Laurent Manivel
email: manivel@math.cnrs.fr

Laurent Gruson

A four-dimensional cousin of the Segre cubic

This note is devoted to a special Fano fourfold defined by a fourdimensional space of skew-symmetric forms in five variables. This fourfold appears to be closely related with the classical Segre cubic and its Cremona-Richmond configuration of planes. Among other exceptional properties, it is infinitesimally rigid and has Picard number six. We show how to construct it by blow-up and contraction, starting from a configuration of five planes in a four-dimensional quadric, compatibly with the symmetry group S 5 . From this construction we are able to describe the Chow ring explicitely.

Introduction

Fano threefolds were classified more that fourty years ago, after some fifty years of efforts. The classification of Fano fourfolds is still elusive and will probably remain so for a long time. There are many ways to construct such manifolds, and a systematic study was launched a few years ago, of those that can be constructed from vector bundles on products of Grassmannians and more general flag manifolds [START_REF] Bernardara | A database of Fano fourfolds, in preparation. 7. M. Bhargava, Higher composition laws. IV. The parametrization of quintic rings[END_REF]; a sample has already appeared in [START_REF] Bernardara | Nested varieties of K3 type[END_REF]. In this database, there is a unique fourfold with maximal Picard number, equal to six: the study of this fourfold is the object of this note.

This study turned out to be related with interesting questions at the intersection of algebraic geometry with Lie theory. Consider two complex vector spaces V 4 and V 5 , of dimension four and five respectively. The action of GL(V 4 ) × GL(V 5 ) on V ∨ 4 ⊗ ∧ 2 V ∨ 5 is known to be prehomogenous, its open orbit being the complement of a degree 40 hypersurface [24, p.98]. It is in fact one of the most complicated prehomogeneous spaces, containing no less than 63 distinct orbits [START_REF] Ozeki | On the microlocal structure of the regular prehomogeneous vector space associated with SL(5) × GL(4)[END_REF][START_REF] Djokovic | Orbit closure diagram for the space of quadruples of quinary alternating forms[END_REF]. An important literature has been devoted to this prehomogeneous space, including some in connections with quintic field extensions, in the spirit of Bhargava's work on higher reciprocity laws [START_REF] Kable | A construction of quintic rings[END_REF][START_REF]On the space of quadruples of quinary alternating forms[END_REF]7].

The Fano fourfold X 4 we are interested in is defined by a generic element of the prehomogeneous space V ∨ 4 ⊗ ∧ 2 V ∨ 5 . It has two natural projections to G(2, 4) ≃ Q 4 and to the six-dimensional G(3, 5) that we describe in some details in section 4.

In particular we show it is a small resolution of a fourfold with ten singular points which appears to be a cousin, or a big brother of the Segre cubic primal; this small resolution contracts ten planes which can be seen as a special subcollection of the classical Cremona-Richmond configuration. We deduce:

Theorem. Consider five general planes in one of the two families of projective planes in Q 4 . They intersect pairwise in ten points. Blow-up these ten points and then the strict transforms of the five planes. Then the strict transforms of the exceptional divisors of the first blowup can be contracted to a smooth Fano fourfold, which is precisely X 4 .

Then we show that the automorphism group is Aut(X 4 ) = S 5 , so that P ic(X 4 ) S5 ≃ Z 2 is generated by the pull-back of the hyperplane classes by the two projections. This suggests to construct the tensor that defines X 4 by reverse-engineering, starting from the representation theory of S 5 ; we show how this leads to a normal form from this tensor. We then use the previous constructions to describe the Chow ring of X 4 completely, including the action of S 5 . We also check that X 4 , as expected, is infinitesimally rigid.

This study can be considered as a warm-up for the more mysterious case of U ∨ 5 ⊗ ∧ 2 V ∨ 5 , directly related to E 8 , which has infinitely many but well-described orbits for the action of GL(U 5 ) × GL(V 5 ) (see [START_REF] Gruson | Moduli of abelian varieties, Vinberg θ-groups, and free resolutions, Commutative algebra[END_REF] for a first approach). Among other nice geometric objects, this representation will give rise to an interesting family of special Fano sixfolds.

Acknowledgements. We thank Marcello Bernardara, Enrico Fatighenti and Fabio Tanturri for sharing our joint project on Fano fourfolds. Thanks also to Pieter Belmans and Igor Dolgachev for their comments and suggestions. We acknowledge support from the ANR project FanoHK, grant ANR-20-CE40-0023.

