Michele Bolognesi
email: michele.bolognesi@umontpellier.fr

Alex Massarenti
email: alexmassarenti@id.uff.br

A VARIATION ON A CONJECTURE OF FABER AND FULTON

Keywords: Mathematics Subject Classification. Primary 14D22, 14H10, 14H37; Secondary 14N05, 14N10, 14N20 GIT quotients, Moduli of curves, Mori Dream Spaces, F-conjecture

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

In one of the most celebrated papers [START_REF] Deligne | The irreducibility of the space of curves of given genus[END_REF] in the history of algebraic geometry Deligne and Mumford proved that there exists an irreducible scheme M g,n coarsely representing the moduli functor of n-pointed genus g smooth curves. Furthermore, they provided a compactification M g,n of M g,n adding the so-called Deligne-Mumford stable curves as boundary points. Afterwards other compactifications of M g,n have been introduced, see for instance [START_REF] Hassett | Moduli spaces of weighted pointed stable curves[END_REF].

In this paper we are interested in the compactification of M 0,n given by the GIT quotient Σ m := (P 1 ) m+3 / /P GL(2) of configurations of n = m + 3 ordered points on P 1 with respect to the symmetric polarization on (P 1 ) n . The aim of our notation is to stress the dimension of the space. In particular, for m = 3 we obtain the celebrated Segre cubic 3-fold Σ 3 ⊂ P 4 [START_REF] Dolgachev | Corrado Segre and nodal cubic threefolds[END_REF].

The moduli spaces M g,n are among the most studied objects in algebraic geometry. Despite this, many natural questions about their biregular and birational geometry remain unanswered.

The F-conjecture, where as far as we know F stays for Fulton and Faber, for M g,n is one of the long-standing conjectures in the field. Its statement is the following:

Conjecture. [GKM02, Conjecture 0.2] A divisor on M g,n is ample if and only if it has positive intersection with all 1-dimensional strata. In other words, any effective curve in M g,n is numerically equivalent to an effective combination of 1-strata.

In other words, as we stated in the abstract, any extremal ray of the Mori cone of effective curves NE(M g,n ) is generated by a one dimensional boundary stratum. In [START_REF] Gibney | Towards the ample cone of Mg,n[END_REF] A. Gibney, S. Keel and I. Morrison managed to reduce the conjecture for any genus g to the genus zero case, that is M 0,n . Anyway, so far the conjecture is known up to n = 7 [KMc96, Thm. 1.2(3)] (see [START_REF] Fontanari | Towards Fultons conjecture[END_REF] for a more recent general account), and the very intricate combinatorics of the moduli space M 0,n for higher n seem an obstacle pretty difficult to avoid if one wants to try his chance.

In this paper we go in a somewhat orthogonal direction. Instead of trying to show the conjecture for n > 7, we modify it slighlty by allowing a little coarser moduli space into the picture. In fact, we consider the GIT quotient Σ m . This quotient offers an alternate compactification of M 0,n , which is a little coarser than M 0,n on the boundary, and in fact it is the target a birational morphism M 0,n → Σ m , that we will recall in the body of the paper, see also [START_REF] Bolognesi | Forgetful linear system on the projective space and rational normal curves over M GIT 0,n[END_REF].

Nevertheless, also Σ m has a stratification of its boundary locus, similar to that of M 0,n , and one can ask exactly the same question of Conjecture . We tackle this problem taking advantage of the fact that Σ m is a Mori Dream Space, while we know that M 0,n is not a Mori Dream Space for n ≥ 10 [CT15, Corollary 1.4], [GK16, Theorem 1.1], [START_REF] Hausen | On blowing up the weighted projective plane[END_REF]Addendum 1.4]. Mori Dream Spaces, introduced by Y. Hu and S. Keel [START_REF] Hu | Mori dream spaces and GIT[END_REF], form a class of algebraic varieties that behave very well from the point of view of the minimal model program. In particular, their cones of curves and divisors are polyhedral and finitely generated.

By Proposition 2.7 if n = m + 3 is even then Pic(Σ m ) ∼ = Z and its cones of curves and divisors are 1-dimensional. On the other hand, if n = m + 3 is odd then Pic(Σ m ) ∼ = Z m+3 and the birational geometry of Σ m gets more interesting. In this case we manage to describe the cones of nef and effective divisors of Σ m and their dual cones of effective and moving curves. The first main result of this paper says that the analogue of the F-conjecture for GIT configurations of points holds.

Theorem 1. If n = m + 3 is odd then the Mori cone NE(Σ m ) is generated by classes of 1-dimensional boundary strata.

In Theorem 2.33 we will also describe precisely what are these 1-dimensional strata. The main ingredients of the proof of Theorem 1 is a construction of our GIT quotients as images Σ m ⊂ P N of rational maps induced by certain linear systems on the projective space P m , due to C. Kumar [START_REF] Kumar | Invariant vector bundles of rank 2 on hyperelliptic curves[END_REF][START_REF] Kumar | Linear Systems and Quotients of Projective Space[END_REF], a careful analysis of the Mori chamber decomposition of the movable cone of certain blow-ups of the projective space and some quite refined projective geometry of the GIT quotients. More precisely C. Kumar realized Σ m as the closure of the image of the rational map induced by the linear system L 2g-1 of degree g hyersurfaces of P 2g-1 having multiplicity g -1 at 2g + 1 general points if m = 2g -1 is odd, and as the closure of the image of the rational map induced by the linear system L 2g of degree 2g + 1 hyersurfaces of P 2g having multiplicity 2g -1 at 2g + 2 general points if m = 2g is even. In particular, N = h 0 (P 2g-1 , L 2g-1 ) if m = 2g -1 is odd, and N = h 0 (P 2g , L 2g ) if m = 2g is even.

The tools developed turn out to be useful in resolving a couple of other problems related to GIT quotients of configuration of points on P 1 . Furthermore, thanks to recent results on the dimension of linear system on the projective space due to M. C. Brambilla, O. Dumitrescu and E. Postinghel [START_REF] Brambilla | On a notion of speciality of linear systems in P n[END_REF][START_REF] Brambilla | On the effective cone of P n blown-up at n + 3 points[END_REF], we obtain some explicit formulas for the Hilbert polynomial of Σ m ⊂ P N . These results are resumed in Corollaries 3.4, 3.5. We would like to mention that an inductive formula for the degree of these GIT quotients had already been given in [START_REF] Howard | The equations for the moduli space of n points on the line[END_REF], while a closed formula for the Hilbert function of GIT quotients of evenly weighted points on the line had been given in [START_REF] Hering | The ring of evenly weighted points on the line[END_REF]. The main results on the geometry of Σ m ⊂ P N in Sections 2.1, 2.9, 4, and Corollaries 3.4, 3.5 can be summarized as follows.

Theorem 2. Let us consider the GIT quotient Σ m ⊂ P N . If m = 2g -1 is odd we have

Pic(Σ 2g-1 ) ∼ = Z, K Σ 2g-1 ∼ = O Σ 2g-1 (-2),
and the the Hilbert polynomial of Σ 2g-1 ⊆ P N is given by

h Σ 2g-1 (t) = gt + 2g -1 2g -1 + g-2 r=0
(-1) r+1 2g + 1 r + 1 t(g -r -1) + 2g -1 -r -1 2g -1 .

In particular deg(Σ 2g-1 ) = g 2g-1 + g-2 r=0

(-1) r+1 2g + 1 r + 1 (g -r -1) 2g-1 .

If m = 2g is even we have

Pic(Σ 2g ) ∼ = Z 2g+3 , K Σ 2g ∼ = O Σ 2g (-1),
and the the Hilbert polynomial of Σ 2g ⊆ P N is given by

h Σ 2g (t) = (2g + 1)t + 2g 2g + 2g-1 2 r=0 (-1) r+1 2g + 2 r + 1 t(2g -2r -1) + 2g -r -1 2g .
In particular

deg(Σ 2g ) = (2g + 1) 2g + 2g-1 2 r=0
(-1) r+1 2g + 2 r + 1 (2g -2r -1) 2g .

Furthermore, the automorphism group of Σ m is isomorphic to the symmetric group on n = m + 3 elements S n for any m ≥ 2.

Plan of the paper. The paper is organized as follows. In Section 1 we recall some wellknown facts and prove some preliminary results on GIT quotient, moduli spaces of weighted pointed rational curves and we clarify the relations between them. In Section 2 we prove the analogue of the F-conjecture for Σ m with m = 2g even. In Section 3 we work out explicit formulas for the Hilbert polynomial a the degree of Σ m ⊂ P N . Finally, in Section 4 we compute the automorphism groups of Σ m .

1. GIT quotients of (P 1 ) n

The main characters of the paper are the GIT quotients (P 1 ) n / /P GL(2), that we review now very quickly. For details there are plenty of very good references on this subject [START_REF] Mumford | Geometric invariant theory[END_REF][START_REF] Dolgachev | Lectures on invariant theory[END_REF][START_REF] Dolgachev | Classical algebraic geometry, a modern view[END_REF][START_REF] Howard | The equations for the moduli space of n points on the line[END_REF][START_REF] Bolognesi | Forgetful linear system on the projective space and rational normal curves over M GIT 0,n[END_REF][START_REF] Dolgachev | Point sets in projective spaces and theta functions[END_REF].

Let us consider the diagonal P GL(2)-action on (P 1 ) n . An ample line bundle L endowed with a linearization for the P GL(2)-action is called a polarization. Such a polarization on (P 1 ) n is completely determined by an n-tuple b = (b 1 , ..., b n ) of positive integers:

L = n i=1 O P 1 (b i ). Now let us set |b| = b 1 + ... + b n . A point x ∈ (P 1
) n is said to be b-semistable if for some k > 0, there exists a P GL(2)-invariant section s ∈ H 0 ((P 1 ) n , L ⊗k ) P GL(2) such that X s := {y ∈ (P 1 ) n : s(y) = 0} is affine and contains x. A semistable point x ∈ (P 1 ) n is stable if its stabilizer under the P GL(2) action is finite and all the orbits of P GL(2) in X s are closed. A categorical quotient of the open set ((P 1 ) n ) ss (b) of semistable points exists, and this is what we normally denote by X(b)/ /P GL(2). We will omit to specify the polarization when it is (1, ..., 1). If b is odd, then H 0 ((P 1 ) n , L ⊗k ) P GL(2) = 0 for odd k, and

(1.0) (P 1 ) n (b)/ /P GL(2) = (P 1 ) n (2b)/ /P GL(2).

Therefore, by replacing b by 2b if necessary, we assume that b is even.

1.1. Linear systems on P n . Throughout the paper we will denote by L n,d (m 1 , ..., m s ) the linear system of hypersurfaces of degree d in P n passing through s general points p 1 , ..., p s ∈ P n with multiplicities respectively m 1 , ..., m s . It was pointed out by Kumar [Ku03, Section 3.3] that the GIT quotients (P 1 ) n (b)/ /P GL(2) can be obtained as the images of certain rational polynomial maps defined on P n-3 .

Theorem 1.2. [Ku03, Theorem 3.4] Let us assume that b i < j =i b j for any i = 1, ..., n, and let p 1 , ..., p n-1 ∈ P n-3 be general points.

-If |b| = 2b is even let

L = L n-3,b-bn (b -b n -b 1 , ..., b -b i -b n , ..., b -b n -b n ) be the linear system of degree b-b n hypersurfaces in P n-3 with multiplicity b-b i -b n at p i .
-If |b| is odd let

L = L n-3,b-2bn (b -2b n -2b 1 , ..., b -2b i -2b n , ..., b -2b n -2b n )
be the linear system of degree b -2b n hypersurfaces in P n-3 with multiplicity b -2b i -2b n at p i . Finally, let φ L : P n-3 P(H 0 (P n-3 , L) * ) be the rational map induced by L. Then φ L maps birationally P n-3 onto (P 1 ) n (b)/ /P GL(2). In other words (P 1 ) n (b)/ /P GL(2) may be realized as the closure of the image φ L in P(H 0 (P n-3 , L) * ).

Example 1.3. For instance, if n = 6 and b 1 = ... = b 6 = 1 then b = 3, and L = L 3,2 (1, ..., 1). In this case the rational map φ L is given by the quadrics in P 3 passing through five general points, and (P 1 ) n (b)/ /P GL(2) ⊂ P 4 is the Segre cubic 3-fold. This is a very well-known classical object, see [START_REF] Dolgachev | Corrado Segre and nodal cubic threefolds[END_REF][START_REF] Alzati | A structure theorem for SUC (2) and the moduli of pointed rational curves[END_REF][START_REF] Bolognesi | Coherent systems and modular subavrieties of SUC (r)[END_REF] for some historic perspective and applications.

If n = 5 and b 1 = ... = b 5 = 1 then L = L 2,3 (1, ..., 1) that is the linear system of plane cubics through four general points. In this case the quotient is a del Pezzo surface of degree five.

