = S t • σ 2|Xt = a t + b t 1 = σ 3,3 • σ 1,1 = S t • σ 1,1|Xt = b t then a t = -1 and b t = 1. Let S := pr * 1 (-σ 4 + σ 2,2 ), then [S t ] = [S |Xt ],
that is S |Xt is eective for all t. Let t ∈ U, then X t is not weak 2-Fano since using [egIQ, Proposition 32]

ch 2 (X t ) • S |Xt = (σ 2|Xt -σ 1,1|Xt ) • (-σ 4|Xt + σ 2,2|Xt ) = -1.
Remark 2.4.10. By Theorems 2.4.2 and 2.2.16 we have H 5 (X t , Z) = 0. By [rtHP, Corollary 3.3] H 4 (X t , Z) is torsion free, then also H 8 (X t , Z) is torsion free by Poincaré duality.

Remark 2.4.11. Let X be a smooth complete intersection of G(2, 5) of type (1, 1) under the Plücker embedding, let Z be the surface given by the union of lines in X through a general point. In [egIQ, Example 30] it is written that Z has homology class equal to σ

∨ 2 + σ ∨ 1,1 . This should be read as 2σ ∨ 1,1 + σ ∨ 2 .
1 his is wellEknown ft for expertsF e good referene is Ott15D roposition QF PFRFQF gomplete intersetion in orthogonl nd sympleti qrssmnniE nsF In this subsection we study the cone Eff 2 (X) in complete intersections in orthogonal and symplectic Grassmannian varieties.

Proposition 2.4.12. Let s, r be positive integers such that 2 ≤ r ≤ s 2 , and s 2 -r = 1, 2 if s is even. Let s = 2r (respectively, s = 2r), let X be a n-dimensional weak 2-Fano complete intersection in a connected component of the orthogonal Grassmann variety OG(r, s) under the Plücker (respectively, half-spinor) embedding, with X very general if X ⊆ OG(2, 7). Then Eff 2 (X) is polyhedral.

Proof. Assume that X is of type (d 1 , ..., d c ). If n > 4, by Theorem 2.4.2 and Lemma 2.2.28, we have b 4 (X) ≤ 2 and we can apply Lemma 2.1.29. Then we have n = 4 and c = r(2s-3r=1) 2 -4. If 2r = s, by [egIQ, Proposition 34] and Remark 2.4.13, we see that X is weak 2-Fano if and only if either d i = 1 and c ≤ 4, or X of type (2). Therefore we get r = 4 and X of type (1, 1). By [egIQ, Proposition 34] we have that

K X = -c 1 (X) = -4H
, where H is the half-spinor embedding. But then, by [uyUQ, Corollary p.37], X is a smooth quadric in P 5 and then b 4 (X) = 2 by [eiUP, p.20], so we apply Lemma 2.1.29.

If 2r = s, since c 1 (OG(r, s)) = (s -r -1)σ 1 we get that c i=1 d i ≤ s -r -2.
It is easy to see that this leads to the following cases

OG(r, s) Type OG(3, 7) (1, 1) OG(2, 7) (1, 1, 1) OG + (2, 6) (1) (2)
But OG(3, 7) ∼ = OG + (4, 8), then the rst case is a quadric. Let X 111 be the variety (1, 1, 1) in OG(2, 7). This is the variety (b8) in the classication given in [u ¤ WS]. Indeed, for the reader's convenience, we point out that X 111 is the zero-locus of a global section of the bundle

∧ 2 S ∨ ⊕3 ⊕ Sym 2 S ∨
where S ∨ is (1, 0; 0, 0, 0, 0, 0) in Küchle's notation (see [u ¤ WS, Section 2.5]). So h 1,3 (X 111 ) > 0 by [u ¤ WS, Theorem 4.8]. Now apply [pWT, Theorem 2] to conclude that the space of algebraic cycles of X 111 is induced by the space of algebraic cycles of OG(2, 7). Then

Z 2 (X 111 ) /Alg 2 (X 111 ) ⊗ R is at most 2-dimensional. Hence Eff 2 (X 111
) is polyhedral by (2.1.1) and Lemma 2.1.29. The last two varieties do not satisfy the condition s 2 -r = 1, 2, anyway, they are not weak 2-Fano by [egIQ,Example 21]. Indeed, OG + (2, 6) is the zero section of the bundle O P 3 (1) ⊕ O P 3 (1) in P 3 × P 3 [uuzIS, Proposition 2.1], and it can easily be seen that the Plücker embedding is given by the divisor (1, 1), then the two varieties are isomorphic to, respectively, a complete intersection of type (1, 1) and (1, 2) in P 3 × P 3 under the embedding given by O P 3 (1) ⊕ O P 3 (1).

• Complete intersection in weighted projective spaces:

-Degree 4 hypersurfaces in P(2, 1, ..., 1) with n > 11; -Degree 6 hypersurfaces in P(3, 2, 1, ..., 1) with n > 23; -Degree 6 hypersurfaces in P(3, 1, ..., 1) with n > 26; -Complete intersections of type (2, 2) in P(2, 1, ..., 1) with n > 14. • G(2, 5).

• OG + (5, 10) and its linear sections of codimension c < 4.

• SG(3, 6). • G 2 /P 2 .
Here G 2 /P 2 is a 5-dimensional homogeneous variety for a group of type G 2 . Using the results in the previous sections we obtain:

Theorem 2.5.3. Let X be a n-dimensional 2-Fano variety with i X ≥ n -2. Then Eff 2 (X) is polyhedral. Also, Eff 3 (X) is polyhedral with the possible exception of the complete intersection of type (2, 2) in P 8 .

In particular, Conjecture 1.1.2 is true for any n-dimensional k-Fano variety with i X ≥ n -2 and k = 2, 3.

Proof. For Eff 2 (X): In the case P n and its complete intersections, we can invoke Theorem 2.4.5. Since none of the complete intersections in P(w) of the list has dimension 4, we can use Proposition 2.4.4. Also G(2, 5), OG + (5, 10), SG(3, 6) and G 2 /P 2 are rational homogeneous varieties, then their cone of pseudoeective 2-cycles is polyhedral by Proposition 2.2.13. Whereas the complete intersections of OG + (5, 10) have polyhedral cone of pseudoeective 2-cycles by Proposition 2.4.12.

For Eff 3 (X): In Theorem 2.5.2, the only complete intersections of dimension 6 in a weighted projective space are the one of type (2, 2) in P 8 and the smooth quadric Q ⊆ P 7 . The rst one is not weak 3-Fano since by [egIQ, Equation (3.1)], ch 3 (X) = -7 6 h 3 |X where h is the class of an hyperplane in P 8 . Then h 3 |X is eective, and ch 3 (X)•h 3

|X = -7 6 h 6 |X < 0.
For the quadric, by [eiUP,p.20] (2) The variety X b is not weak 2-Fano.