Models

According to the classical Borel-Weil theorem, one can interprete the representation V ∨ 4 ⊗ ∧ 2 V ∨ 5 as a space of global sections of an irreducible homogeneous vector bundle over a homogeneous space, and this in more than one way:

V ∨ 4 ⊗ ∧ 2 V ∨ 5 = Γ(G(2, V 4 ) × P(V 5 ), U ∨ ⊠ Q ∨ (1)) = Γ(P(V 4 ) × P(V ∨ 5 ), O(1) ⊠ ∧ 2 V ∨ ) = Γ(G(2, V 5 ), V ∨ 4 ⊗ ∧ 2 V ∨ ) = Γ(P(V 4 ) × P(V 5 ), O(1) ⊠ Q ∨ (1)) = Γ(G(2, V 4 ) × G(3, V 5 ), U ∨ ⊠ ∧ 2 V ∨ ) = Γ(P(V 4 ) × G(3, V 5 ), O(1) ⊠ ∧ 2 V ∨ ) = Γ(P(V 4 ) × G(2, V 5 ), O(1) ⊠ ∧ 2 V ∨ ) = Γ(G(2, V 4 ) × G(2, V 5 ), U ∨ ⊠ ∧ 2 V ∨ ).
Here U and V denote tautological bundles on Grassmannians (with some abuse of notations since we use the these symbols several times for distinct bundles on different Grassmannians). As a consequence, consider a general element θ in V ∨ 4 ⊗ ∧ 2 V ∨ 5 . Interpreting it as a global section of a vector bundle in these seven different ways, we obtain smooth subvarieties of codimensions equal to the ranks of the vector bundles in question, that we respectively denote as follows (the notation is such that X d has dimension d):

X 0 ⊂ G(2, V 4 ) × P(V 5 ), X 1 ⊂ P(V 4 ) × P(V ∨ 5 ), X 2 ⊂ G(2, V 5 ), X 3 ⊂ P(V 4 ) × P(V 5 ), X 4 ⊂ G(2, V 4 ) × G(3, V 5 ), X 6 ⊂ P(V 4 ) × G(3, V 5 ), X 8 ⊂ P(V 4 ) × G(2, V 5 ), X ′ 8 ⊂ G(2, V 4 ) × G(2, V 5 ).
Another obvious thing to do is to consider θ as a general morphism from V 4 to ∧ 2 V ∨ 5 . The image of P(V 4 ) inside P(∧ 2 V ∨ 5 ) is then a generic projective three-plane, that has to meet the Grassmannian G(2, V ∨ 5 ) along a set Y 0 of five reduced points (the degree of the Grassmannian being equal to five). Correspondingly, we get a set P 0 of five points in P(V 4 ), and a set Π 0 of five planes in P(V 5 ), all in general position. Concretely, if we choose a basis e 1 , . . . , e 4 of V 4 , with dual basis e ∨ 1 , . . . , e ∨ 4 of V ∨ 4 and decompose θ accordingly as

θ = e ∨ 1 ⊗ θ 1 + e ∨ 2 ⊗ θ 2 + e ∨ 3 ⊗ θ 3 + e ∨ 4 ⊗ θ 4 then the contraction θ(v) = v 1 θ 1 + v 2 θ 2 + v 3 θ 3 + v 4 θ 4 has rank two when [v] belongs to P 0 ; that is, it decomposes as f ∨ 1 ∧ f ∨ 2 for two linear forms f ∨ 1 , f ∨ 2
whose kernels intersect along the corresponding plane in P(V 5 ). We will denote the five two-forms of rank two (defined up to scalars) obtained by contracting θ as ω 1 , . . . , ω 5 . It would be natural then to impose the normalization ω 1 + • • • + ω 5 = 0, and decompose θ as

θ = u ∨ 1 ⊗ ω 1 + u ∨ 2 ⊗ ω 2 + u ∨ 3 ⊗ ω 3 + u ∨ 4 ⊗ ω 4 + u ∨ 5 ⊗ ω 5 for some linear forms u ∨ 1 , . . . , u ∨ 5 such that u ∨ 1 + • • • + u ∨ 5 = 0.

Notations.

P 0 = {p 1 , . . . , p 5 } is a set of five points in P(V 4 ), in natural bijection with the set {ω 1 , . . . , ω 5 }, of five decomposable two-forms in ∧ 2 V ∨ 5 , that define five points in G(2, V ∨ 5 ) ≃ G(3, V 5 ), hence five planes P 1 , . . . , P 5 in P(V 5 ). They also define five planes π 1 , . . . , π 5 in G(2, V 4 ), where π k is the set of planes in V 4 that contain p k .

L 0 is the set of pairs of points in P 0 . According to the previous identifications, it is in natural bijection with a set of ten lines in P(V 4 ), a set of ten points in P(V 5 ), and a set of ten points in G(2, V 4 ).