1.4. Moduli of weighted pointed curves. In [START_REF] Hassett | Moduli spaces of weighted pointed stable curves[END_REF], Hassett introduced moduli spaces of weighted pointed curves. Given g ≥ 0 and rational weight data

A[n] = (a 1 , ..., a n ), 0 < a i ≤ 1, satisfying 2g -2 + n i=1 a i > 0, the moduli space M g,A[n]
parametrizes genus g nodal n-pointed curves {C, (x 1 , ..., x n )} subject to the following stability conditions:

-each x i is a smooth point of C, and the points

x i 1 , . . . , x i k are allowed to coincide only if k j=1 a i j ≤ 1, -the twisted dualizing sheaf ω C (a 1 x 1 + • • • + a n x n ) is ample.
In particular, M g,A[n] is one of the compactifications of the moduli space M g,n of genus g smooth n-pointed curves.

1.5. For fixed g, n, consider two collections of weight data A[n], B[n] such that a i ≥ b i for any i = 1, ..., n. Then there exists a birational reduction morphism 1.6. Furthermore, for any g, consider a collection of weight data

ρ B[n],A[n] : M g,A[n] → M g,B[n] associating to a curve [C, s 1 , ..., s n ] ∈ M g,A[n] the curve ρ B[n],A[n] ([C, s 1 , ..., s n ])
A[n] = (a 1 , ..., a n ) and a subset A[r] := (a i 1 , ..., a ir ) ⊂ A[n] such that 2g -2 + a i 1 + ... + a ir > 0. Then there exists a forgetful morphism π A[n],A[r] : M g,A[n] → M g,A[r] associating to a curve [C, s 1 , ..., s n ] ∈ M g,A[n] the curve π A[n],A[r] ([C, s 1 , ..., s n ]
) obtained by collapsing components of C along which ω C (a i 1 s i 1 + ... + a ir s ir ) fails to be ample.

One of the most elegant aspects of the theory of rational pointed curves is the relation with rational normal curves and their projective geometry. This has been outlined by Kapranov in [START_REF] Kapranov | Veronese curves and Grothendieck-Knudsen moduli spaces M0,n[END_REF]. Here below we briefly recall this, and his construction of M 0,n as an iterated blow-up of P n-3 . Kapranov's blow-up construction. We follow [START_REF] Kapranov | Veronese curves and Grothendieck-Knudsen moduli spaces M0,n[END_REF]. Let (C, x 1 , ..., x n ) be a genus zero n-pointed stable curve. The dualizing sheaf ω C of C is invertible, see [START_REF] Knudsen | The projectivity of the moduli space of stable curves II: the stack Mg,n[END_REF]. By [Kn83, Corollaries 1.10 and 1.11] the sheaf ω C (x 1 +...+x n ) is very ample and has n-1 independent sections. Then it defines an embedding φ : C → P n-2 . In particular, if

C ∼ = P 1 then deg(ω C (x 1 + ... + x n )) = n -2, ω C (x 1 + ... + x n ) ∼ = φ * O P n-2 (1) ∼ = O P 1 (n -2), and φ(C) is a degree n -2 rational normal curve in P n-2 . By [Ka93, Lemma 1.4] if (C, x 1 , ..., x n ) is stable the points p i = φ(x i ) are in linear general position in P n-2 .
This fact combined with a careful analysis of limits in M 0,n of 1-parameter families contained in M 0,n are the key for the proof of the following theorem [Ka93, Theorem 0.1].

Theorem 1.8. Let p 1 , ..., p n ∈ P n-2 be points in linear general position, and let V 0 (p 1 , ..., p n ) be the scheme parametrizing rational normal curves through p 1 , ..., p n . Consider V 0 (p 1 , ..., p n ) as a subscheme of the Hilbert scheme H parametrizing subschemes of P n-2 . Then

-V 0 (p 1 , ..., p n ) ∼ = M 0,n . -Let V (p 1 , ..., p n ) be the closure of V 0 (p 1 , ..., p n ) in H. Then V (p 1 , ..., p n ) ∼ = M 0,n .
Kapranov's construction allows to translate many questions about M 0,n into statements on linear systems on P n-3 . Consider a general line L i ⊂ P n-2 through p i . There exists a unique rational normal curve C L i through p 1 , ..., p n , and with tangent direction L i in p i . Let [C, x 1 , ..., x n ] ∈ M 0,n be a stable curve, and let Γ ∈ V 0 (p 1 , ..., p n ) be the corresponding rational normal curve. Since p i ∈ Γ is a smooth point, by considering the tangent line T p i Γ we get a morphism (1.9)

f i : M 0,n -→ P n-3 [C, x 1 , ..., x n ] -→ T p i Γ
Furthermore, f i is birational and defines an isomorphism on M 0,n . The birational maps

f j • f -1 i M 0,n P n-3 P n-3 f j •f -1 i f j f i
are standard Cremona transformations of P n-3 [Ka93, Proposition 2.12]. For any i = 1, ..., n the class Ψ i is the line bundle on M 0,n whose fiber on [C, x 1 , ..., x n ] is the tangent line T p i C. From the previous description we see that the line bundle Ψ i induces the birational morphism f i : M 0,n → P n-3 , that is Ψ i = f * i O P n-3 (1). In [START_REF] Kapranov | Veronese curves and Grothendieck-Knudsen moduli spaces M0,n[END_REF] Kapranov proved that Ψ i is big and globally generated, and that the birational morphism f i is an iterated blow-up of the projections from p i of the points p 1 , ..., pi , ...p n and of all strict transforms of the linear spaces that they generate, in order of increasing dimension.

Construction 1.10. Fix (n -1)-points p 1 , ..., p n-1 ∈ P n-3 in linear general position:

(1) Blow-up the points p 1 , ..., p n-1 , (2) Blow-up the strict transforms of the lines p i 1 , p i 2 , i 1 , i 2 = 1, ..., n -1, . . .

(k) Blow-up the strict transforms of the (k-1)-planes p i 1 , ..., p i k , i 1 , ..., i k = 1, ..., n-1, . . .

(n -4) Blow-up the strict transforms of the (n -5)-planes p i 1 , ..., p i n-4 , i 1 , ..., i n-4 = 1, ..., n -1. Now, consider the moduli spaces of weighted pointed curves X k [n] := M 0,A[n] for k = 1, ..., n -4, such that -a i + a n > 1 for i = 1, ..., n -1, -a i 1 + ... + a ir ≤ 1 for each {i 1 , ..., i r } ⊂ {1, ..., n -1} with r ≤ n -k -2, -a i 1 + ... + a ir > 1 for each {i 1 , ..., i r } ⊂ {1, ..., n -1} with r > n -k -2. While apologizing for the new notation, we try to justify it by remarking that X k [n] is isomorphic to the variety obtained at the k th step of the blow-up construction. The composition of these blow-up morphism here above is the morphism f n : M 0,n → P n-3 induced by the psi-class Ψ n . Identifying M 0,n with V (p 1 , ..., p n ), and fixing a general (n -3)-plane H ⊂ P n-2 , the morphism f n associates to a curve C ∈ V (p 1 , ..., p n ) the point

T pn C ∩ H.
In [Has03, Section 2.1.2] Hassett considers a natural variation of the moduli problem of weighted pointed rational stable curves by considering weights of the type A[n] = (a 1 , ..., a n ) such that a i ∈ Q, 0 < a i ≤ 1 for any i = 1, ..., n, and n i=1 a i = 2. By [Has03, Section 2.1.2] we may construct an explicit family of such weighted curves C( A) → M 0,n over M 0,n as an explicit blow-down of the universal curve over M 0,n . Furthermore, if a i < 1 for any i = 1, ..., n we may interpret the geometric invariant theory quotient (P 1 ) n / /P GL(2) with respect to the linearization O(a 1 , ..., a n ) as the moduli space M 0, A[n] associated to the family C( A).

Remark 1.11. Note that we may interpret the GIT quotient (P 1 ) n (b)/ /P GL(2) as a moduli space M 0, A[n] by taking the weights a i = 2 |b| b i . Conversely, given the space M 0, A[n] with (a 1 , ..., a n ) = ( α 1 β 1 , ..., αn βn ) such that n i=1 a i = 2 we may consider the GIT quotient (P 1 ) n (b)/ /P GL(2) with b i = a i M , where M = LCM (β i ).

Remark 1.12. Let M 0, A[n] be a moduli space with weights a i summing up to two, and let M 0,A[n] be a moduli space with weights a i ≥ a i for any i = 1, ..., n. By [Has03, Theorem 8.3] there exists a reduction morphism ρ A[n],A[n] : M 0,A[n] → M 0, A[n] operating as the standard reduction morphisms in 1.5. Proposition 1.13. Let φ L : P n-3 (P 1 ) n (b)/ /P GL(2) ⊂ P(H 0 (P n-3 , L) * ) be the rational map in Theorem 1.2, and let f i : M 0,n → P n-3 in (1.9). Then there exits a reduction morphism ρ : M 0,n → (P 1 ) n (b)/ /P GL(2) making the following diagram

M 0,n P n-3 (P 1 ) n (b)/ /P GL(2) fn φ L ρ commutative.
Proof. As observed in [START_REF] Kumar | Linear Systems and Quotients of Projective Space[END_REF] via the theory of associated points, each point x ∈ P n-3 which is linearly general with respect to the n -1 fixed points in P n-3 defines a configuration of points on the unique rational normal curve of degree n-3 passing through the (n-1)+1 = n points. Moreover, this configuration is the image of x in (P 1 ) n (b)/ /P GL(2) via φ L . Via the identification V 0 (p 1 , . . . , p n ) ∼ = M 0,n of Theorem 1.8, one easily obtains the claim.

From the next section on, we will always omit the vector b of the polarization since it will always be either (1, . . . , 1) or (2, . . . , 2) when we consider an odd number of points, according to (1.0).

2. Birational geometry of GIT quotients of (P 1 ) n and the F-conjecture

In this section we will study some birational aspects of the geometry of the GIT quotients we introduced. In particular, we will describe their Mori cone and show that its extremal rays are generated by 1-dimensional strata of the boundary.

Let us now fix a suitable notation for the GIT quotients. We will denote by Σ m the GIT quotient (P 1 ) m+3 / /P GL(2).

2.1. The odd dimensional case. We start by studying the case where the GIT quotients parametrize an even number of points, that is Σ 2g-1 , for g ≥ 2. This reveals to be less complicated than the even dimensional one, that we will consider eventually. In any case it is worth looking at it. In fact it will allow us to prove another interesting result in Theorem 2.7 that, as far as we know, does not seem to have appeared in the literature. First of all, we need a preliminary result.

Let us define L 2g-1 := L 2g-1,g (g -1, ..., g -1) as the linear system of degree g forms on P 2g-1 vanishing with multiplicity g at 2g + 1 general points p 1 , ..., p 2g+1 ∈ P 2g-1 . In [Ku00, Theorem 4.1] Kumar proved that L 2g-1 induces a birational map (2.2) σ g : P 2g-1 P(H 0 (P 2g-1 , L 2g-1 ) * ) and that the GIT quotient Σ 2g-1 , that is the Segre g-variety, in Kumar's paper [START_REF] Kumar | Invariant vector bundles of rank 2 on hyperelliptic curves[END_REF], is obtained as the closure of the image of σ g in P(H 0 (P 2g-1 , L 2g-1 ) * ).

Proposition 2.3. Let p 1 , ..., p 2g+1 ∈ P 2g-1 be points in general position, and let X 2g-1 g-2 be the variety obtained the step g of Construction 1.10. Then we have the following commutative diagram

X 2g-1 g-2 P 2g-1 Σ 2g-1 ⊂ P N . f σg σg
That is, the blow-up morphism f : X 2g-1 g-2 → P 2g-1 resolves the rational map σ g .

Proof. By Construction 1.10 the blow-up X 2g-1 g-2 may be interpreted as the moduli space

M 0,A[2g+2] with A[2g + 2] = 1 g , ..., 1 g , 1 . Furthermore, by Remark 1.11 Σ 2g-1 is the singular moduli space M 0, A[2g+2] with weights A[2g + 2] = 1 g+1 , ..., 1 g+1 , 1
g+1 , and by Proposition 1.13 the morphism σ g :

X 2g-1 g-2 → Σ g is exactly the reduction morphism ρ A[2g+2],A[2g+2] : M 0,A[2g+2] → M 0, A[2g+2]
defined just by lowering the weights.

This fairly simple result has some interesting consequences, that we will illustrate in the rest of this section. Anyway, first we need a technical, and probably well-known, lemma.

Lemma 2.4. Let X be a normal projective variety and f : X Y a birational map of projective varieties not contracting any divisor. Then

K X ∼ f -1 * K Y . Proof. Since X is normal f is defined in codimension one. Let U ⊆ X be
a dense open subset whose complementary set has codimension at least two where f is defined. Since f does not contract any divisor then its exceptional set has at least codimension two, hence we may assume that f |U is an isomorphism onto its image. Therefore K X|U ∼ (f -1 * K Y ) |U , and since U is at least of codimension two we get the statement.

The first consequence of Proposition 2.3 is the following.