Suppose now that Eff 1 (Y b ) = Nef 1 (Y b ) is a round cone. Then (3) The cone Eff 2 (X b ) is not polyhedral. Proof. Let f : X b → P 4 be the blow-up. It follows that -K X b = 5f * H -E,
where H is the class of an hyperplane section of P 4 , E is the exceptional divisor. Let L 1 be the class of a ber of the map π : E → Y b . Let L 2 be the class of the strict transform of a line L 2 contained in the projective space P 3 ⊇ Y b . Note that L 1 and L 2 are eective. We claim the following.

Claim. The two divisors

D 1 = f * H, D 2 = 4f * H -E
are nef and extremal in Nef(X b ).

Proof of the Claim. The vector space N 1 (X b ) is 2-dimensional because it is generated by f * H and E. So a nef divisor is extremal if and only if it has zero intersection with an eective 1-cycle. Since H is nef in P 4 , its pull-back D 1 is nef. Since D 1 • L 1 = 0 because L 1 is f -exceptional, we get that D 1 is extremal. Note that D 2 is base-point free, hence nef, by [rrUU,Example II.7.17.3] since f is the blow up of the base ideal of the linear system of quartics containing Y b . Furthermore, L 2 meets Y b in four points, so

D 2 • L 2 = (4f * H -E) • L 2 = 4H • f * (L 2 ) -E • L 2 = 4 -4 = 0.
Then -K X b is the sum of two extremal nef divisors, then it must be an internal point of Nef(X b ). By Kleiman's theorem, -K X b is ample, so X b is Fano. Anyway, X b is not weak 2-Fano. This is because for any Z ⊆ P n such that codim P n Z > 1, we have that Bl Z P n is not weak 2-Fano [dtHT, p.5].

Let us prove the point (3). In [hivII, Example 6.10] we have the following decomposition of N 2 (X b ) (2.5.1)

N 2 (X b ) = f * (H 2 ) R ⊕ N 1 (Y b ).
Let γ ∈ Eff 1 (Y b ) be an eective cycle. Let us remember the commutative diagram of the blow-up

E π i / / X b f Y b j / / P 4
By Projection Formula [pulWV, Proposition 8.3(c)]

(i * π * γ)•f * (H 2 ) = f * (i * π * γ) • f * (H 2 ) = (f * i * π * γ)•H 2 = (j * π * π * γ)•H 2 = (j * γ)•H 2 = 0
the last intersection being zero for dimensional reasons.

Then i * π * Eff 1 (Y b ) is in the boundary of Eff 2 (X b ).
Moreover, the decomposition of (2.5.1) suggests that i

* π * Eff 1 (Y b ) injects in Eff 2 (X b ).
Then Eff 2 (X b ) has a (partially) round boundary.

La science, mon garçon, est faite d'erreurs, mais ce sont des erreurs qu'il est utile de faire, parce qu'elles conduisent peu à peu à la vérité. Jules Verne, Voyage au centre de la Terre.

CHAPTER 3

The indeterminacy locus of the Voisin map

Le peu de temps que j'ai eu a été cause de l'un et de l'autre. Je n'ai l'ait celle-ci plus longue que parce que je n'ai pas eu le loisir de la faire plus courte... Blaise Pascal, Lettre XVI (to the Jesuits).

QFIF qenerl fts out rtionl mps Definition 3.1.1. Let f : X Y be a rational map. We say that f is dened at a point x ∈ X if there exists a morphism

f 1 : U 1 → Y with class f such that x ∈ U 1 .
The domain of f is the largest open subset dom(f ) ⊆ X of points where f is dened.

A point of X is called a point of indeterminacy of f , if it is not contained in dom(f ). We set Ind(f ) to be the closed subscheme of the points of indeterminacy of f . We call

Ind(f ) the indeterminacy locus of f . If g : Y W is a rational map and f (dom(f )) is dense in Y , then it is well-dened the composition g • f : X W.
Lemma 3.1.2. Let X, Y and W be varieties sitting inside the following commutative diagram

X f h / / W Y g > >
where f and h are rational maps and g is a morphism. Then Ind(h) ⊆ Ind(f ).

Proof. Since the composition g • f is dened in the domain of f , then h is dened in the domain of f by commutativity. Definition 3.1.3. Let f : X Y be a rational map. We will denote by f : X → Y a resolution of the indeterminacy of the map f , i.e. a commutative diagram (3.1.1)

X π f X f / / Y
where X is a non-singular variety and π is a birational morphism that is an isomorphism outside Ind(f ).

The existence of such X is a consequence of the problem of the elimination of points of indeterminacy. A solution was found by Hironaka [rirTR, I.Question (E) p.140]. Indeed, he proved that for any rational map f : X Y of smooth varieties there exists a resolution like in Denition 3.1.3, where π may be obtained as a sequence of blow-ups along smooth subvarieties.

QFPF qenerl fts out the vriety of lines

In this subsection we will study the Fano variety of lines on a smooth cubic fourfold.

In particular we will be interested in cubic fourfolds not containing a plane. For the reader's convenience, let us prove that they exists. Proposition 3.2.1. The general cubic fourfold in P 5 does not contain a plane.

Proof. Consider the incidence variety

J := {(L, Y ) ∈ G(3, 6) × P(H 0 (P 5 , O P 5 (3)))/L ⊂ Y }
and the projections pr 1 to G(3, 6) and pr 2 to P(H 0 (P 5 , O P 5 (3))). The dimension of the ber pr -1 1 (L) is given by the dimension of the variety of cubics containing L. From the exact sequence

0 → I L/P 5 (3) → O P 5 (3) → O P 2 (3) → 0 since L is a complete intersection, we get h 0 (P 5 , I L/P 5 (3)) = h 0 (P 5 , O P 5 (3)) -h 0 (P 2 , O P 2 (3)) = 46.
Then the dimension of pr -1 1 (L) ∼ = P 45 is 45. In particular it is never empty, in other words pr 1 is surjective. Then J is irreducible by [hWR, Chap. I, 6.4, Theorem 8] and we have

dim J = dim G(3, 6) + dim pr -1 1 (L) = 9 + 45 = 54.
Observe that the dimension of P(H 0 (P 5 , O P 5 (3))) is 55, so pr 2 cannot be dominant, which means that the general cubic does not contain a plane.

In order to introduce the variety of lines on a cubic fourfold, let us give a brief introduction to the Hilbert scheme. Theorem 3.2.2. Let M = ⊕ l∈Z M l be a nitely generated graded C[x 0 , ..., x n ]-module.

Then there is a unique polynomial P M (z) ∈ Q[z] such that dim C M l = P M (l) for all l 0. Furthermore, deg P M (z) = dim Z(Ann M ), where Z denotes the zero set in P n of a homogeneous ideal. The polynomial P M is called the Hilbert polynomial of M .

Proof. [rrUU,Theorem.I.7.5].