Small dimensions

Most results in this section are classical. Our purpose is mainly to set up the scene for the main character, which will make its entry in the next section. Proposition 3.1. X 0 consists in 10 points of G(2, V 4 ) × P(V 5 ), in natural bijection with L 0 . Proof. By definition, a point (A 2 , B 1 ) belongs to X 0 if and only if we can decompose θ in such a way that A 2 is cut out by the linear forms e ∨ 3 , e ∨ 4 and the skew-symmetric forms θ 1 , θ 2 have the same kernel B 1 ⊂ V 5 . Otherwise said, θ 1 and θ 2 belong to ∧ 2 B ⊥ 1 . Since in the latter space, decomposable tensors are parametrized by a quadric, we can make a change of basis in A ⊥ 2 and suppose that θ 1 and θ 2 are indeed decomposable. Concretely, this means that we can write θ in the form

θ = e ∨ 1 ⊗ f ∨ 1 ∧ f ∨ 2 + e ∨ 2 ⊗ f ∨ 3 ∧ f ∨ 4 + e ∨ 3 ⊗ θ 3 + e ∨ 4 ⊗ θ 4 .
Then [e 1 ] belongs to P 0 , the associated plane in P(V 5 ) being

P 1 = f 1 , f 2
⊥ , and also [e 2 ] belongs to P 0 , the associated plane being P 2 = f 3 , f 4 ⊥ . In particular A 2 = e 1 , e 2 and B 1 = P 1 ∩ P 2 , as claimed. Proposition 3.2. X 1 is the union of five disjoint lines, in natural bijection with P 0 . Proof. By definition, a point in X 1 is a pair (A 1 , B 4 ) such that θ(v) vanishes on B 4 when v generates A 1 . But then θ(v) must have rank two, of the form f ∨ 1 ∧ f ∨ 2 . In particular A 1 must correspond to one of the five points of P 0 , and the hyperplane

B 4 can move in the pencil f ∨ 1 , f ∨ 2 . Proposition 3.3. X 2 ⊂ G(2, V 5 ) is a del Pezzo surface of degree five. Proof. Obvious.
Recall that the del Pezzo surface of degree five contains 10 lines. Since the embedding in G(2, V 5 ) is anticanonical, this means in our setting that there exists ten flags A 1 ⊂ A 3 ⊂ V 5 such that θ(v, w) = 0 for any v ∈ A 1 , w ∈ A 3 . It is easy to see that these ten flags are in natural bijection with L 0 , the ten points [A 1 ] in P(V 5 ) being exactly the intersections of the planes P 1 , . . . , P 5 .

Proposition 3.4. The projection of X 3 to P(V 4 ) is the blow-up of the five points of P 0 . The projection to P(V 5 ) is a small resolution of a Segre cubic primal C 3 , ten lines being contracted to the ten singular points of C 3 defined by L 0 .

Proof. (Well known.) By definition, X 3 parametrizes the pairs (A 1 = [v], B 1 ) such that B 1 is contained in the kernel of θ(v). Generically this two-form has rank four and the kernel is one-dimensional, which implies that X 3 projects birationally to P(V 4 ). The projection has non trivial fibers when the rank of θ(v) drops, that is, over one of the five points in P 0 . Then the kernel has dimension three and the fiber is a projective plane, as it has to be.

No we turn to the projection to P(V 5 ). By definition, the fibers are linear subspaces defined by the image of the morphism Q(-1) → V ∨ 4 ⊗ O P(V5) induced by θ. In particular the fibers are non trivial over the corresponding determinantal locus C 3 , which is a cubic threefold since det(Q(-1)) = O P(V5) (-3). This threefold becomes singular exactly when the rank drops to two. If w ∈ V 5 generates B 1 , this means that the morphism from V 4 to V ∨ 5 sending e i to θ i (w, •) has rank two. So we may suppose after a change of basis that θ 1 (w, •) = θ 2 (w, •) = 0. In other words, θ 1 and θ 2 have the same kernel B 1 , and after another change of basis if necessary we have already seen that we can suppose they are decomposable. So they define two points in P 0 , in such a way that B 1 is the point obtained as the intersection of the corresponding planes in P(V 5 ), while the line contracted to this point is the span of the corresponding points in P(V 4 ).

As a result, C 3 is a cubic threefold with 10 nodes. (In fact C 3 is the image of the rational map from P(V 4 ) to P(V 5 ) sending [v] to the kernel of the two-form θ(v), and essentially by definition this is a Segre cubic primal [START_REF] Segre | nodal cubic threefolds, From classical to modern algebraic geometry[END_REF].