Lemma 2.5. The canonical sheaf of Σ 2g-1 ⊂ P N is K Σ 2g-1 ∼ = O Σ 2g-1 (-2).
Proof. Let X 2g-1 1 be the blow-up of P 2g-1 at 2g + 1 general points p 1 , ..., p 2g+1 , and let f g : X 2g-1 1 Σ 2g-1 be the birational map induced by the morphism σ g : X 2g-1 g-2 → Σ 2g-1 in Proposition 2.3. We will denote by H the pull-back in X 2g-1 1 of the hyperplane section of P 2g-1 , and by E 1 , ..., E 2g+1 the exceptional divisors.

The interpretation of σ g as a reduction morphism in the proof of Proposition 2.3 yields that f g does not contract any divisor. Indeed f g contracts just the strict transforms of the (g -1)-planes generated by g of the blown-up points. Since σ g is induced by the linear system L 2g-1 , the pull-back via f g of a hyperplane section of Σ 2g-1 is the strict transform of a hypersurface of degree g in P 2g-1 having multiplicity g -1 at p 1 , ..., p 2g+1 . Since

-K X 2g-1 1 ∼ 2gH -(2g -2) 2g+1 i=1 E i = 2 gH -(g -1) 2g+1 i=1 E i we have that -K X 2g-1 1 ∼ f * g O Σ 2g-1 (2)
. Now, in order to conclude it is enough to apply Lemma 2.4 to the birational map f g :

X 2g-1 1 Σ 2g-1 . Proposition 2.6. The divisor class group of Σ 2g-1 is Cl(Σ 2g-1 ) ∼ = Z 2g+2 . Furthermore Pic(Σ 2g-1 ) is torsion free.
Proof. The classical case of the Segre cubic Σ 3 ⊂ P 4 has been treated in [Hu96, Section 3.2.2]. Hence we may assume that g ≥ 2.

Let Y = H ∩Σ 2g-1 be a general hyperplane section of Σ 2g-1 . Since dim(Sing(Σ 2g-1 )) = 0 by Bertini's theorem, see [START_REF] Hartshorne | Algebraic geometry[END_REF]Corollary 10.9] and [Har77, Remark 10.9

.2], Y is smooth. Note that X = ( σ g ) -1 (Y ) ⊂ X 2g-1
g-2 is the strict transform via f of a general element of the linear system L 2g-1 inducing σ g . Therefore, X is smooth and

σ g|X : X → Y is a divisorial contraction between smooth varieties. Since dim(X) > 2, the Grothendieck-Lefschetz theorem (see for instance [Ba78, Theorem A]) yields that the natural restriction morphism Pic(X 2g-1 g-2 ) → Pic(X) is an isomorphism. Therefore Pic(X) ∼ = Pic(X 2g-1 g-2 ) ∼ = Z h , with h = 1 + (2g + 1) + 2g+1 2 + ... + 2g+1 g
. By the interpretation of σ g as a reduction morphism in Proposition 2.3 we know that the codimension one part of the exceptional locus of σ g consists of the exceptional divisors the blown-up positive dimensional linear subspaces of P 2g-1 . Therefore, Pic(Y ) ∼ = Z 2g+2 . Indeed Pic(Y ) is generated by the images via σ g|X of the pull-back of the hyperplane section of P 2g-1 and of the exceptional divisors over the 2g + 1 blown-up points. However, since Σ 2g-1 is not smooth we can not conclude by the Grothendieck-Lefschetz theorem that its Picard group is isomorphic to Z 2g+2 as well.

On the other hand, thanks to a version for normal varieties of the Grothendieck-Lefschetz theorem [RS06, Theorem 1], we get that Cl(Σ 2g-1 ) ∼ = Cl(Y ), and since Y is smooth we have Cl(Y ) ∼ = Pic(Y ).

Finally, by [Kl66, Corollary 2] we have that, even when the ambient variety is singular, the restriction morphism in the Grothendieck-Lefschetz theorem is injective. Therefore, we have an injective morphism Pic(Σ 2g-1 ) → Pic(Y ) ∼ = Z 2g+2 , and hence Pic(Σ 2g-1 ) is torsion free.

This allows us to compute the Picard group of Σ 2g-1 . By different methods, this computation was also carried out in [START_REF] Moon | On the Sn-invariant F -conjecture[END_REF].

Proposition 2.7. The Picard group of the GIT quotient

Σ 2g-1 ⊂ P N is Pic(Σ 2g-1 ) ∼ = Z H
, where H is the hyperplane class. In particular, we have that

Nef(Σ 2g-1 ) ∼ = R ≥0 . Proof. Let σ g : X 2g-1 g-2 → Σ 2g-1 ⊂ P N be the resolution of the birational map σ g : P 2g+1 Σ 2g-1 in Proposition 2.3. Note that Pic(X 2g-1 g-2 ) ∼ = Z ρ(X 2g-1 g-2 ) , where ρ(X 2g-1 g-2 ) = 1 + (2g + 1) + 2g+1 2 + ... + 2g+1
g-1 , and that σ g contracts all the exceptional divisors of the blow-up f : X 2g-1 g-2 → P 2g-1 over positive dimensional linear subspaces. Now, let E i ⊂ X 2g-1 0 be the exceptional divisor over the point p i ∈ P 2g-1 , and let E i its strict transform in X 2g-1 g-2 . Furthermore, denote by e i the class of a general line in E i ⊂ X 2g-1 0 . By Proposition 2.3 we know that, besides the divisors over the positive dimensional linear subspaces, σ g also contracts the strict transforms S 1 , ..., S r , with r = 2g+1 g , in X 2g-1 g-2 of the (g -1)-planes p i 1 , ..., p ig . Note that the contraction of S i is given by the contraction of the strict transforms of the degree g -1 rational normal curves in p i 1 , ..., p ig passing through p i 1 , ..., p ig . In fact, any degree g -1 rational normal curve through p i 1 , ..., p ig is contained in p i 1 , ..., p ig and a morphism contracts S i to a point if and only if it contracts all these curves to a point. We may write the class in the cone

N 1 (X 2g-1 g-2 ) R ∼ = R ρ(X 2g-1 g-2 )
of the strict transform of such a rational normal curve as

(2.8) (g -1)l -e i 1 -... -e ig
where l is the pull-back of a general line in P 2g-1 . Note that the classes in (2.8) generate the hyperplane

   gl + 2g+1 j=1 (g -1)e i = 0    in N 1 (X 2g-1 1 ) R ∼ = R 2g+2 , where X 2g-1 1
is the blow-up of P 2g-1 at p 1 , ..., p 2g+1 . Therefore, the birational morphism σ g : X 2g-1 g-2 → Σ 2g-1 contracts the locus spanned by classes of curves generating a subspace of dimension 2g+1 2 + ...

+ 2g+1 g-1 + 2g + 1 of N 1 (X 2g-1 g-2 ) R ∼ = R ρ(X 2g-1 g-2 )
, and then

ρ(Σ 2g-1 ) = ρ(X 2g-1 g-2 ) - 2g + 1 2 + ... + 2g + 1 g -1 + 2g + 1 = 1.
Since by Proposition 2.6 the Picard group of Σ 2g-1 is torsion free and Σ 2g-1 ⊂ P N contains lines we conclude that Pic(Σ 2g-1 ) = Z H , where H is the hyperplane class.

2.9. The even dimensional case. In this section we investigate the geometry of the counterpart of Σ 2g-1 , that is the GIT quotient (P 1 ) n (b)/ /P GL(2) with b i = 1 for i = 1, ..., n and n odd. The quotient here has even dimension 2g, for a positive integer g, and hence we will denote it by Σ 2g . Of course, we have n = 2g + 3. Note that in this case all the semistable points are indeed stable, and then Σ 2g is smooth. Recall that, by Theorem 1.2, Σ 2g ⊂ P(H 0 (P 2g , L 2g ) * ) is the closure of the image of the rational map induced by the linear system L 2g , where we define L 2g as given by degree 2g + 1 hypersurfaces in P 2g with multiplicity 2g -1 at p i for i = 1, ..., 2g + 2. We will denote by µ g :

P 2g Σ 2g ⊂ P(H 0 (P 2g , L 2g ) * ) = P N this rational map.
This case is far more complicated and interesting than the odd dimensional one, and we will need a good amount of preliminary results.

Proposition 2.10. Let X 2g g-1 be the variety obtained at the step g of Construction 1.10. Then there exists a morphism µ g : X 2g

g-1 → X(b)/ /P GL(2) making the following diagram

X 2g g-1 P 2g Σ 2g ⊂ P N f µg µg commute, where f : X 2g g-1 → P 2g is the blow-up morphism.
Proof. By Construction 1.10 X 2g

g-1

∼ = M 0,A[2g+3] with A[2g + 3] = 2 2g+2 , ..., 2 2g+2 , 1 , and by Remark 1.11 Σ 2g ∼ = M 0, A[2g+3] with A[2g +3] = 2 2g+3 , ..., 2
2g+3 . Therefore we may take

µ g = ρ A[2g+3],A[2g+3] : M 0,A[2g+3] → M 0, A[2g+3]
and argue as in the proof of Proposition 2.3.

Remark 2.11. Note that arguing as in the proof of Proposition 4.3 we see that the standard Cremona transformation of P 2g induces an automorphism of Σ 2g ⊂ P N . Indeed the automorphism induced by the Cremona and the group S 2g+2 permuting the points p 1 , ..., p 2g+2 ∈ P 2g generates the symmetric group S 2g+3 acting on Σ 2g by permuting the marked points.

2.12. Linear subspaces of dimension g in Σ 2g ⊂ P N . Let H I = H i 1 ,...,i g+1 be the g-plane generated by p i 1 , ..., p i g+1 ∈ P 2g . The linear system L 2g|H I is given by the hypersurfaces of degree 2g + 1 in H I ∼ = P g with multiplicity 2g -1 at p i 1 , ..., p i 2g+2 . Now, let H J ⊂ H I be a (g -1)-plane generated by the points indexed by a subset J ⊂ I with |J| = g. Then the general element of L 2g|H I must contain H J with multiplicity g(2g -1) -(g -1)(2g + 1) = 1. This means that the divisor D equals {J⊂I, | |J|=g} H J . Note that deg(D) = g + 1 and mult p i j D = g for any i j ∈ I. Therefore, L 2g|H I is the linear system of hypersurfaces of degree 2g + 1 -(g + 1) = g in H I ∼ = P g having multiplicity 2g -1 -g = g -1 at p i j for any i j ∈ I. This is the linear system of the standard Cremona transformation of P g . Therefore, µ g|H I (H I ) ⊂ Σ 2g ⊂ P N is a linear subspace of dimension g. Now, let E g-1 I be the exceptional divisor over the strict transform of a (g -1)-plane of P 2g generated by g of the p i 's. Note that the reduction morphism µ g : 's respectively. Note that: -on any δ i we have g+2 distinguished points determined by the intersections with the g -1 of the γ j 's coming from g-planes in P 2g containing the (g -1)-plane associated to δ i , -on any γ i , say coming from H I , we have g + 2 distinguished points as well: g + 1 of them coming from the exceptional divisors E g-1 J with J ⊂ I, |J| = g, and another one determined as the image of the point H I ∩ H I c , where I c = {1, ..., 2g + 2} \ I. Note that the g + 2 distinguished points on γ i and δ j are in linear general position. This is clear for the γ i 's. In order to see that it is true for the δ j 's as well, notice that way map any δ j to any γ i just by acting with a suitable permutation in S 2g+3 involving the standard Cremona transformation as in Remark 2.11.

X 2g g-1 ∼ = M 0,A[2g+3] → Σ 2g ∼ = M 0, A[2g+3] in Proposition 2.10 contracts E g-1 I to a g-plane µ g (E g-1 I ) ⊂ Σ 2g ⊂ P N .
Finally note that on any γ i we have g + 2 distinguished points and one of them is the intersection point of γ i with a γ j . We call such a γ j the complementary of γ i , and we denote it by γ j = γ i c . Therefore γ i and γ i c determine 2(g + 2) -1 distinguished points for any i = 1, ..., 1 2 2g+2 g+1 . Summing up we have

(2.14) 1 2 2g + 2 g + 1 (2(g + 2) -1) = (2g + 3)! 2((g + 1)!) 2
distinguished points in the configuration of g-planes C ∪ D.

Remark 2.15. Arguing as in 4.9 and using the description of the sections of L 2g in [Ku03, Section 3.3] we get that in Σ 2g ⊂ P(H 0 (P 2g , L 2g ) * ) = P N there are N +2 of the distinguished points described in 2.13 that are in linear general position in P N .

Let us consider the following diagram:

X 2g g-1 X 2g 1 Σ 2g µg h ψ where h : X 2g g-1 → X 2g
1 is the composition of blow-ups in Construction 1.10. Note that, by interpreting the varieties appearing in the diagram as moduli spaces of weighted pointed curves, and h and µ g as reduction morphisms as in the proof of Proposition 2.10, we see that the rational map ψ : X 2g 1 Σ 2g is a composition of flips of strict transforms of linear subspaces generated by subsets of {p 1 , ..., p 2g+2 } up to dimension g-1. In particular, ψ is an isomorphism in codimension one and Pic(Σ 2g ) ∼ = Z 2g+3 is the free abelian group generated by the strict transforms via ψ of H, E 1 , ..., E 2g+2 . The Picard groups was also computed in [START_REF] Moon | On the Sn-invariant F -conjecture[END_REF]. Our computation has the advantage of showing explicit generators.