Definition 3.2.3 ([rrUU,p.52]). If X ⊆ P n is an algebraic set of dimension r, we dene the Hilbert polynomial of X to be the Hilbert polynomial P X of its homogeneous coordinate ring. (By the theorem, it is a polynomial of degree r.) We dene the degree of X to be r! times the leading coecient of P X . is irreducible, it follows that U is irreducible, then Γ ρ |U is irreducible, then also Γ ρ is irreducible. It follows that

(p 1 × p 2 ) -1 (U ) = Γ ρ |U then Γ ρ |U is an open subset of I. Then dim I = dim Γ ρ |U = dim U = dim Γ ρ
and, by Lemma 3.2.9 I = Γ ρ . Since the ber (p 1 × p 2 ) -1 (P, P ) of any point of I G is not a point, it follows that U = U ρ .

Remark 3.2.12. As Γ ρ is birational to G(a+1, n+1)×G(b+1, n+1), it follows that Γ ρ is the blow up of G(a+1, n+1)×G(b+1, n+1) with respect to a sheaf of ideals supported in I G [rrUU,Theorem II.7.17,Exercise II.7.11c]. Let X ⊆ G(a+1, n+1)×G(b+1, n+1) be any irreducible subvariety such that X I G . Then the inclusion induces a rational map X G(a + b + 2, n + 1), and its graph Γ X is the strict transform of U ∩ X in Γ ρ . By [rrUU,Corollary II.7.15], Γ X is the blow up of X along a sheaf of ideals supported in X ∩ I G .

Notice that (F × F ) ∩ I G F × F . Proposition 3.2.13. The indeterminacy locus of the restricted rational map

ρ |F ×F : F × F G(4, 6)
is the variety I.

Proof. In the setting of Remark 3.2.12, with X = F ×F , the closed subset X ∩I G = I is of codimension 2 by Lemma 3.2.8. Then Γ F ×F is not isomorphic to F × F exactly at the points of I. It follows that F × F G(4, 6) cannot be extended at any point of

I.
QFQF qenerl fts out the vv vriety Definition 3.3.1. Let X be a compact Kähler manifold. We say that X is hy-perKähler, if it is simply connected and the space of its global holomorphic two-forms H 0 (X, ∧ 2 Ω X ) is spanned by a nowhere degenerate symplectic form.

Remark 3.3.2. The symplectic form induces an isomorphism between the vector bundle Ω X and T X = Ω ∨ X . It follows that for every k ≥ 0 we have

c k (Ω X ) = c k (X) = (-1) k c k (Ω X ).
Then the odd Chern classes are zero, i.e. c 2k+1 (X) = 0. Let us see some other example of hyperKähler varieties. Let S be a complex surface. For any positive integer n, we denote by S [n] the Hilbert scheme of dimension 0 and length n subschemes of S. That is, the general point of S [n] represents the union of n general points of X. Let S (n) be the quotient of the direct product S ×n by the action of the symmetric group Σ n . That is

S (n) := S ×n /Σn.
It can be seen that there exists a birational morphism S [n] → S (n) .

Theorem 3.3.4 ([feVQ,Théorème 3]). Let S be a K3 surface. Then S [n] is a hyperKähler variety of dimension 2n.

Definition 3.3.5. Let X be a hyperKähler variety. Suppose that X is deformation equivalent to S [n] for some K3 surface S. We say that X is of K3 [n] -type.

If A is an Abelian surface, the operation of A induces a map n+1) and A (n+1) → A. Let K n A be a generalized Kummer variety, that is K n A = f -1 (0). We have the following.

A (n) → A. Let f : A [n+1] → A be the composition of A [n+1] → A (
Theorem 3.3.6 ([feVQ,Théorème 4]). The variety K n A is a hyperKähler variety of dimension 2n.

Definition 3.3.7. Let X be a hyperKähler variety. Suppose that X is deformation equivalent to K n A for some Abelian surface A. We say that X is of K n A-type.

Remark 3.3.8. hyperKähler varieties of K n A-type and K3 [n] -type are the only families of hyperKähler varieties in any complex even dimension. O'Grady in [y9qHQD y9qWW] constructed two new examples in dimension 6 and 10 of hyperKähler varieties that are not deformation of known types.

Theorem 3.3.9 ([fhVS]). Let Y ⊆ P 5 a smooth cubic fourfold. Let F be its variety of lines. Then

(1) The variety F is hyperKähler of K3 [2] -type.

(2) If Y is Pfaan, then F is isomorphic to S [2] for some K3 surface S.

We are now going to study the variety Z dened in [vvvIU]. This variety is constructed using the Hilbert scheme of twisted cubic curves. Let us give a brief introduction to this Hilbert scheme. Definition 3.3.10. A rational normal curve of degree 3, or twisted cubic for short, is a smooth curve C ⊆ P 3 that is projectively equivalent to the image of P 1 under the Veronese embedding P 1 → P 3 of degree 3.

Piene and Schlessinger [VS] showed that Hilb 3z+1 (P 3 ) = H 0 ∪ H 1 , where H 0 is a 12-dimensional smooth component such that the general point is a rational normal curve, and H 1 is a 15-dimensional smooth component such that the general point is a curve C such that C red is a plane cubic. A generalized twisted cubic is a curve with class in H 0 .

We dene Hilb gtc (P 3 ) as the H 0 component of Hilb 3z+1 (P 3 ) and

Hilb gtc (P 5 ) := {[Γ] ∈ Hilb 3z+1 (P 5 )/∃P 3 ⊆ P 5 , [Γ] ∈ Hilb gtc (P 3 )}.

Definition 3.3.11. Let Y ⊆ P 5 be a smooth cubic fourfold that does not contain a plane. We set

F 3 (Y ) := Hilb gtc (P 5 ) ∩ Hilb 3z+1 (Y ). By [vvvIU, Theorem 4.7] F 3 (Y ) is a smooth variety of dimension 10. An element [Γ] ∈ F 3 (Y )
is the class of a one dimensional subscheme Γ of Y , with Hilbert polynomial equal to 3z+1. The linear span of Γ is a 3-dimensional space Γ ∼ = P 3 , and [Γ] ∈ Hilb gtc ( Γ ). Then the span induces a morphism

F 3 (Y ) → G(4, 6)
and, as stated by [vvvIU,p.113 and Theorem 4.8] and [vvwIU,(1.1.1)], there is a commutative diagram (3.3.1)

F 3 (Y ) φ / / G(4, 6) Z g 9 9
where Z is a smooth irreducible projective variety [vvvIU,Theorem 4.8]. The diagram (3.3.1) has the following remarkable properties:

• The morphism φ is a P 2 -bration.