As Han-Bom Moon has observed in a personal communication, Σ 2g-1 is not Q-factorial, and many boundary divisors are just Weil divisors. This is basically the reason why the rank of Pic(Σ 2g-1 ) drops so dramatically from the rank of Pic(Σ 2g ).

Lemma 2.17. The effective cone Eff(X 2g 1 ) ⊂ R 2g+3 of X 2g 1 is the polyhedral cone generated by the classes of the exceptional divisors E i and of the strict transforms H -i∈I E i , I ⊂ {1, ..., 2g + 2}, with |I| = 2g, of the hyperplanes generated by 2g of the p i 's.

Proof. The faces of Eff(X 2g 1 ) are described in [CT06, Lemma 4.24], and in [BDP16, Corollary 2.5]. It is straightforward to compute the extremal rays of Eff(X 2g 1 ) by intersecting its faces.

Lemma 2.18. The canonical sheaf of Σ 2g ⊂ P N is isomorphic to O Σ 2g (-1).

Proof. Let X 2g

1 be the blow-up of P 2g at 2g + 2 general points p 1 , ..., p 2g+2 , and let h g : X 2g 1 Σ 2g be the birational map induced by the morphism µ g : X 2g

g-1 → Σ 2g in Proposition 2.10. As usual we will denote by H the pull-back to X 2g 1 of the hyperplane section of P 2g , and by E 1 , ..., E 2g+2 the exceptional divisors. In order to conclude, it is enough to note that

-K X 2g 1 ∼ (2g + 1)H -(2g -1) 2g+2 i=1 E i
is an element of the linear system L 2g inducing µ g : P 2g

Σ 2g , and to argue as in the proof of Lemma 2.5.

As we have seen in 2.16,t Pic(Σ 2g ) ∼ = Z 2g+3 . In the next section our aim will be to describe the cones of curves and divisors of Σ 2g .

2.19. Mori Dream Spaces and chamber decomposition. Let X be a normal projective variety. We denote by N 1 (X) the real vector space of R-Cartier divisors modulo numerical equivalence. The nef cone of X is the closed convex cone Nef(X) ⊂ N 1 (X) generated by classes of nef divisors. The movable cone of X is the convex cone Mov(X) ⊂ N 1 (X) generated by classes of movable divisors. These are Cartier divisors whose stable base locus has codimension at least two in X. The effective cone of X is the convex cone Eff(X) ⊂ N 1 (X) generated by classes of effective divisors. We have inclusions:

Nef(X) ⊂ Mov(X) ⊂ Eff(X).
We say that a birational map f : X X into a normal projective variety X is a birational contraction if its inverse does not contract any divisor. We say that it is a small Q-factorial modification if X is Q-factorial and f is an isomorphism in codimension one. If f : X X is a small Q-factorial modification, then the natural pullback map f * : N 1 (X ) → N 1 (X) sends Mov(X ) and Eff(X ) isomorphically onto Mov(X) and Eff(X), respectively. In particular, we have f * (Nef(X )) ⊂ Mov(X).

Definition 2.20. A normal projective Q-factorial variety X is called a Mori dream space if the following conditions hold:

-Pic (X) is finitely generated, -Nef (X) is generated by the classes of finitely many semi-ample divisors, -there is a finite collection of small Q-factorial modifications f i : X X i , such that each X i satisfies the second condition above, and

Mov (X) = i f * i (Nef (X i )).
The collection of all faces of all cones f * i (Nef (X i )) above forms a fan which is supported on Mov(X). If two maximal cones of this fan, say f * i (Nef (X i )) and f * j (Nef (X j )), meet along a facet, then there exists a commutative diagram:

X i X j Y h i h j ϕ
where Y is a normal projective variety, ϕ is a small modification, and h i and h j are small birational morphisms of relative Picard number one. The fan structure on Mov(X) can be extended to a fan supported on Eff(X) as follows.

Definition 2.21. Let X be a Mori dream space. We describe a fan structure on the effective cone Eff(X), called the Mori chamber decomposition. We refer to [HK00, Proposition 1.11] and [Ok16, Section 2.2] for details. There are finitely many birational contractions from X to Mori dream spaces, denoted by

g i : X → Y i . The set Exc(g i ) of exceptional prime divisors of g i has cardinality ρ(X/Y i ) = ρ(X) -ρ(Y i ).
The maximal cones C of the Mori chamber decomposition of Eff(X) are of the form:

C i = Cone g * i Nef(Y i ) , Exc(g i ) .
We call C i or its interior C

• i a maximal chamber of Eff(X). Let m > 1 be an integer. Let X m m+2 be the blow-up of

P m at m + 2 general points. It is well-known that X m m+2 is a Mori Dream Space [CT06, Theorem 1.3], [AM16, Theorem 1.3].
In what follows we will describe the cones of divisors of X m m+2 , as well as its Mori chamber decomposition. Once this will be done, we will concentrate on the case where m = 2g in order to use these results to compute the cone of curves of Σ 2g . We will consider the standard bases of Pic(X m m+2 ) R ∼ = R m+3 given by the pull-back H of the hyperplane section and the exceptional divisors E 1 , ..., E m+2 .

Theorem 2.22. Let X m m+2 be the blow-up of P m at m + 2, and write the class of a general divisor

D ∈ Pic(X m m+2 ) R as D = yH + m+2 i=1 x i E i . Then -The effective cone Eff(X m m+2
) is defined by

y + x i ≥ 0 i ∈ {1, ..., m + 2}, my + m+2 i=1 x i ≥ 0. -The Mori chamber decomposition of Eff(X m m+2
) is defined by the hyperplane arrangement

(2.23) (2 -k)y -i∈I x i = 0 if I ⊆ {1, ..., m + 2}, |I| = k -1, (m -k + 1)y -i∈I x i + m+2 i=1 x i = 0 if I ⊆ {1, ..., m + 2}, |I| = k, with 2 ≤ k ≤ m+3 2 . -The movable cone Mov(X m m+2 ) is given by        (m -1)y -i∈I x i + m+2 i=1 x i ≥ 0 if I ⊆ {1, ..., m + 2}, |I| = 2, x i ≤ 0 i ∈ {1, ..., m + 2}, y + x i ≥ 0 i ∈ {1, ..., m + 2}, my + m+2 i=1 x i ≥ 0.
-All small Q-factorial modifications of X are smooth. Let C and C be two adjacent chambers of Mov(X), corresponding to small Q-factorial modifications of X, f : X X and f : X X , respectively. These chambers are separated by a hyperplane

H I in (2.23), with 3 ≤ k ≤ m+3 2 and |I| ∈ {k -1, k}. Assume that ϕ(C) ⊂ (H I ≤ k) and ϕ(C ) ⊂ (H I ≥ k). Then the birational map f •f -1 : X X flips a P k-2 into a P m+1-k .
-Let C be a chamber of Mov(X), corresponding to small Q-factorial modification X of X. Let σ ⊂ ∂C be a wall such that σ ⊂ ∂ Mov(X), and let f : X → Y be the corresponding elementary contraction. Then either σ is supported on a hyperplane of the form (y + x i = 0) or (my -m+2 i=1 x i = 0) and f : X → Y is a P 1 -bundle, or σ is supported on a hyperplane of the form (x i = 0) or (m-1)y-i∈I x i + m+2 i=1 x i = 0, I ⊆ {1, ..., m + 2}, |I| = 2, and f : X → Y is the blow-up of a smooth point, and the exceptional divisor of f is the image in X of either an exceptional divisor E i or a divisor of the form H -i∈I E i with I ⊂ {1, ..., m + 2}, |I| = m.

Proof. The statement about the effective cone is in Lemma 2.17. By [Ok16, Theorem 1.2] the inequalities for the movable cones and for the Mori chamber decomposition follow by removing from the analogous systems of inequalities for the blow-up of P m in m + 3 points in [AM16, Theorem 1.3] the inequalities involving the exceptional divisor E m+3 . Similarly, by [Ok16, Theorem 1.2] the last two claims follow easily from the last two items in [AM16, Theorem 1.3].

Remark 2.24. Theorem 2.22 allows us to find explicit inequalities defining the cones Eff(X m m+2 ), Mov(X m m+2 ), and Nef( X), for any small Q-factorial modification X of X m m+2 . For instance, Nef(X m m+2 ) is given by

x i ≤ 0 i ∈ {1, ..., m + 2}, y + x i + x j ≥ 0 i, j ∈ {1, ..., m + 2}, i = j.
2.25. The Fano model of X m m+2 . First, let us consider the case when m = 2g is even. Then the anti-canonical divisor

-K X 2g 2g+2 ∼ (2g + 1)H -(2g -1) 2g+2 i=1 E i
lies in the interior of the chamber C F ano defined by (2.26)

(1 -g)y -i∈I x i = 0 if I ⊆ {1, ..., m + 2}, |I| = g, gy -i∈I x i + m+2 i=1 x i = 0 if I ⊆ {1, ..., m + 2}, |I| = g + 1, that is the chamber in (2.23) of Theorem 2.22 for k = g + 1. By Theorem 2.22 the chamber C F ano corresponds to a smooth small Q-factorial modification X 2g F ano of X 2g 2g+2 whose nef cone is given by (2.26). By Lemma 2.18 we get that (2.27)

X 2g F ano ∼ = Σ 2g . If m = 2g -1 is odd the class -K X 2g-1 2g+1 ∼ 2gH -(2g -2) 2g+1 i=1 E i
lies in the intersection of the hyperplanes

(2.28) (1 -g)y -i∈I x i = 0 I ⊆ {1, ..., 2g + 1}, |I| = g, (g -1)y -i∈I x i + 2g+1 i=1 x i = 0 I ⊆ {1, ..., 2g + 1}, |I| = g + 1.
The variety corresponding to the intersection (2.28) is a small non Q-factorial modification of X 2g-1 2g+1 , and by Lemma 2.5 we can identify this variety with the GIT quotient Σ 2g-1 .

Corollary 2.29. The Mori cone NE(X 2g F ano ) of X 2g F ano has 2g+2 g + 2g+2 g+1 = 2g+3 g+1 extremal rays.

Proof. Since X 2g

F ano is a Mori Dream Space, NE(X 2g F ano ) is polyhedral and finitely generated. Furthermore, NE(X 2g F ano ) is dual to Nef(X 2g F ano ). Therefore, the number of extremal rays of NE(X 2g F ano ) is equal to the number of faces of Nef(X 2g F ano ). Now, the statement follows from (2.26).

Fibrations on X 2g

F ano . Let us stick to the situation where m = 2g is the dimension of the GIT quotient. By the isomorphism (2.27) we may identify the Fano model X 2g with B[2g + 2] = 2 2g+2 , ..., 2 2g+2 . Therefore, for any i = 1, ..., 2g + 3 we have a forgetful morphism

π i : M 0, B[2g+3] ∼ = X 2g F ano → M 0, B[2g+2] ∼ = Σ 2g-1 .
Now, consider the g-planes in X 2g F ano described in (2.13). Note that in C ∪ D we have

(2.31) 2g + 2 g + 1 + 2g + 2 g = 2g + 3 g + 1 g-planes.
From the modular point of view these g-planes parametrize configurations of 2g + 3 points (P 1 , x 1 , ..., x 2g+3 ) in P 1 with g + 1 points coinciding. Let us fix a marked point, say x 2g+3 . For any choice of g + 1 points in (x 1 , ..., x 2g+2 ) we get a g-plane H in C ∪ D and its complementary g-plane H c . For instance, if H is given by (x 1 = ... = x g+1 ) then H c is defined by (x g+2 = ... = x 2g+2 ). Note that H and H c intersects a point representing the configuration (x 1 = ... = x g+1 , x g+2 = ... = x 2g+2 , x 2g+3 ), and that the morphism

π 2g : M 0, B[2g+3] ∼ = X 2g F ano → M 0, B[2g+2] ∼ = Σ 2g-1 contracts H ∪ H c to the singular point of Σ 2g-1 representing the configuration (x 1 = ... = x g+1 , x g+2 , ..., x 2g+2 ) = (x 1 , ..., x g+1 , x g+2 = ... = x 2g+2 , x 2g+3 ).
Thanks to the modular interpretation of π 2g+3 we see that π -1 2g+3 (p) ∼ = P 1 for any p ∈ Σ 2g-1 \ Sing(Σ 2g-1 ), while π -1 2g+3 (p j ) is the union of two g-planes in C ∪ D intersecting in one point for any p j ∈ Sing(Σ 2g-1 ). Indeed

1 2 2g + 2 g + 1 = 2g + 1 g
is exactly the number of singular points of Σ 2g-1 . More generally for any i = 1, ..., 2g + 3 we may consider the set {x 1 , ..., x i-1 , x i+1 , ..., x 2g+3 }, and for any subset I of cardinality g + 1 of {x 1 , ..., x i-1 , x i+1 , ..., x 2g+3 } we have a g-plane H I defined by requiring the points marked by I to coincide, and the complementary g-plane H I c defined as the locus parametrizing configurations where the marked points in I c coincide. Then the morphism

π i : M 0, B[2g+3] ∼ = X 2g F ano → M 0, B[2g+2] ∼ = Σ 2g-1 contracts the unions H ∪ H c to the singular points of Σ 2g-1 .
For any g-plane H I in C ∪ D we will denote by L I the class of a line in H I . Note that L I is the class of 1-dimensional boundary stratum of Σ 2g corresponding to configurations where the points in I coincide, other g points coincide as well, and the remaining two points are different.

Proposition 2.32. The classes L I , L I c of lines in H I , H I c ∼ = P g described above generate the extremal rays of NE(X 2g F ano ). Proof. From the above discussion we have that for any L I there is a complementary class L I c , and a forgetful morphism

π i : M 0, B[2g+3] ∼ = X 2g F ano → M 0, B[2g+2] ∼ = Σ 2g-1 , such that
the fibers of π i over Σ 2g-1 \ Sing(Σ 2g-1 ) are isomorphic to P 1 , and H I ∪ H I c is contracted by π i onto a singular point of Σ 2g-1 . Our aim is to compute the relative Mori cone NE(π i ) of the morphism π i . Let C ⊂ X 2g F ano be a curve contracted by π i . Then C is either a fiber of π i over a smooth point of Σ 2g-1 or a curve in H I ∪ H I c . In the latter case the class of C may be written as a combination with non-negative coefficients of the classes L I and L I c . Now, for simplicity of notation assume that i = 2g + 3, and consider the surface S ⊆ X 2g

F ano parametrizing configurations of points of the type (x 1 = ... = x g , x g+1 , x g+2 = ... = x 2g+1 , x 2g+2 , x 2g+3 ). Then the image of π 2g+3|S : S → Σ 2g-1 is the line Γ passing through a singular point p ∈ Σ 2g-1 parametrizing the configurations (x 1 = ... = x g , x g+1 , x g+2 = ... = x 2g+1 , x 2g+2 , x 2g+3 ). Furthermore, π -1 2g+3|S (q) ∼ = P 1 for any q ∈ Γ \ {p}, and π -1 2g+3|S (q) = L I ∪ L I c . Therefore, the general fiber of π 2g+3 is numerically equivalent to L I + L I c . This means that NE(π i ) is generated by the classes L I , L I c for {x 1 , ..., x i-1 , x i+1 , ..., x 2g+3 } with |I| = g+1. Finally, since by [De01, Proposition 1.14] the subcone NE(π i ) of NE(X 2g F ano ) is extremal, we get that L I , L I c generate extremal rays of NE(X 2g F ano ). We are now in the position to claim and prove the F-conjecture for Σ 2g .

Theorem 2.33. The Mori cone of the GIT quotient Σ 2g is generated by the classes L I , L I c of 1-dimensional boundary strata.

Proof. By Proposition 2.32 the classes L I , L I c generate extremal rays of NE(Σ 2g ). By (2.31) these are 2g+3 g+1 rays which by Corollary 2.29 is exactly the number of extremal rays of NE(Σ 2g ).

Remark 2.34. Consider the Mori chamber decomposition in Theorem 2.22. Note that in order to go from the chamber corresponding to Nef(X 2g 2g+2 ) to the chamber corresponding to Nef(X 2g

F ano ) we must cross the walls in (2.23) for any 3 ≤ k ≤ 2g + 1. Indeed by the modular description of the small modification ψ :

X 2g 2g+2 X 2g
F ano we see that it factors as:

X 0 = X 2g 2g+2 ψ 1 X 1 ψ 2 X 2 ψ 3 ... ψ g-1 X g-1 = X 2g F ano
where ψ i : X i-1 X i is the flip of the strict transform in X i-1 of the i-planes in P 2g generated by i + 1 among the blown-up points. These strict transforms are disjoint i-planes in X i-1 , while the flipped locus in X i is a disjoint union of (2g -1 -i)-planes. In particular the 2g+2 g+1

+ 2g+2 g = 2g+3 g+1
extremal rays in (2.31) correspond to the 2g+2 g+1 g-planes coming as strict transforms of the g-planes in P m generated by g + 1 of the marked points, plus the g-planes that are the flipped loci of the (g -1)-planes in P n generated by g of the marked points.

2.35. The cone of moving curves of Σ 2g . In this section we describe extremal rays of the cone of moving curves Mov 1 (Σ 2g ) of Σ 2g . Recall that an irreducible curve C on a projective variety X is called a moving curve if C is a member of an algebraic family of curves covering a dense subset of X. By [BDPP13, Theorems 2.2 and 2.4] the cone of moving curves is dual to the cone of pseudoeffective divisor classes which is spanned by classes that appear as limits of sequences of effective Q-divisors.

By Theorem 2.22 we have that Eff(Σ 2g ) is closed and therefore Mov 1 (Σ 2g ) is the dual cone of Eff(Σ 2g ). In particular, by the description of Eff(X 2g 2g+2 ) in Theorem 2.22 we get that Mov 1 (Σ 2g ) has exactly 2g + 3 extremal rays.

Theorem 2.36. The cone of moving curves Mov 1 (Σ 2g ) of Σ 2g is generated by the the classes of the fiber of the forgetful morphisms π i : Σ 2g → Σ 2g-1 for i = 1, ..., 2g + 3.

Proof. By the description of the faces of Eff(X 2g 2g+2 ) in Theorem 2.22 we see that the 2g + 3 extremal rays of Mov 1 (X 2g 2g+2 ) = Eff(X 2g 2g+2 ) ∨ are generated by the class of the strict transform L i of a line through the blown-up point p i ∈ P 2g for i = 1, ..., 2g + 2, and by the class of the strict transform of a degree 2g rational normal curve C through p 1 , ..., p 2g+2 . Now, consider the following diagram where ψ : X 2g 2g+2 Σ 2g is the sequence of flips in Remark 2.34, ρ : M 0,2g+3 → Σ 2g is the reduction morphism, f : M 0,2g+3 → X 2g 2g+2 is the Kapranov's blow-up morphism in Construction 1.10, π i : Σ 2g → Σ 2g-1 is a forgetful morphism, and π i : M 0,2g+3 → M 0,2g+2 is the corresponding forgetful morphism on M 0,2g+3 . Now, by [START_REF] Kapranov | Veronese curves and Grothendieck-Knudsen moduli spaces M0,n[END_REF] the fibers of the forgetful morphism π i : M 0,2g+3 → M 0,2g+2 are either the strict transforms of the lines through a point p i or the strict transforms of the degree 2g rational normal curves through p 1 , ..., p 2g+2 . Now, note that ρ, and hence ψ, map a general fiber of π i onto a general fiber of π i .

Finally, to conclude it is enough to observe that since the L i 's and C generate extremal rays of Mov 1 (X 2g 2g+2 ) and ψ : X 2g 2g+2 Σ 2g is a sequence of flips of small elementary contractions [Bar08, Proposition 3.14] yields that the fibers of the forgetful morphisms π i : Σ 2g → Σ 2g-1 generates extremal rays of Mov 1 (Σ 2g ).

2.37. Analogy with the geometric invariant theory of moduli of weighted pointed curves. Finally, we would like to stress another link between the GIT quotients we are studying and moduli of weighted curves. Consider the map:

(2.38) φ : Eff(X 2g 2g+2 ) ⊂ Pic(X 2g 2g+2 ) Q -→ Q 2g+3 (y, x 1 , ..., x 2g+2 ) -→ (a 1 , ..., a 2g+3 ),
where

a j = y + x j (2g + 1)y + 2g+2 i=1 x i , for j = 1, ..., 2g + 2,
and

a 2g+3 = 2 - 2g+2 i=1 a i .
Note that φ maps Eff(X 2g 2g+2 ) onto the hypercube [0, 1] 2g+3 ⊂ R 2g+3 . Furthermore, via φ the walls of the Mori chamber decomposition in (2.23) translate into i∈I a i = 1 for I ⊂ {1, ..., 2g + 3}, with |I| ∈ {k -1, k} and 2 ≤ k ≤ 2g+3 2 . These walls are exactly the ones used in [START_REF] Hassett | Moduli spaces of weighted pointed stable curves[END_REF]Section 8] to describe variations of some GIT quotients of products of P 1 in terms of moduli of weighted curves. Therefore, interpreting the rational numbers (a 1 , ..., a 2g+3 ) as the weights of pointed curves, we have that the map φ in (2.38) translates the Mori chamber decomposition of Theorem 2.22 into the GIT chamber decomposition from [Has03, Section 8].

For instance, φ(-K X 3 5 ) = φ(4, -2, ..., -2) = 1 3 , ..., 1 3 and by Remark 1.11 these are the weights of the Segre cubic threefold. Similarly, taking D = 3H -E 1 -... -E 5 , which is ample on X 3 5 , we have that φ(D) = φ(3, -1, ..., -1) = 2 7 , ..., 2 7 , 4 7 , and by Construction 1.10 the moduli space with these weights is isomorphic to X 3 5 itself.

Degrees of projections of Veronese varieties

In this section, we use part of the preliminary results developed in the preceding ones to describe some projective geometry of the GIT quotients. In particular we make massive use once again of the linear systems on the projective space of Theorem 1.2, in order to compute the Hilbert polynomial and the degree of Σ 2g-1 and Σ 2g in their natural embeddings in the spaces of invariants. A formula for the degree of the GIT quotients already appeared in [START_REF] Hausen | On blowing up the weighted projective plane[END_REF]. Our formula has the advantage of not being recursive, as it is that in [START_REF] Hausen | On blowing up the weighted projective plane[END_REF]. Similar methods allow us to show that M 0,6 is a weak Fano variety.

For any integer 0 ≤ r ≤ s -1 and for any multi-index I(r) = {i 1 , ..., i r+1 } ⊆ {1, ..., s} set

(3.1) k I(r) = max{m i 1 + ... + m i r+1 -rd, 0}.
In [BDP15, Definition 3.2] the authors define the linear virtual dimension of the linear system L n,d (m 1 , ..., m s ) on P n as the number

(3.2) n + d d + s-1 r=0 I[r]⊆{1,...,s} (-1) r+1 n + k I[r] -r -1 n
Furthermore, they define the linear expected dimension of L n,d (m 1 , ..., m s ), denoted by ldim(L), as follows: if the linear system L n,d (m 1 , ..., m s ) is contained in a linear system whose linear virtual dimension is negative then we set ldim(L) = -1; otherwise we define ldim(L) to be the maximum between the linear virtual dimension of L n,d (m 1 , ..., m s ) and -1.

Proposition 3.3. Let φ L : X P(H 0 (P n , L) * ) be the rational map induced by the linear system L := L n,d (m 1 , ..., m s ). Assume that φ L is birational, and let

X L = φ L (P n ). If s ≤ n + 2 then the Hilbert polynomial of X L is given by h X L (t) = dt + n n + s-1 r=0 I[r]⊆{1,...,s} (-1) r+1 n + tk I[r] -r -1 n .
In particular

deg(X L ) = d n + s-1 r=0 I[r]⊆{1,...,s} (-1) r+1 k n I[r] .
Proof. Polynomials of degree t ∈ N on P(H 0 (P n , L) * ) correspond to degree td polynomials on P n vanishing with multiplicity m i at p i . Therefore, the Hilbert polynomial of X L is given by h X L (t) = h 0 (P n , tL). Now, since tL is effective for any t ≥ 0, and s ≤ n + 1 [BDP15, Theorem 4.6] yields that

h 0 (P n , tL) = ldim(tL)
where ldim(tL) is the linear expected dimension of tL defined by (3.2). To get the formula for the Hilbert polynomial it is enough to observe that, in the notation of (3.1), for the linear system tL g we have

k I[r] (tL) = max{t(m i 1 + ... + m i r+1 -rd), 0} = tk I[r] (L).
Finally, note that h X L (t) may be written as

h X L (t) = d n + s-1 r=0 I[r]⊆{1,...,s} k n I[r] n! t n + P (t)
where P (t) is a polynomial in t of degree deg(P ) ≤ n -1. Therefore, the volume of the big linear system L is given by

Vol(L) = lim sup t →+∞ h X L (t) t n /n! = d n + s-1 r=0 I[r]⊆{1,...,s} k n I[r]
and Vol(L) is exactly the degree X L ⊆ P(H 0 (P n , L) * ).

Recall from Section 2.1 the we define L 2g-1 = L 2g-1,g (g -1, ..., g -1) as the linear system of degree g forms on P 2g-1 vanishing with multiplicity g at 2g + 1 general points p 1 , ..., p 2g+1 ∈ P 2g-1 and that it induces a birational map σ g : P 2g-1 P(H 0 (P 2g-1 , L 2g-1 ) * ). The GIT quotient Σ 2g-1 is the closure of the image of σ g in P(H 0 (P 2g-1 , L 2g-1 ) * ).