• The morphism g is nite on the open subset g -1 (W ADE ) =: V ADE ⊆ Z where

W ADE := {P ∈ G(4, 6)/P ∩ Y has ADE singularities}.
• The degree of g on V ADE is 72 by [vvvIU, Theorem 2.1 and Table 1]. Moreover, by [vvvIU,Theorem 4.11 and Proposition 4.5] there exists a divisorial contraction

Z σ P(T Y ) ? _ o o Z Y ? _ o o
making Z be the blow up of a variety Z over a subvariety canonically isomorphic to Y . Z is an hyperKähler variety [vvvIU, Theorem 4.19] of K3 [4] -type [evIU, Corollary] (or also [vehIS,Corollary 6.3]).

Obviously, both Z and Z depend on Y , so they should be denoted by Z (Y ) and Z(Y ). We choose to keep the same notation as [vvvIU]. So, when no confusion is possible, we will simply write Z and Z.

QFRF he oisin mp

In [oiIT, Proposition 4.8] Voisin dened a rational map ψ : F × F Z using the following nice geometric argument. Let (l, l ) ∈ F × F be a general point, that is l and l are the classes of two disjoint lines L and L such that the following surface S l,l := L, L ∩ Y is smooth. The point (l, l ) denes a linear system in S l,l given by the divisor

D l,l = L -L -K S l,l . Since O S l,l (1) = O S l,l (-K S l,l ), in |D l,l | there is a curve of the form L ∪ C x , where x is any point of L and C x is the unique conic such that x, L ∩ Y = L ∪ C x .
Then any member of this linear system is a generalized twisted cubic [rrVP, Section 1.b p. 39] contained in Y . Voisin denes the map ψ by setting ψ(l, l ) to be the class in Z of any member of |D l,l |. The degree of the map is obtained as follows. It can be seen that D l,l denes a morphism ϕ D l,l : S l,l → P 2 that contracts exactly 6 lines. The members of |D l,l | are pull-back of lines in P 2 . The line L is the inverse image of a blown up point, thus it is a component of the pull-back of any line through that point. We can see, by intersection theory in S l,l , that L is the strict transform of a conic through the other ve points. Then we have 6 lines that are components of some rational normal curve in |D l,l |, so we have 6 possible choices of pairs of lines R, R ⊆ S l,l such that |D l,l | = |D r,r |.

We follow a slightly dierent approach: we construct a degree 6 rational map

ψ : F × F Z ,
and then we will check that ψ = σ • ψ . First of all, we notice that if L is a line and C is a conic in a projective space, with L not contained in the plane dened by C and L ∩ C = {x}, then L ∪ C is a limit of rational normal curves [rrVP, Section 1.b p. 39]. If both L and C are contained in Y , we have

[L ∪ C] ∈ F 3 (Y ).
As already pointed out in [oiIT, Proposition 4.8], there is a rational map

ψ 1 : P × F F 3 (Y )
dened as follows. Let (l, l ) be not in I, let x ∈ L be a point and let C x be the unique conic such that

x, L ∩ Y = L ∪ C x .
Then

ψ 1 (l, x, l ) := [L ∪ C x ].
Consider

U ADE := (l, l ) ∈ F × F |(l, l ) / ∈ I, S l,l := L, L ∩ Y has ADE singularities . Pick (l, l ) ∈ U ADE . Since the linear span of L ∪ C x is L, L for each x ∈ L, then the image of the curve Γ l,l = {[L ∪ C x ]/x ∈ L ∼ = P 1 }
under the span map

F 3 (Y ) → G(4, 6)
is the point L, L . In other words the curve Γ l,l is contracted by g

• φ. By construction, g • φ(Γ l,l ) ⊂ W ADE then φ(Γ l,l ) ⊂ V ADE .
The curve Γ l,l must be contracted by φ, since φ(Γ l,l ) is in the set where g is nite. Let

p 1 : P × F → F × F
be the canonical P 1 -bundle. Then the restricted map

φ • ψ 1 : p -1 1 (U ADE ) → Z
contracts all the bers of

p 1 : p -1 1 (U ADE ) → U ADE Since p 1 is a linear P d -bundle, we can consider an open set U ⊆ U ADE trivializing p 1 [fWS, Section 3.2]. Then the diagram (3.4.1) p -1 1 (U ) ∼ = P 1 × U p 1,U
U ADE → Z which point-wise is (l, l ) → φ([L ∪ C x ]).
Because we know that this map does not depend on the choice of x ∈ L. We have therefore dened a rational map

ψ : F × F Z (l, l ) → φ([L ∪ C x ]), for all (l, l ) in the open subset U ADE of F × F .
Proposition 3.4.1. The rational map ψ : F × F Z dened above is dominant, has of degree 6 and Ind(ψ ) = I.

Proof. The composition of ψ with g : Z → G(4, 6) gives rise to a commutative diagram (3.4.2)

F × F ψ ρ |F ×F / / G(4, 6) Z g 9 9
where ρ is the span map. The inclusion I ⊆ Ind(ψ ) is an application of Lemma 3.1.2 and Proposition 3.2.13 to the diagram (3.4.2). To prove the other inclusion, consider the diagram (3.4.3)

P × F p 1 ψ 1 / / F 3 (Y ) φ F × F ψ / / Z that is commutative in p -1 1 (U ADE ). Indeed, if (l, x, l ) ∈ p -1 1 (U ADE ), then we know that ψ 1 is dened in (l, x, l ) as (l, l ) / ∈ I. Then φ(ψ 1 (l, x, l )) = φ([L ∪ C x ]) = φ(Γ l,l ) = ψ (l, l ) = ψ (p 1 (l, x, l )).
Consider a section s : U → P × F of p 1 dened in a open subset U ⊆ F × F \I, and let

ψ := φ • ψ 1 • s
be the rational map dened on U . By commutativity of (3.4.3), ψ coincides with ψ on U ∩ U ADE . Since U is arbitrary, it follows that ψ can be extended to every point of F × F \I. We have then proved that

Ind(ψ ) ⊆ I. Let M ∈ G(4, 6), then ρ -1 (M ) = (l, l ) ∈ F × F \I/ L, L = M .
In particular, the pairs (l, l ) ∈ ρ -1 (M ) represent pairs of disjoint lines contained in the cubic surface Let ψ be the Voisin map. We will denote by ψ : F × F → Z a resolution of the indeterminacy of the map ψ as in Denition 3.1.3. Then there is a commutative diagram (3.4.4)

F × F π ψ " " F × F ψ / / Z
where F × F is a non-singular variety, and π is a birational morphism that is an isomorphism outside Ind(ψ). We will denote by E the support of the exceptional divisor of π.

Remark 3.4.2. Notice that we have the following commutative diagram.

(3.4.5) For the reader's convenience, we collect the following.

P × F p 1 ψ 1 / / F 3 (Y ) φ•σ φ | | Z σ # # F × F
Lemma 3.4.3 ([oiIT,Remark 4.10]). The map ψ : F × F Z is étale of degree 6 where it is dened. Furthermore, the image of the exceptional divisor of the resolution of ψ is a divisor. Now we want to check that (3.4.12) I W, where W ⊆ F × F is the closed subset dened in (3.4.8). Since (l, l ) ∈ I, it is enough to prove the following.

Claim. The point (l, l ) is not in W .

Proof of the Claim. If (l, l ) ∈ W , by denition of W there is some (3.4.13)

u ∈ ψ -1 (Y )
such that (l, l ) = π(u ). Again by uniqueness in (3.4.10), u = u . Now (3.4.13) contradicts (3.4.11).

We get that the open subset I\(W ∩ I) of I is not empty by (3.4.12). If we apply Lemma 3.1.2 to diagram (3.4.9) we get

Ind(ρ |F ×F \W ) ⊆ Ind(ψ |F ×F \W ) ⇒ I\(W ∩ I) ⊆ Ind(ψ)\(W ∩ Ind(ψ)).
In particular, we have a dense subset of I contained in Ind(ψ), then I ⊆ Ind(ψ). Hence by (3.4.7) we get I = Ind(ψ). This contradicts the assumption Ind(ψ) I and we are done.

  avec l'application de Voisin. Enn on montre que. Théorème 0.0.19. Le lieu d'indétermination de l'application de Voisin ψ est la variété I des droites concourantes dans Y . issunto Questa tesi è divisa in due parti distinte.

  Example 2.1.10. The Chern characters are very interesting examples of homogeneous polynomials in the Chern classes with Q-coecients. In [pulWV, Example 15.1.2(b)] we have the following closed formula, where

  is maximal solvable [roHU, Theorem p.341]. Then it is the Lie algebra of a Borel subgroup B containing T by [roHU, Denition-Proposition 2 p.356].

  and we have to choose a µ such that b = 2 in order to have λ = λ ∪ {2} which implies λ = (m -1, m -2, m -4, ...).

  Proposition2.5.4 ([hivII, Example 6.10]). Let X b be the blow-up of P 4 along a smooth quartic surface Y b ⊆ P 3 . Then(1) The variety X b is Fano.

  Example 3.3.3. The only examples of hyperKähler varieties in dimension two are the K3 surfaces. That is, a surface S such that c 1 (S) = 0 and H 1 (S, O S ) = 0.

Z

  Hence we have another denition of the Voisin map in [oiIT, Proposition 4.8] as the composition σ • ψ .

  Qual è `l geomètra che tutto s'age per misurar lo cerchio, e non ritrova, pensando, quel principio ond'elli indige, Dante Alighieri, La Divina Commedia, Paradiso XXXIII, v.133-135 Bibliography egIP grolin erujo nd enEwri gstrvetD Polarized minimal families of rational curves and higher Fano manifoldsD emerF tF wthF 134 @PHIPAD noF ID VU!IHUF w PVUTIRH egIQ D Classication of 2-Fano manifolds with high indexD e elertion of lgeri geE ometryD gly wthF roFD volF IVD emerF wthF oFD rovideneD sD PHIQD ppF I!QTF w QIIRWQR evIU xiols eddington nd wnfred vehnD On the symplectic eightfold associated to a Pfaan cubic fourfoldD tF eine engewF wthF 731 @PHIUAD IPW!IQUF w QUHWHTP fgrwIH guher firkrD olo gsiniD ghristopher hF ronD nd tmes wuernnD Existence of minimal models for varieties of log general typeD tF emerF wthF oF 23 @PHIHAD noF PD RHS!RTVF w PTHIHQW fhVS ernud feuville nd on hongiD La variété des droites d'une hypersurface cubique de dimension 4D gF F edF iF ris érF s wthF 301 @IWVSAD noF IRD UHQ!UHTF w VIVSRW feVQ ernud feuvilleD Variétés Kähleriennes dont la première classe de Chern est nulleD tF hifE ferentil qeomF 18 @IWVQAD noF RD USS!UVP @IWVRAF w UQHWPT feHU D On the splitting of the Bloch-Beilinson ltrationD elgeri yles nd motivesF olF PD vondon wthF oF veture xote erFD volF QRRD gmridge nivF ressD gmridgeD PHHUD ppF QV!SQF w PIVUIRV frSV eF forel nd pF rirzeruhD Characteristic classes and homogeneous spaces. ID emerF tF wthF 80 @IWSVAD RSV!SQVF w HIHPVHH fogUR pF eF fogomolovD The decomposition of Kähler manifolds with a trivial canonical classD wtF F @xFFA 93(135) @IWURAD SUQ!SUSD TQHF w HQRSWTW fouTV xF fourkiD Éléments de mathématique. Fasc. XXXIV. Groupes et algèbres de Lie. Chapitre IV: Groupes de Coxeter et systèmes de Tits. Chapitre V: Groupes engendrés par des réexions. Chapitre VI: systèmes de racinesD etulités ienti(ques et sndustriellesD xoF IQQUD rermnnD risD IWTVF w HPRHPQV fTP eF forel nd F emmertD über kompakte homogene Kählersche MannigfaltigkeitenD wthF ennF 145 @IWTIGIWTPAD RPW!RQWF w HIRSSSU fWS wuro gF feltrmetti nd endrew tF ommeseD The adjunction theory of complex projective varietiesD he qruyter ixpositions in wthemtisD volF ITD lter de qruyter 8 goFD ferlinD IWWSF w IQIVTVU gsHT ginzi gsgrndeD Mori Dream Spaces and Fano varietiesD notes for miniourse given t qeq @PHHTAF gosII szzet goskunD Restriction varieties and geometric branching rulesD edvF wthF 228 @PHIIAD noF RD PRRI!PSHPF w PVQTIPU gosIV D Restriction varieties and the rigidity problemD huert rietiesD iquivrint goE homology nd ghrteristi glsses @PHIVAD RW!WSF gHW szzet goskun nd tson trrD Rational curves on smooth cubic hypersurfacesD sntF wthF esF xotF swx @PHHWAD noF PRD RTPT!RTRIF w PSTRQUH gutHH F hle gutkoskyD Irrational asymptotic behaviour of Castelnuovo-Mumford regularityD tF eine engewF wthF 522 @PHHHAD WQ!IHQF w IUSVSUU deIS fel vus de errudD On varieties of lines on linear sections of grassmanniansD rivXISHSFHTRVV @PHISAF heHI ylivier herreD Higher-Dimensional Algebraic GeometryD niversitextD pringerEerlgD xew orkD PHHIF w IVRIHWI hivII ylivier herreD vwrene iinD oert vzrsfeldD nd glire oisinD Pseudoeective and nef classes on abelian varietiesD gomposF wthF 147 @PHIIAD noF TD IUWQ!IVIVF w PVTPHTQ himWP elexndru himD Singularities and topology of hypersurfacesD niversitextD pringerEerlgD xew orkD IWWPF w IIWRIVH dtHT eF tF de tong nd tson wihel trrD A note on fano manifolds whose second chern character is positiveD rivXmthGHTHPTRR @PHHTAF dtHU eF tF de tong nd tson trrD Higher Fano manifolds and rational surfacesD huke wthF tF 139 @PHHUAD noF ID IUQ!IVQF w PQPPTUW ihrQR ghrles ihresmnnD Sur la topologie de certains espaces homogènesD ennF of wthF @PA 35 @IWQRAD noF PD QWT!RRQF w ISHQIUH pwWS F pultonD F whersonD pF ottileD nd fF turmfelsD Intersection Theory on Spherical VarietiesD tF elgeri qeomF 4 @IWWSAD noF ID IVI!IWQF w IPWWHHV pujVP ko pujitD Classication of projective varieties of ∆-genus oneD roF tpn edF erF e wthF iF 58 @IWVPAD noF QD IIQ!IITF w TTRSRW pujVP D On polarized varieties of small ∆-generaD ohoku wthF tF @PA 34 @IWVPAD noF QD QIW!QRIF w TUTIIQ pulWS illim pultonD Algebraic TopologyD qrdute exts in wthemtisD volF ISQD pringerE erlgD xew orkD IWWSD e (rst ourseF w IQRQPSH pulWV D Intersection TheoryD seond edFD irgenisse der wthemtik und ihrer qrenzgeieteF QF polgeF e eries of wodern urveys in wthemtis esults in wthemtis nd elted eresF Qrd eriesF e eries of wodern urveys in wthemtisD volF PD pringerEerlgD ferlinD IWWVF w ITRRQPQ qrUV hillip qri0ths nd toseph rrrisD Principles of Algebraic GeometryD ileyEsntersiene tohn iley 8 onsD xew orkD IWUVD ure nd epplied wthemtisF w SHUUPS qodUI glude qodillonD Éléments de topologie algébriqueD rermnnD risD IWUIF w HQHIUPS qIH lrih qörtz nd orsten edhornD Algebraic geometry ID edvned vetures in wthemtE isD ieweg C eunerD iesdenD PHIHD hemes with exmples nd exerisesF w PTUSISS rrUU oin rrtshorneD Algebraic GeometryD pringerEerlgD xew orkEreidelergD IWUUD qrdE ute exts in wthemtisD xoF SPF w HRTQISU rrVP toe rrrisD Curves in projective spaceD éminire de wthémtiques upérieures eminr on righer wthemtisD volF VSD resses de l9niversité de wontrélD wontrelD ueFD IWVPD ith the ollortion of hvid iisenudF w TVSRPU rtHP ellen rtherD Algebraic TopologyD gmridge niversity ressD gmridgeD PHHPF w IVTUQSR rirTR reisuke rironkD Resolution of singularities of an algebraic variety over a eld of characteristic zero. 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  Le premier exemple de ce phénomène a été donné dans[hivII]. Ottem a montré que si la cubique lisse Y est très générale, alors Eff 2 (F ) Nef 2 (F ). De plus, la deuxième classe de Chern c 2 (F ) est nef mais elle n'est pas eective [yttIS, Theorem 1].Un autre exemple de variété hyperkählérienne a été donné récemment par C. Lehn, M. Lehn, Sorger et van Straten dans[vvvIU]. Ils ont observé que si F 3 (Y ) est une variété compacte des cubiques rationnelles dans Y , où Y est une cubique de dimension 4 lisse qui ne contient pas de plan, alors il y a une bration en P 2 φ : F 3 (Y ) → Z , où Z est une variété lisse. Il y a aussi un diviseur sur Z qui peut être contracté, la contraction obtenue σ : Z → Z produit la variété hyperkählérienne Z, qui est de type K3[4]