Corollary 3.4. The Hilbert polynomial of Σ 2g-1 ⊆ P(H 0 (P 2g-1 , L 2g-1 ) * ) is given by

h Σ 2g-1 (t) = gt + 2g -1 2g -1 + g-2 r=0 (-1) r+1 2g + 1 r + 1 t(g -r -1) + 2g -1 -r -1 2g -1 .
In particular

h 0 (P 2g-1 , L 2g-1 ) = 3g -1 2g -1 + g-2 r=0 (-1) r+1 2g + 1 r + 1 3g -2r -3 2g -1 and deg(Σ 2g-1 ) = g 2g-1 + g-2 r=0 (-1) r+1 2g + 1 r + 1 (g -r -1) 2g-1 .
Proof. By Theorem 1.2 Σ 2g-1 = σ g (P 2g-1 ), where σ g is the map in (2.2). Then polynomials of degree t ∈ N on P(H 0 (P 2g-1 , L 2g-1 ) * ) correspond to degree tg polynomials on P 2g-1 vanishing with multiplicity t(g -1) at p 1 , ..., p 2g+1 . Note that in the notation of (3.1) for the linear system tL 2g-1 we have

k I[r] = t(g -r -1) if r ≤ g -2, 0 if r ≥ g -1.
Furthermore, note that in (3.2) we have

n + k I[r] -r -1 n = 2h -1 + t(g -r -1) -r -1 2g -1 = 0
for t 0 if and only if r ≤ g -2. Now, in order to conclude it is enough to observe that for any r = 0, ..., g -2 we have 2g+1 r+1 subsets of {1, ..., 2g+1} of the form I[r]. Then the formulas for the Hilbert polynomial and the degree of Σ 2g-1 follow from Proposition 3.3. In particular, the dimension of the linear system L 2g-1 is then given by h 0 (P 2g-1 , L 2g-1 ) = h Σ 2g-1 (1).

Corollary 3.5. Let us consider the GIT quotient Σ 2g and let n = 2g + 3 be the number of points on P 1 that it parametrizes.The Hilbert polynomial of Σ 2g is given by

h Σ 2g (t) = (2g + 1)t + 2g 2g + 2g-1 2 r=0 (-1) r+1 2g + 2 r + 1 t(2g -2r -1) + 2g -r -1 2g .
In particular

deg(Σ 2g ) = (2g + 1) 2g + 2g-1 2 r=0 (-1) r+1 2g + 2 r + 1 (2g -2r -1) 2g .
Proof. By Theorem 1.2 Σ 2g ⊂ P(H 0 (P 2g , L 2g ) * ) is the closure of the image of the rational map induced by the linear system L 2g of degree 2g +1 hypersurfaces in P 2g with multiplicity 2g -1 at p i for i = 1, ..., 2g + 2. Now, to conclude it is enough to observe that for the linear system tL 2g we have

k I[r] = t(2g -1 -2r) if r ≤ 2g-1 2 , 0 if r > 2g-1 2 .
and to argue as in the proof of Corollary 3.4.

3.6. M 0,6 is weak Fano. In this section we prove that M 0,6 is weak Fano that is -K M 0,6 is nef and big.

Construction 3.7. For any effective divisor D in L n,d (m 1 , ..., m s ), we denote by D h the strict transform of D in the space X n h obtained as the blow-up of of P n along the linear base locus of D up to dimension h, with h ≤ n -1. That is:

-X n 0 is the blow-up P n at the points p 1 , ..., p s ; -X n 1 is the blow-up of X n 0 along the strict transforms of the lines p i 1 , p i 2 ; . . . -X n h is the blow-up of X n h-1 along the strict transforms of the h-planes p i 1 , ..., p i h+1 . Note that the number k I[r] defined in (3.1) is the multiplicity of a general element of L n,d (m 1 , ..., m s ) along an r-plane p i 1 , ..., p i r+1 . Therefore, the strict transform D h of D in X n h may be written as:

D h = dH - h r=1 I[r]⊆{1,...,s} k I[r] E I[r]
where H is the pull-back of the hyperplane section of P n , and E I[r] , with I[r] = {i 1 , ..., i r+1 }, is the exceptional divisor over the r-plane p i 1 , ..., p i r+1 . Finally, let D := D n-1 .

Proposition 3.8. Let us denote by φ L : P n P N the birational map induced by

L := L n,d (m 1 , ..., m s ) with image X L = φ L (P n ). If D is nef then (3.9) deg(X L ) = d n + s-1 r=0 I[r]⊆{1,...,s} (-1) r+1 k n I[r]
In particular, if m 1 = ... = m s = m and r is the maximal dimension of a linear subspace contained in the base locus of L then

(3.10) deg(X L ) = d n + r r=0 (-1) r+1 s r + 1 ((r + 1)m -rd) n
Proof. Polynomials of degree t ∈ N on P N correspond to degree td polynomials on P n vanishing with multiplicity tm i at p i for i = 1, ..., s. Therefore, the Hilbert polynomial of X L is given by h X L (t) = h 0 (P n , tL). By [DP14, Theorem 1.5] we have that

h 0 (P n , tL) = ldim(tD) + n i=1 (-1) i h i (X n-1 , t D)
Now, since by hypothesis D is nef, the asymptotic Riemann-Roch theorem [La04, Theorem 1.4.40] yields that h i (X n-1 , t D) = O(t n-i ) Now, we may compute the degree of X L as the volume of the big linear system L: Finally, note that if m 1 = ... = m s = m and r is the maximal dimension of a linear subspace contained in the base locus of L then

Vol(L) = lim sup t →+∞ ldim(tD) + n i=1 O(t n-i ) t n /n! =
k I[r] = (r + 1)m -rd if r ≤ r, 0 if r > r.
Now, to get formula (3.10) in the statement, it is enough to plug these values of k I[r] in formula (3.9), and to notice that we have exactly s r+1 r-planes of type p i 1 , ..., p i r+1 in the base locus of L.

In Proposition 3.8, when L does not have fixed components and D is base-point-freeso in particular D is nef -deg(X L ) may be also computed as the top self-intersection of D. In the rest of this section we will work out the case n = 3, and while doing this we will get a simple and direct argument proving that M 0,6 is a weak Fano variety, that is the anti-canonical divisor -K M 0,6 is nef and big.

Let us recall Construction 1.10 for M 0,6 : let p 1 , ..., p 5 ∈ P 3 be points in linear general position, and consider -π 1 : X → P 3 the blow-up of p 1 , ..., p 5 , -π 2 : Y → X the blow-up of the strict transforms of the lines p i , p j , i, j = 1, ..., 5, Then Y ∼ = M 0,6 , and the morphism f 6 = π 1 • π 2 : M 0,6 → P 3 is induced by the psi-class Ψ 6 on M 0,6 . By [KMc96, Theorem 1.2] the Mori Cone NE(M 0,6 ) of M 0,6 is generated by classes of vital curves. Let us denote by E i and E i,j the exceptional divisors over p i and the strict transform of p i , p j respectively. In the first blow-up X the strict transforms of the lines p i , p j intersects the exceptional divisor E i over p i in four points q j for j = i. Therefore, after blowing-up all the strict transforms of the lines, the divisor E i in M 0,6 is isomorphic to the blow-up of P 2 in four points. We denote by L i h,k the strict transform in E i of the line spanned by q h and q k , and by R i h the exceptional divisor over q h . So, in any exceptional divisor, we get 10 vital curves: 6 of type L i h,k and 4 of type R i h . Now, for any line p i , p j ⊂ P 3 we have three planes p i , p j , p k for k = i, j containing this line. The strict transforms of the three planes intersects the exceptional divisor E i,j in three vital curves σ k i,j . Therefore, we have 5 2 • 3 = 30 of them. Now remark that E i,j is isomorphic to P 1 × P 1 . Note moreover that R i j is numerically equivalent to R j i for any i, j because they are fibers of the same ruling of E i,j . Furthermore, the σ k i,j 's for k = i, j are all numerically equivalent because they are fibers of the other ruling of E i,j . We conclude that N E(M 0,6 ) is a polyhedral cone generated by 50 extremal rays.

Lemma 3.11. For any i we have H 2 E i = HE 2 i = 0, E 3 i = 1. Furthermore, HE 2 i,j = -1, H 2 E i,j = 0 for any i, j, and

E i E 2 h,k = -1 if i ∈ {h, k}, 0 if i / ∈ {h, k}.
Finally E 2 i E h,k = 0 for any i, h, k, and E 3 i,j = 2. Proof. We will denote by E i both the exceptional divisor over p i in X and its strict transform in Y . Let H i be the strict transform of a general plane through p i . Then H i = H -E i and

H 3 i = H 3 -3H 2 E i + 3HE 2 i -E 3 i , H 3 i = H 2 E i = HE 2 i = 0 yield E 3 i = H 3 = 1. Now,
let us consider the following diagram, where L i,j is the strict transform of the line p i , p j .

E i,j Y L i,j X j i π π E
where π E = π |E i,j . We have (H -E i,j ) 2 = H 3 -H 2 (H -E i,j ) = 0. Therefore,

HE 2 i,j = π * Hj * E 2 i,j = j * (E 2 i,j π * E i * H) = -1, H 2 E i,j = π * H 2 j * E i,j = j * (E i,j π * E i * H 2 ) = 0, E i E 2 i,j = π * E i E 2 i,j = j * (E 2 i,j π * E i * E i ) = -1, E 2 i E i,j = π * E 2 i E i,j = j * (E i,j π * E i * E 2 i ) = 0. Finally (H -E i -E j -E i,j ) 3 = H 3 -E 3 i -E 3 j -E 3 ij + 3HE 2 ij -3E i E 2 ij -3E j E 2 ij = 0 yields E 3 i,j = 2.
Proposition 3.12. The moduli space M 0,6 is weak Fano.

Proof. The anti-canonical bundle is given by

-K M 0,6 = 4H -2 5 i=1 E i - 5 i,j=1 E i,j .
First we consider the curves of type L i h,k . We have

L i h,k E t = -1 if i = t, 0 if i = t.
Furthermore,

L i h,k E s,t =
1 if s = i and t ∈ {h,k}, 0 otherwise.

Finally, L i h,k H = 0, and -K M 0,6 L i h,k = -2(-1) -(1 + 1) = 0. Now, let us consider a curve of type R i j . Then R i j H = R i j E k = 0 for any i, j, k, and

R i j E h,k = -1 if {i,j} = {h,k}, 0 otherwise.
This yields -K M 0,6 R i j = 1. Finally, we consider a curve of type σ i,j . Note that the normal bundle of the strict transform of a line L i,j = p i , p j is N L i,j = O P 1 (-1) ⊕ O P 1 (-1). Therefore, O E i,j (E i,j ) = O E i,j (-1, -1). This yields

σ i,j E h,k = -1 if {i,j} = {h,k}, 0 otherwise.
Furthermore σ i,j H = 1 and

σ i,j E h = 1 if h ∈ {i,j}, 0 otherwise.
Therefore -K M 0,6 σ i,j = 4 -2(1 + 1) -(-1) = 1. This means that -K M 0,6 is nef. Now, by the formulas in Lemma 3.11 we get that (-K M 0,6 ) 3 > 0 which implies that -K M 0,6 is big.

We would like to stress that by [KMc96] M 0,n is not even log Fano for n ≥ 7.

Example 3.13. Under the hypothesis of Proposition 3.8 let us consider the case n = 3,

m 1 = ... = m s = m then k I[0] = m, k I[1] = 2m -d and D = dH - s i=1 mE i - i =j (2m -d)E ij
where E i is the exceptional divisor over p i , and E ij is the exceptional divisor over the strict transform L ij in X 3 0 of the line p i , p j ⊂ P 3 . By Lemma 3.11 we have

D 3 = d 3 H 3 +3d i =j (2m-d) 2 HE 2 ij -3 s i=1 m(2m-d) 2 i =j E i E 2 ij - s i=1 m 3 E 3 i - i =j (2m-d) 3 E 3 ij and then D 3 = d 3 -sm 3 + s 2 (2m-d) 3
, which is exactly formula (3.10) for n = 3 and r = 1. For instance, if d = 3, m = 2 and s = 4 then D 3 = 1. Indeed, in this case L = L 3,3 (2, 2, 2, 2) is the linear system inducing the standard Cremona transformation of P 3 .