  Allora, b 4 (X) ≤ 2. In particolare Eff 2 (X) è poliedrale. 0.0.13. Sia Y = G(2, 5) o G(2, 6), e sia X intersezione completa di tipo(1, 1) in Y con la mappa di Plücker. Allora X non è 2-Fano debole.

	Queste proposizioni sono state ottenute utilizzando vari risultati in [egIQ], dove ven-
	gono date varie condizioni su una varietà X per essere 2-Fano debole. Inoltre rispondiamo
	a [egIQ, Question 39 and 40], infatti dimostriamo il seguente.
	Per provare il teorema si è utilizzato sia [deIS, Corollary 5.1] (che implica che se X
	è generale allora non è 2-Fano), sia [yttIS, Proposition 3]. Questo teorema completa
	la classicazione delle 2-Fano deboli di indice i X ≥ n -2 in [egIQ, Theorem 4]. Nella
	dimostrazione abbiamo utilizzato il seguente.
	Lemma 0.0.14. Sia X 11 intersezione completa liscia di tipo (1, 1) in G(2, 5) con la
	mappa di Plücker. Allora b 4 (X) = 2.

Teorema

  is either a half-line or it is spanned by two extremal rays, depending on the rank of N k (X) R . 2.1.30. A smooth cubic surface S ⊆ P 3 is a Fano variety ad it is a classical object in Algebraic Geometry. The surface S is isomorphic to the blow-up of P 2 centered at six points in general position[rrUU, Corollary II.4.7]. Mori's Cone of S is a polyhedral cone in a 7-dimensional vector space, and it is spanned by 27 extremal rays spanned by the classes of the 27 lines of S [heHI, p.149].

Example

  Definition 2.2.1([roHU, p.170]). A subgroup H ⊆ GL n (k) is called a linear group. A Zariski closed subgroup H ⊆ GL n (k) is a linear algebraic group. 2.2.2 ([roHU, p.188]). A linear algebraic group is called reductive if it does not contain any closed unipotent normal subgroup. A linear algebraic group is called semisimple if it is connected and its solvable radical is trivial. 2.2.3([roHU, p.190]). A subgroup of an algebraic group is called a maximal torus if it is a closed subgroup, a torus as an algebraic group, and maximal with respect to this property. A subgroup of an algebraic group is called a Borel subgroup if it is closed, connected and solvable, and maximal with respect to this property.It is a consequence of[roHU, Lemma p.207] that any maximal torus is contained in some Borel subgroup of G. Furthermore, we have the following. 2.2.4([roHU, p.190]). All maximal tori are conjugate. All Borel subgroups are conjugate.

Definition Definition

Theorem

  ∆ [roHU, Remark p.352]. Let P ∆ be the subgroup with Lie algebra p ∆ . The subgroup P ∆ is parabolic [roHU, Corollary of the proof p.355]. On the other hand, any parabolic subgroup containing B is equal to P ∆ for some ∆ ⊆ ∆ [roHU, Corollary of the proof p.355]. 2.2.8. A projective variety X is homogeneous if there exists a group variety which acts on X transitively.

Definition

  Z) is injective with torsion free cokernel by Theorem 2.4.2 and [vzHR, Example 7.1.2]. Since b 4 (Y ) = b 4 (X t ) = 2 by Lemma 2.4.6, we have that i * : H 4 (Y, Z) → H 4 (X t , Z) is an isomorphism. By [deIS, Corollary 5.1], for a general t there exists a surface S t such that

  b 6 (Q) = 2, then Eff 3 (X) is polyhedral by Lemma 2.1.29. For the other complete intersections we can use Proposition 2.4.4, whilst for the rational homogeneous varieties we can use Proposition 2.2.13. Also for the complete intersections in OG + (5, 10) we have b 6 (X) = 2, because b 6 (OG + (5, 10)) = 2 by Lemma 2.2.29 and we can use Theorem 2.4.2.

  satises the hypothesis of the Rigidity Lemma[qIH, Proposition 16.54]. Indeed: U is reduced, Z is separated and P 1 is reduced, connected and proper. It follows that there exists a unique morphism U → Z making (3.4.1) commutative. We can cover U ADE by trivializing open subsets, repeat that argument and get a map

	φ•ψ 1 / / Z
	U

  M ∩ Y . If M is suciently general, then M ∩ Y is smooth and contains 27 lines, each of them meeting exactly 10 other lines[qrUV, p. 485]. It can easily be seen that there are 27 • (27 -11) = 432 pairs of lines contained in ρ -1 (M ), and therefore ρ is generically nite of degree 432. All the manifolds appearing in the commutative diagram (3.4.2) are 8-dimensional manifolds. Then as ρ is generically nite is also dominant, this implies that also g and ψ are dominant and generically nite. Since deg ρ = deg ψ deg g by commutativity of (3.4.2), the degree of ψ is

	deg ψ =	432 72	= 6.

given by the generator σ (s-r,...,s-r) .

Example 2.2.22. Let us compute some products of cycles which will be useful later. Using Pieri's formula for the cycles of G(2, 5), we have:

While in the case G(2, 6) we have

Remark 2.2.23. It can be shown that the the Plücker line bundle generates Pic(G(r, s)). Then due to (2.2.5), we have -K G(r,s) = O(s) where O(1) gives the Plücker embedding.

Remark 2.2.24. By Serre duality

Then we have by [noVT,Theorem p.p. 165,169]. By Lemma 2.2.21 and Remark 2.2.17 we have (2.2.8)

Remark 2.2.25. In order to compute the values of χ(Ω p G(2,s) (m)) we can use the explicit formulas given in [uII, Chapter 4] and get the same results of Remark 2.2.24. Roughly speaking: for any s consider a sequence of r integer λ = (λ 1 , ..., λ r ), set By [vzHR,Example 7.1.5], all rows of the Hodge Diamond of X, except the middle row, are equal to those of the Hodge Diamond of G = G(2, 5). Since X is Fano, h 0,4 (X) = 0 then

(2.4.3)

Note that by Serre duality and adjunction formula, for any integer m

then by Theorem 2.1.31,

and tensor it by Ω G

(2.4.5)

then, by Remark 2.2.24,

If we tensor (2.4.5) by O G (-1) we have

From the canonical sequence (2.4.6)

we get, by Remark 2.2.24,

By [rrUU, Exercise II.5.16d] and (2.4.6) we get

Then by (2.4.2) and (2.4.3) we get h 2,2 (X) = 2 and b 4 (X) = 2.

Proposition 2.4.7. Let X be a n-dimensional weak 2-Fano complete intersection in a Grassmann variety G(r, s) under the Plücker embedding. Then, b 4 (X) ≤ 2. In particular Eff 2 (X) is polyhedral.

Remark 2.4.13. In [egIQ, Proposition 34], it is stated that the smooth complete intersection of OG + (k, 2k) of type (2, 2) under the Plücker embedding is a weak 2-Fano variety. This should be read as (2).

Proposition 2.4.14. Let X be a n-dimensional weak 2-Fano complete intersection in a symplectic Grassmann variety SG(r, s) under the Plücker embedding. Then b 4 (X) ≤ 2. In particular Eff 2 (X) is polyhedral.

Proof. Assume that X is of type (d 1 , ..., d c ). If n > 4, by Theorem 2.4.2 and Lemma 2.2.32, we have b 4 (X) = b 4 (SG(r, s)) ≤ 2 and we can apply Lemma 2.1.29. If n = 4, since c 1 (SG(r, s)) = (s -r + 1)σ 1 we have the following conditions:

It is easy to see that this leads to the following cases:

The variety

, as we said in Remark 2.2.31. Thus the last two case are, respectively, (1, 1, 1, 1) and(1, 1, 1, 2) in G(2, 6). The rst two cases are not weak 2-Fano by [egIQ, Proposition 36], the last two by [egIQ,Proposition 32(i)].

PFSF pno mnifolds of dimension

Fano varieties of high index have been classied: [uyUQ] proved that i X ≤ n + 1, i X = n + 1 if and only if X = P n , and i X = n if and only if X ⊆ P n+1 is a smooth hyperquadric.

Furthermore the case i X = n-1 (the so called Del Pezzo varieties) has been classied by Fujita in [pujVPD pujVP], and the case i X = n -2 (the so called Mukai varieties) by Mukai (see [wukVW] and [sWW]).

Araujo and Castravet [egIQ, Theorem 3] succeeded to classify 2-Fano Del Pezzo and Mukai varieties. They proved: Theorem 2.5.2. Let X be a 2-Fano variety of dimension n ≥ 3 and index i X ≥ n-2.

Then X is isomorphic to one of the following.

• P n .

• Complete intersection in projective spaces: -Quadric hypersurfaces X ⊆ P n+1 with n > 2; -Complete intersections of type (2, 2) in P n+2 with n > 5; -Cubic hypersurfaces X ⊆ P n+1 with n > 7; -Quartic hypersurfaces X ⊆ P n+1 with n > 15; -Complete intersections of type (2, 3) in P n+2 with n > 11; -Complete intersections of type (2, 2, 2) in P n+3 with n > 9.

Theorem 3.2.4 (Grothendieck). Let X ⊆ P n be a projective C-scheme and P ∈ Q[z].

The functor

which are proper and flat over B and have Hilbert polynomial P is representable by a pair (Hilb P (X), Univ P (X)), where Hilb P (X) is a projective complex scheme.

Proof. [uolWT,1.4 Theorem]. This theorem implies that there exists a scheme parametrizing all the subschemes of X with a given Hilbert polynomial. If V ⊆ X is a subscheme, we set [V ] to be its point in Hilb P (X). Furthermore, there exists a universal family Univ P (X) ⊆ X × Hilb P (X). The morphism Univ P (X) → Hilb P (X) is at, the ber over a point [V ] is

We are now ready to dene the variety of lines on a cubic fourfold. Let L ⊆ P n be a line, that is a degree 1 rational curve. Then by [rrUU, Proposition.IV.1.1] its Hilbert polynomial is P L (z) = z + 1. Definition 3.2.5. Let Y ⊆ P 5 a smooth cubic fourfold. We denote by F the projective scheme Hilb z+1 (Y ), and by P the projective scheme Univ z+1 (Y ). In particular we have the following universal family (3.2.1) We can deduce by [uolWT,4.3 Theorem] that F is smooth and connected of dimension 4. Furthermore, by what we said in Subsection 2.2.2, we have Hilb z+1 (P n ) = G(2, n + 1). Then F is canonically embedded in G(2, 6). Definition 3.2.6. We denote by I the subscheme in F ×F , endowed with the reduced structure, of intersecting lines, i.e.