Symmetries of GIT quotients

In this section, in analogy with Section 2.12, we describe a special arrangement of linear spaces contained in Σ 2g-1 , and exploiting these arrangements we manage to compute the automorphism group of the GIT quotients Σ 2g and Σ 2g-1 . In several cases, automorphisms of moduli spaces tend to be modular, in the sense that they can be described in terms of the objects parametrized by the moduli spaces themselves. See for instance, [START_REF] Bruno | The automorphism group of M 0,n[END_REF], [START_REF] Massarenti | The automorphism group of M g,n[END_REF], [START_REF] Massarenti | On the automorphisms of moduli spaces of curves[END_REF], [START_REF] Massarenti | On the automorphisms of Hassett's moduli spaces[END_REF], [START_REF] Fantechi | On the rigidity of moduli of curves in arbitrary characteristic[END_REF], [START_REF] Massarenti | On the biregular geometry of Fulton-MacPherson configuration spaces[END_REF], [START_REF] Fantechi | On the rigidity of moduli spaces of weighted pointed stable curves[END_REF], [START_REF] Lin | Configuration spaces of C and CP 1 : some analytic properties[END_REF], [START_REF] Lin | Algebraic functions, configuration spaces, Teichmüller spaces, and new holomorphically combinatorial invariants[END_REF], [START_REF] Royden | Automorphisms and isometries of Teichmüller spaces, Advances in the theory of Riemann surfaces[END_REF] for moduli spaces of pointed and weighted curves, [START_REF] Biswas | Automorphisms of moduli spaces of vector bundles over a curve[END_REF] for moduli spaces of vector bundles over a curve, and [START_REF] Biswas | Automorphisms of the generalized quot schemes[END_REF] for generalized quot schemes. We confirm this behavior also for the GIT quotients Σ 2g and Σ 2g-1 . We would like to stress that while the results in the above cited paper relies on arguments coming from birational geometry and moduli theory, in this case we use fairly different techniques based on explicit projective geometry.

4.1. The odd dimensional case. Let p 1 , ..., p n+1 ∈ P n be general points, and X n n+1 be the blow-up of P n at p 1 , ..., p n+1 . We may assume that p 1 = [1 : 0 : ... : 0], ..., p n+1 = [0 : ... : 0 : 1]. Let us consider the standard Cremona transformation:

ψ n : P n P n [x 0 : ... : x n ] -→ [ 1 x 0 : ... : 1 xn ] Note that ψ n • ψ n = Id P n , and ψ -1 n = ψ n . Let H 1 , .
.., H n+1 be the coordinate hyperplanes of P n . Then ψ n is not defined on the locus 1≤i<j≤n+1 H i ∩ H j . Furthermore, ψ n is an isomorphism off of the union 1≤i≤n+1 H i . Now, ψ n induces a birational transformation ψ n : X n n+1

X n n+1 and we have the following commutative diagram:

X n n+1 X n n+1 P n P n ψn ψn
Note that, since ψ n contracts the hyperplane H i spanned by the n points p 1 , ..., pi , ..., p n+1 onto the point p i , the map ψ n maps the strict transform of H i onto the exceptional divisor E i . Therefore ψ n is an isomorphism in codimension one. Indeed, it is a composition of flops. In particular ψ n induces an isomorphism Pic(X n n+1 ) → Pic(X n n+1 ). Now, the linear system on P n associated to the standard Cremona transformation ψ n is

H = O P n (n) ⊗ I (n-1)(p 1 +...+p n+1 )
, that is H is the linear system of hypersurfaces in P n of degree n having points of multiplicity at least n -1 in p 1 , ..., p n+1 . Therefore, the inverse image of a general hyperplane of P n via ψ n is an hypersurface of degree n with points of multiplicity n -1 in p 1 , .

ψ n * D = d(nH -n+1 i=1 E i ) -n+1 i=1 m i (H -j =i E j ) = dnH -d n+1 i=1 (n -1)E i -n+1 i=1 H + n+1 i=1 m i j =i E j = (dn -n+1 i=1 m i )H -n+1 i=1 (d(n -1) -j =i m j )E j .
which gives exactly the statement.

Proposition 4.3. The standard Cremona transformation ψ 2g-1 : P 2g-1 P 2g-1 induces an automorphism of the odd dimensional GIT quotient Σ 2g-1 .

Proof. Recall that Σ 2g-1 is closure of the image of the map σ g : P 2g-1 P(H 0 (P 2g-1 , L 2g-1 ) * ) induced by the linear system L 2g-1 of degree g hypersurfaces having multiplicity g -1 at p 1 , ..., p 2g and p 2g+1 = [1 : ... : 1]. Note that ψ 2g-1 (p 2g+1 ) = p 2g+1 . Now, let D ∈ H 0 (P 2g-1 , L 2g-1 ) be a section. By Lemma 4.2 we get deg(ψ 2g-1 (D)) = g(2g-1)-2g i=1 m i = g(2g-1)-2g(g-1) = g and mult p i ψ 2g-1 (D) = g(2g-2)j =i m j = g(2g -2) -(2g -1)(g -1) = g -1 for i = 1, ..., 2g.

Furthermore, since ψ 2g-1 is an isomorphism in a neighborhood of p 2g+1 we have that mult p 2g+1 ψ 2g-1 (D) = g -1 as well.

Therefore, ψ 2g-1 acts on the sections of L 2g-1 , and hence it induces an automorphism ψ 2g-1 of P(H 0 (P 2g-1 , L 2g-1 ) * ) that keeps Σ 2g-1 stable.

Example 4.4. The famous Segre cubic 3-fold is the image of the map σ : P 3 P 4

given by

σ(x 0 , x 1 , x 2 , x 3 ) = [x 0 x 2 -x 0 x 1 : x 0 x 3 -x 0 x 1 : x 1 x 2 -x 0 x 1 : x 1 x 3 -x 0 x 1 : x 2 x 3 -x 0 x 1 ]
and the standard Cremona transformation of P 3 may be written as

φ 3 (x 0 , x 1 , x 2 , x 3 ) = [x 0 x 1 x 2 : x 0 x 1 x 3 : x 0 x 2 x 3 : x 1 x 2 x 3 ]
Let z 0 , ..., z 4 be the homogeneous coordinates on P 4 . Then the equation of the Segre cubic is

(4.5) z 0 z 1 z 2 -z 0 z 1 z 3 -z 0 z 2 z 3 + z 1 z 2 z 3 -z 1 z 2 z 4 + z 0 z 3 z 4 = 0 and φ 3 (z 0 ) = φ 3 (x 0 x 2 -x 0 x 1 ) = x 0 x 1 x 2 x 3 (x 0 x 2 -x 0 x 1 ) φ 3 (z 1 ) = φ 3 (x 0 x 3 -x 0 x 1 ) = x 0 x 1 x 2 x 3 (x 1 x 2 -x 0 x 1 ) φ 3 (z 2 ) = φ 3 (x 1 x 2 -x 0 x 1 ) = x 0 x 1 x 2 x 3 (x 0 x 3 -x 0 x 1 ) φ 3 (z 3 ) = φ 3 (x 1 x 3 -x 0 x 1 ) = x 0 x 1 x 2 x 3 (x 1 x 3 -x 0 x 1 ) φ 3 (z 2 ) = φ 3 (x 2 x 3 -x 0 x 1 ) = x 0 x 1 x 2 x 3 (x 2 x 3 -x 0 x 1 )
Therefore, ψ 3 induces the automorphism ψ 3 (z 0 , z 1 , z 2 , z 3 , z 4 ) = [z 0 : z 2 : z 1 : z 3 : z 4 ] of P 4 which clearly preserves Equation (4.5).

4.6. Linear subspaces of dimension g in Σ 2g-1 . In this section we will study a particular configuration of g-planes contained in Σ 2g-1 , and then we will exploit this configuration to compute the symmetries of Σ 2g-1 .

4.7. Let H I = H i 1 ,...,i g+1 be the g-plane in P 2g-1 , linear span of the points p i 1 , ..., p i g+1 , and let L 2g-1|H I be the restriction to H I of the linear system L 2g-1 inducing σ g . Then L 2g-1|H I is the linear system of degree g hypersurfaces in H I ∼ = P g having multiplicity g -1 at p i 1 , ..., p i g+1 . This means that σ g|H I is the standard Cremona transformation of P g . Therefore, σ g (H I ) is a g-plane in Σ g passing through the singular points given by the contractions of the (g -1)-planes generated by subsets of cardinality g of {p i 1 , ..., p i g+1 }. Now let Π I c the (g -1)-plane generated by the points in {p 1 , ..., p 2g+1 } \ {p i 1 , ..., p i g+1 }.

Note that H I c intersects H I in one point, hence σ g (H I ) passes through the singular point σ g (H I c ) as well. We conclude that there are g + 1 + 1 = g + 2 singular points of Σ 2g-1 lying on the g-plane σ g (H I ). Now, let E g-2 I ⊂ X 2g-1 g-2 be the exceptional divisor over the (g -2)-plane H g-2 I generated by the p i 's for i ∈ I. Then E g-2 I is a P g -bundle over the strict transform of H g-2 I in X 2g-1 g-3 . For any j / ∈ I let H g-1 I∪{j} be the (g -1)-plane generated by the p i 's for i ∈ I and p j . Note that the strict transform of H g-1

I∪{j} intersects E g-2 I along a section s which is mapped by the blow-up morphism isomorphically onto H g-2

I

. Since the strict transform of H g-1 I∪{j} is contracted to a point by σg , the section s must be contracted to a point as well. Therefore, σg (E g-1

I

) is a g-plane passing through g + 2 singular points of Σ 2g-1 . So far we have found 2g+1 g+1 + 2g + 1 linear spaces of dimension g in Σ 2g-1 , and each of them contains at least g + 2 of the 2g+1 g singular points of Σ 2g-1 . We will divide the gplanes inside Σ 2g-1 passing through a singular point p ∈ Σ 2g-1 into two families according to their mutual intersection.

4.8. The singular locus of Σ 2g-1 consists of 2g+1 g points corresponding to the 2g+1 g linear subspaces p i 1 , ..., p ig with {i 1 , ..., i g } ⊂ {1, ..., 2g + 1} contracted by σ g . Now, let p ∈ Σ 2g-1 be a singular point. So far, we found 2g + 2 linear subspaces of dimension g in Σ 2g-1 passing through p ∈ Σ g . These g-planes may be divided in two families:

A p = {α 1 , ..., α g+1 }, B p = {β 1 , ..., β g+1 } with the following properties:

-α i ∩ α j = β i ∩ β j = {p} for any i, j = 1, ..., g + 1, -α i ∩ β j = p, q ij for any i, j = 1, ..., g + 1, where q ij ∈ Σ 2g-1 is a singular point q ij = p.

The configuration is summarized in the following picture:

β 1 β 2 β j β g+1 α 1 α 2 α i α g+1 q ij
where the black dots should all be interpreted as representing the singular point p ∈ Σ 2g-1 . By Proposition 3.4 we have h 0 (P 2g-1 , L 2g-1 ) = 2g g -2g g-2 . Now, set N = 2g g -2g g-2 -1, and consider the expressions s i = I∈R a i I x I for i = 0, ..., N . Let H 1 , ..., H N +2 be (g -1)planes in P 2g-1 generated by subsets of cardinality g of {[1 : 0 : ... : 0], ..., [0 : ... : 0 : 1]} ⊂ P 2g-1 . Note that imposing [s 1 (H i ) : ... : s N (H i )] = [0 : ... : 0 : 1 : 0 : ... : 0], with the non-zero entry in the i-th position, for i = 1, ..., N + 1 we get N (N + 1) equations. Furthermore, by setting [s 1 (H N +2 ) : ... : s N (H N +2 )] equal to [1 : ... : 1] we get N more equations. Recall that by (4.10) for each i = 0, ..., N there are 2g g-2 relations among the a i I 's. Therefore, we get 2g g-2 (N +1) more constraints. Summing up we have a linear system of N (N + 1) + N + 2g g-2 (N + 1) homogeneous equations in the 2g g (N + 1) indeterminates a i I . Note that 2g g (N + 1) -N (N + 1) + N + 2g g -2 (N + 1) = 1, hence there exists a non-trivial solution. Let s 0 , ..., s N be the sections of H 0 (P 2g-1 , L 2g-1 ) associated to such a solution. These sections yield an explicit realization of the map σ g : P 2g-1 Σ 2g-1 ⊂ P(H 0 (P 2g-1 , L 2g-1 ) * ) = P N , σ g (x) = [s 0 (x) : ... : s N (x)]. By construction and by the description of the singular locus of Σ 2g-1 in 4.8, we see that, with respect to this expression for σ g , the points [1 : 0 : ... : 0], ..., [0 : ... : 0 : 1], [1 : ... : 1] ∈ P N are singular points of Σ 2g-1 . Now we need two technical lemmas. Lemma 4.11. Let p ∈ Σ 2g-1 ⊂ P N be a singular point. The tangent cone of Σ 2g-1 at p is a cone with vertex p over the Segre product P g-1 × P g-1 . In particular, Σ 2g-1 has an ordinary singularity of multiplicity (2g-2)! ((g-1)!) 2 at p ∈ Σ 2g-1 .

Proof. The statement follows from [HMSV09, Lemma 4.3].