Note that I can be dened as

Remark 3.2.7. Let x ∈ Y be a point. Let C x be the subvariety of F parametrizing lines through x. Take a system of coordinates of P 5 such that x = [0 : 0 : 0 : 0 : 0 : 1].

Then the equation of Y is x 2 5 q 1 + x 5 q 2 + q 3 = 0, where the polynomial q i is homogeneous of degree i. Since Y is smooth, q 1 is not the zero polynomial. The variety C x can now be seen in P 4 as given by q 1 = q 2 = q 3 = 0. Then C x can be seen as q 2 = q 3 = 0 in P 3 . Thus it is connected [rrUU,Exercise II.8.4c]. Consider the projection to the rst component pr 1 : I → F , the bers are

It is known that C x is a curve for x ∈ Y , except for nitely many x i ∈ Y such that C x i is a surface [gHW, Proposition 2.4]. Hence pr -1 1 (l) is a surface for all l ∈ F . Moreover,

is connected for all l ∈ F . Indeed, pr -1 1 (l) is the union of connected subvarieties C x all of them meeting at the point (l, l). Furthermore, pr -1 1 (l) is smooth for general l ∈ F by [oiVT, Section 3 Lemma 1], hence irreducible.

I thank Mingmin Shen for suggesting me the following proof.

Lemma 3.2.8. The variety I ⊆ F × F is irreducible of dimension 6.

Proof. Let J := (q × q) -1 (∆ Y ). Then J is locally dened by four equations in P × P , hence each component of J has dimension at least 6. The map p × p : J → I is surjective and only contracts ∆ P to ∆ F . Hence p × p is birational and each component of I has dimension at least 6. Since pr -1 1 (l) is a surface for all l ∈ F by Remark 3.2.7, each component of I has dimension 6. Moreover, pr -1 1 (l) is irreducible for general l ∈ F . It follows that only one component of I maps surjectively over F . Indeed, let I 1 be any irreducible component of I such that pr 1 (I 1 ) = F , then we have a surjective map pr 1|I 1 : I 1 → F between two irreducibles varieties of dimension, respectively, 6 and 4. The bers of this map are pr -1 1|I 1 (l) = pr -1 1 (l) ∩ I 1 . Thus, for dimensional reasons, the general ber of pr 1|I 1 is a surface, then pr -1 1 (l) ∩ I 1 is a component of pr -1 1 (l). As pr -1

is irreducible for general l ∈ F , it follows that pr -1 1 (l) has only one component, then

We have proved that pr -1 1 (l) ⊆ I 1 for general l ∈ F , then if I 2 is another irreducible component of I such that pr 1 (I 2 ) = F , the same argument implies that pr -1 1 (l) ⊆ I 2 for general l ∈ F . It follows that pr -1 1 (l) ⊆ I 1 ∩ I 2 for general l ∈ F , then I 1 = I 2 as they are both irreducible.

Any other component of I dierent from I 1 maps over a closed subset of F , then for dimensional reasons this component must have dimension at most 5, then I = I 1 . It follows that I is irreducible and dim I = 6.

If X 1 , X 2 are subvarieties in the same projective space, we denote by X 1 , X 2 their linear span.

Let a, b, n ∈ Z ≥0 such that a+b+1 < n. Let P 1 ⊂ P n and P 2 ⊂ P n be linear subspaces of dimension, respectively, a and b. By denition, if

It is know by linear algebra that

Let us consider the incidence correspondence

with the canonical projections

Lemma 3.2.9. The scheme I is irreducible.

Proof. Consider M ∈ G(a + b + 2, n + 1). The ber of the map

and the natural projection

is an isomorphism. Then we have the following chain of isomorphisms

Hence I is irreducible by [hWR, Chap. I, 6.4, Theorem 8].

Definition 3.2.10. Let

If P 1 ∩ P 2 = ∅, then P 1 , P 2 ∈ G(a + b + 2, n + 1). Then there exists a rational map

We denote by Γ ρ the closure of Γ ρ |U inside G(a+1, n+1)×G(b+1, n+1)×G(a+b+2, n+1).

Lemma 3.2.11. Γ ρ = I, and Ind(ρ) = I G .

Proof. Let (P 1 , P 2 ) ∈ Γ ρ |U , then ∃!M P 1 ,P 2 such that (P 1 , P 2 , M P 1 ,P 2 ) ∈ Γ ρ |U , that is P 1 , P 2 = M P 1 ,P 2 . Then we have P 1 , P 2 ⊆ M P 1 ,P 2 so (P 1 , P 2 , M P 1 ,P 2 ) ∈ I. This implies that Γ ρ |U ⊆ I, but I is closed, then

To prove that Ind(ψ) = I G we argue as follows. Let U ρ be the domain of ρ. By [heHI,1.39

Proof. The map ψ is dominant of degree 6 because it is the composition of a dominant degree 6 rational map and of a blow up. Let R ψ be the ramication divisor of ψ, that is the divisor supported in the subset of points of F × F where the induced map d ψ : T F ×F → ψ * T Z is not an isomorphism. The scheme structure is given locally by the vanishing of the Jacobian determinant det d ψ [pulWV,Example 3.2.20]. Thus we have the formula

and since the rst Chern class of F and Z is trivial, E = R ψ. This implies that the ramication locus of ψ is E = SuppE , then the Jacobian matrix is of maximal rank outside E. Let D = ψ(E) be the image of the exceptional divisor. Let G = ψ-1 (D) ⊇ E be the inverse image of D. By the properties of the ramication, the maps We are now ready for the following. We point out that outside the support of the exceptional divisor, ψ is nite by Lemma 3.4.3 and the fact that

is an isomorphism. Then there is a commutative diagram of rational maps (3.4.9) We argue by contradiction. Suppose Ind(ψ) I, and set T := π(π -1 (I)\( ψ -1 (Y ) ∩ π -1 (I))).

Claim. The set T is dense in I. that is dense in the strict transform of I, since π is an isomorphism outside Ind(ψ). Now I\( ψ -1 (Y ) ∩ I) is dense in I since it is open and not empty. Indeed, if it were empty then I ⊆ ψ -1 (Y ), so Claim. The point u is not in ψ -1 (Y ), i.e. (3.4.11) u / ∈ ψ -1 (Y ).

Proof of the Claim. Since (l, l ) ∈ T , by denition of T and by uniqueness of u, we have u ∈ π -1 (I)\( ψ -1 (Y ) ∩ π -1 (I)).

In particular (3.4.11) holds.