By Corollary 4.12 φ is induced by a linear automorphism of the ambient projective space. Then φ must map g-planes through p to g-planes through φ(p). In particular, since φ stabilizes Sing(Σ 2g-1 ), it maps A p ∪ B p to A φ(p) ∪ B φ(p) . In order to do this, we have the following 2((g + 1)!) 2 possibilities: α i,p → α j,φ(p) β i,p → β k,φ(p) or α i,p → β j,φ(p) β i,p → α k,φ(p)

Summing up, we have 2((g + 1)!) 2 2g + 1 g = (2g + 2)! possibilities. Now, assume that φ(p) = p, and that φ maps α i,p to α i,p , and β i,p to β i,p for any i = 1, ..., g + 1. Then φ must fix all the singular points q ij determined by A p ∪ B p . Now, let us take into account one of these points, say q g+1,g+1 . Since φ fixes g + 2 nodes in linear general position on α g+1 , and g + 2 nodes in linear general position on β g+1 we have that φ is the identity on both α g+1 and β g+1 . On the other hand, α g+1 and β g+1 are elements of the configuration A q g+1 ∪ B q g+1 , therefore all the singular points of Σ 2g-1 determined by A q g+1 ∪ B q g+1 are fixed by φ as well. Proceeding recursively this way, we see that φ must then fix all the singular points of Σ 2g-1 ⊂ P N .

Note that, with respect to the expression for σ g given in 4.9, the points [1 : 0... : 0], [0 : ... : 0 : 1], [1 : ... : 1] ∈ Σ 2g-1 ⊂ P(H 0 (P 2g-1 , L * 2g-1 )) = P N are singular points. Hence the automorphism of P N inducing φ fixes N + 2 points in linear general position. Therefore, it is the identity and then φ = Id Σ 2g-1 . This completes the proof for Σ 2g-1 , in the next section we will go through to case of Σ 2g . 4.14. The even dimensional case. For the reader's relief, in this section we will make large use of results that we have already proven in the preceding sections. Let us recall shortly the notation. Recall that Σ 2g parametrizes ordered configurations of n = 2g + 3 points on P 1 , with the democratic polarization. By Theorem 1.2, Σ 2g ⊂ P(H 0 (P 2g , L 2g ) * ) is the closure of the image of the rational map induced by the linear system L 2g , see also Section 2.9. We denote by µ g : P 2g

Σ 2g ⊂ P(H 0 (P 2g , L 2g ) * ) = P N the rational map induced by this linear system. Lemma 4.15. Any automorphism of Σ 2g ⊂ P N is induced by an automorphism of P N . Proof. Recall that by 2.16 Pic(Σ 2g ) ∼ = Z 2g+3 . Then it is enough to argue as in the proof of Corollary 4.12 using Lemma 2.18 instead of Lemma 2.5.

Theorem 4.16. The automorphism group of the GIT quotient Σ 2g is the symmetric group on n elements:

Aut(Σ 2g ) ∼ = S 2g+3 for any g ≥ 1.

Proof. Let φ ∈ Aut(Σ 2g ) be an automorphism. By the discussion in 2.16, φ induces a pseudo-automorphism, that is an automorphism in codimension two,

θ := ψ -1 • φ • ψ : X 2g 1 X 2g
1 . The pseudo-automorphism θ must preserve the set of the extremal rays of Eff(X 2g 1 ). Let D ⊂ X 2g 1 be the union of the exceptional divisors E i and of the strict transforms of the

  obtained by collapsing components of C along which ω C (b 1 s 1 + ... + b n s n ) fails to be ample, where ω C denotes the dualizing sheaf of C.

2. 13 .

 13 We found 2g+2 g+1 + 2g+2 g linear subspaces of dimension g in Σ 2g ⊂ P N . We will denote by C = {γ 1 , ..., γ c }, D = {δ 1 , ..., δ d } where c = 2g+2 g+1 and d = 2g+2 g , the families of the g-planes coming from the H I 's and the E g-1 I

F

  ano with the GIT quotient Σ 2g , which in turn, by Remark (1.11) is isomorphic to the moduli space M 0, A[2g+3] with weights A[2g + 3] = 2 2g+3 , ..., 2 2g+3 . Now, let us consider the moduli space M 0, B[2g+3] with weights B[2g + 3] = 2 2g+2 , ..., 2 2g+2 , and the reduction morphism ρ A[2g+3], B[2g+3] : M 0, B[2g+3] → M 0, A[2g+3] . Note that k 2 2g+2 > 1 if and only if k ≥ g + 1, and k ≥ g + 2 implies that k 2 2g+3 > 1. Therefore, [Has03, Corollary 4.7] yields that ρ A[2g+3], B[2g+3] is an isomorphism, and we may identify X 2g F ano with the moduli space M 0, B[2g+3] . Recall that by Remark 1.11 we may interpret Σ 2g-1 as the Hassett space M 0, B[2g+2]

4. 9 .

 9 Let R = {I ⊂ {1, ..., 2g} | |I| = g}, S = {J ⊂ {1, ..., 2g} | |J| = g -2}, and x I = x i 1 ...x ig where I = {i 1 , ..., i g }, and the x i 's are homogeneous coordinates on P 2g-1 . By [Ku00, Theorem 4.1] we have that (4.10) H 0 (P 2g-1 , L 2g-1 ) = I∈R a I x I | J⊂I∈R a I = 0 ∀ J ∈ S .

  lim sup

						ldim(tD)
					t →+∞	t n /n!
	Now, in order to conclude is enough to note that by (3.2) we have
	lim sup t →+∞	ldim(tD) t n /n!	= d n +	s-1 r=0 I[r]⊆{1,...,s}	(-1) r+1 k n I[r]

  .., p n+1 , and ψ * n H = nH -(n -1)(E 1 + ... + E n+1 ). Furthermore, since ψ n contracts the hyperplane H i spanned by the n points p 1 , ..., pi , ..., p n+1 onto the point p i we haveψ * n E i = H -E 1 -... -Êi -... -E n+1 .Lemma 4.2. Let D ⊂ P n be a hypersurface of degree d having points of multiplicities m 1 , ..., m n+1 in p 1 , ..., p n+1 , and let ψ n : P n P n be the standard Cremona transformation of P n . Then deg(ψ n (D)) = dn -Bl p 1 ,...,p n+1 P n , and ψ n : X n n+1 X n n+1 be the birational map induced by ψ n . The strict transform of D in X n n+1 X n n+1 can be written as D ∼ = dH -n+1 i=1 m i E i . Now, since ψ n * H = nH -n+1 i=1 (n -1)E i , and ψ n * E i = Hj =i E i we get the formula

	n+1
	m i
	i=1
	and
	mult p i ψ n (D) = d(n -1) -	m j
	j =i
	for any i = 1, ..., n + 1.
	Proof. Let X n n+1 =

Michele Bolognesi, Alex Massarenti. A variation on a conjecture of Faber and Fulton. 2019. hal-

Acknowledgments. We thank Carolina Araujo, Cinzia Casagrande, Han-Bom Moon and Elisa Postinghel for useful conversations.

The authors are members of the Gruppo Nazionale per le Strutture Algebriche, Geometriche e le loro Applicazioni of the Istituto Nazionale di Alta Matematica "F. Severi" (GNSAGA-INDAM).

Corollary 4.12. Any automorphism of the GIT quotient Σ 2g-1 ⊂ P N is induced by an automorphism of P N .

Proof. Recall that, by Proposition 2.6, Pic(Σ 2g-1 ) is torsion free. Let φ be an automorphism of Σ 2g-1 . Then φ * K Σ 2g-1 ∼ K Σ 2g-1 . Lemma 2.5 yields that φ * O Σ 2g-1 (1) ∼ O Σ 2g-1 (1), that is φ is induced by an automorphism of P N .

Theorem 4.13. The automorphism group of the GIT quotient Σ 2g-1 is the symmetric group on 2g + 2 elements:

for any g ≥ 2.

Proof. Let p ∈ Σ 2g-1 ⊂ P N be a singular point, and H a g-plane inside Σ 2g-1 passing through p. Then H is contained in the tangent cone of Σ 2g-1 at p, and by Lemma 4.11 H must be a linear space generated by p and (g -1)-plane in the Segre embedding of P g-1 × P g-1 , and such a (g -1)-plane must be either of the form {pt} × P g-1 or of the form P g-1 × {pt}.

Assume that H intersects the g-planes of the family A p in the line p, q ij , where q i,j = α i ∩ β j is a singular point of Σ 2g-1 , and the planes of the family B p in p. Our aim is to prove that then H must be one of the β j 's.

By Proposition 2.3, the resolution σ g has a modular interpretation as the reduction morphism

. The only g-planes in Σ 2g-1 that are images of subvarieties contained in exceptional locus of the blow-up f : M 0,A[2g+2] → P 2g-1 are the σ g (E g-1 I ) described in Section 4.7. Therefore, we may assume that H is the closure of the image via σ g of a g-dimensional variety Z ⊂ P 2g-1 . We may assume that p = σ g ( p 1 , ..., p g ), and consider Π = σ g ( p g+1 , ..., p 2g+1 ). Let ξ i = σ g ( p g+1 , ..., pi , ..., p 2g+1 ) be the other g + 1 singular points of Σ 2g-1 lying on Π, and denote by π p : TC p Σ g → P g-1 × P g-1 the projection from the tangent cone of Σ 2g-1 at p onto its base, and let π i : P g-1 × P g-1 → P g-1 be the projection onto the factors for i = 1, 2. Then H must be of the form (π 1 • π p ) -1 (ξ i ) for some i = 1, ..., g + 1. Therefore, Z is of the form p 1 , ..., p g , p i for some i = g + 1, ..., 2g + 1. Hence σ g|Z : Z H is the standard Cremona transformation of P g and H is one of the β i 's. Now, assume that H is a g-plane inside Σ 2g-1 through p intersecting the g-planes of the family B p in the line p, q ij , where q i,j = α i ∩ β j is a singular point of Σ 2g-1 , and that it also intersects the planes of the family A p in p. By Proposition 4.3, the standard Cremona transformation of P 2g-1 induces an automorphism φ Cr : Σ 2g-1 → Σ 2g-1 , set q = φ Cr (p). Note that φ Cr maps the family A p to the family B q , and the family B p to the family A q . Since in our argument p is an arbitrary singular point of Σ 2g-1 , proceeding as in the previous case we can show that H must be one of the α i 's.

Clearly, since Σ 2g-1 ∼ = M 0, A[2g+2] with symmetric weights A[2g+2] = 1 g+1 , ..., 1 g+1 , 1 g+1 , the symmetric group S 2g+2 acts on Σ 2g-1 by permuting the marked points. Our aim is now to show that Σ 2g-1 has at most (2g + 2)! automorphisms. Now, let φ : Σ 2g-1 → Σ 2g-1 be an automorphism. If p ∈ Σ 2g-1 is a singular point then φ(p) must be a singular point as well. Therefore, we have at most | Sing(Σ 2g-1 )| = 2g+1 g choices for the image of p.

hyperplanes generated by 2g of the p i 's. By Lemma 2.17 any irreducible component of D generates an extremal ray of Eff(X 2g 1 ). Furthermore, any of these irreducible components is the unique element in its linear equivalence class. Therefore, θ must keep D stable.

Set D = ψ(D). Then the automorphism φ stabilizes D. As we did in Section 2.12, we will now consider g-planes; check that section for the needed definitions. Note that any g-plane γ i ∈ C is the intersection of g+1 g-1 divisors of type ψ(H -i∈I E i ), and that any g-plane δ i ∈ D is the intersection of g divisors of type ψ(E i ). Therefore φ stabilizes the configuration of g-planes C ∪ D. Now, let p ∈ Σ 2g be one of the distinguished points in 2.14. Then q = φ(p) must be a distinguished point as well. Let us denote by H 1 , H 2 the two g-planes intersecting in {p}, and by Π 1 , Π 2 the two g-planes intersecting in {q}. Therefore, by 2.14 we have (2g+3)! 2((g+1)!) 2 choices for the image of p.

Let p, p 1 1 , ..., p 1 g+1 and p, p 2 1 , ..., p 2 g+1 be the distinguished points on H 1 and H 2 respectively. Similarly, we denote by q, q 1 1 , ..., q 1 g+1 and by q, q 2 1 , ..., q 2 g+1 the distinguished points on Π 1 and Π 2 , respectively. Now, for the image of p 1 1 we have 2(g + 1) possibilities, namely q 1 1 , ..., q 1 g+1 , q 2 1 , ..., q 2 g+1 . Once this choice is made, it is determined whether the image of H 1 via φ is either Π 1 or Π 2 . Therefore, for the image of p 1 2 we have g possibilities, for p 1 3 we have g -1 possibilities, and so on until we are left with just one possibility for p 1 g+1 . Summing up, for the images of p 1 1 , ..., p 1 g+1 we have 2(g + 1)! possibilities. Similarly, we have also (g + 1)! possibilities for the images of p 2 1 , ..., p 2 g+1 . Finally, we have 2(g + 1)!(g + 1)! (2g + 3)! 2((g + 1)!) 2 = (2g + 3)! = |S 2g+3 | possibilities. Now, assume that φ fixes the points p, p 1 1 , ..., p 1 g+1 , p 2 1 , ..., p 2 g+1 . Since by Corollary 4.12 φ is induced by an automorphism of P N and the points p, p 1 1 , ..., p 1 g+1 ∈ H 1 , p, p 2 1 , ..., p 2 g+1 ∈ H 2 are in linear general position, then φ restricts to the identity on both H 1 and H 2 .

Then, by the description of the configuration C∪D in 2.13, φ must fix all the distinguished points in C ∪ D, and then Remark 2.15 yields that the automorphism φ must be the identity.