Reminder on the Segre cubic primal. Recall that the Segre primal can be defined, if x 0 , . . . , x 5 are homogeneous coordinates on P 5 , by the two equations

x 0 + • • • + x 5 = 0, x 3 0 + • • • + x 3 5 = 0.
This presentation exhibits an S 6 symmetry, and it is known that Aut(C 3 ) = S 6 . Classically, the Segre primal contains 15 planes. (See [START_REF] Dolgachev | Classical algebraic geometry[END_REF]Chapter 9] for much more information.)

The Segre cubic primal admits a classical modular interpretation, according to which C 3 ≃ (P 1 ) 6 //SL 2 . Moreover M0,6 is a resolution of its singularities (that just blows-up the singular points) and according to Kapranov it can be constructed by blowing-up five general points in P 3 , plus the strict transforms of the ten lines that join them [START_REF] Kapranov | Chow quotients of Grassmannians[END_REF]. (Note also that M0,6 compactifies the moduli space of genus 2 curves.) Note also that C 3 is known to be G-birationally rigid, and even G-birationally superrigid, when A 5 ⊂ G ⊂ S 6 [START_REF] Avilov | Automorphisms of singular three-dimensional cubic hypersurfaces[END_REF].

Blowing-up the ten singular points in C 3 we get ten exceptional divisors isomorphic to P 1 × P 1 , each of which is contracted to P 1 in X 3 . According to [START_REF] Finkelnberg | Small resolutions of the Segre cubic[END_REF] any of the rulings of P 1 × P 1 can in fact be contracted, yielding 2 10 = 1024 small resolutions of the singularities of C 3 , falling into 13 orbits of S 6 , including 6 for which the resolution is projective. Homological Projective Duality for the Segre cubic is discussed in [START_REF] Beckmann | Homological projective duality for the Segre cubic[END_REF].

On the planes in the Segre cubic. In coordinates, the 15 planes on the Segre cubic are given by three equations

x a + x b = x c + x d = x e + x f = 0,
for (abcdef ) a permutation of (123456); we denote such a plane by (ab|cd|ef ). Together with the 15 points in the hyperplane x 0 +• • •+x 5 = 0 with four coordinates equal to zero, they form a (15 3 , 15 3 ) configuration classically known as the Cremona-Richmond configuration: each plane contains three of the 15 points and each of those points belongs to three planes of the configuration. But beware that two planes may meet along a single point, or a projective line; the second possibility occurs when their symbols have a common pair. Proposition 3.5. There are exactly 6 collections of five planes among the fifteen planes in C 3 , meeting pairwise along single points. These collections are exchanged transitively by the action of S 6 . Each one has for stabilizer a copy of S 5 , embedded in S 6 in a non standard way.

To understand the last sentence, recall that S 6 has the exceptional property that its outer automorphism group is non trivial: there exists a unique outer automorphism, and a non standard embedding of S 5 in S 6 is the composition of a standard embedding by such an outer automorphism. Note that this outer automorphism of S 6 exchanges the two conjugacy classes consisting of transpositions on one hand, and products of three disjoint transpositions on the other hand; the former corresponds to points, the latter to planes in the Cremona-Richmond configuration, which is for this reason self-dual.

Proof. Suppose given a collection of five planes, any two of which meet at a single point. This means that each plane is represented by three pairs, none of which being shared with another plane. So we have a total amount of 15 distinct pairs; necessarily, all the 15 pairs of integers from 1 to 6 must appear exactly once.

Up to permutation, we can assume that one of our planes is (12|34|56). Then the plane containing ( 13) is either (13|25|46) or (13|26|45) and up to permuting 5 and 6 we can suppose it is the first one. Then the other planes are determined. For example, for the one containing ( 14), we must split (2356) into two pairs, and since (25) and ( 56) have already been used the only possibility is (14|26|35). This also shows that we have three choices for the plane containing [START_REF]Quartic surfaces with icosahedral symmetry[END_REF], then two choices for the plane containing [START_REF] Dolgachev | Geometry of the Wiman-Edge pencil, I: algebrogeometric aspects[END_REF], and then no more choices; this means there are exactly six possibilities. Explicitely, they are the following:

(12|34|56) (12|34|56) (12|35|46) (12|35|46) (12|36|45) (12|36|45) (13|25|46) (13|26|45) (13|24|56) (13|26|45) (13|25|46) (13|24|56) (14|26|35) (14|25|36) (14|25|36) (14|23|56) (14|23|56) (14|26|35) (15|24|36) (15|23|46) (15|26|34) (15|24|36) (15|26|34) (15|23|46) (16|23|45) (16|24|35) (16|23|45) (16|25|34) (16|24|35) (16|25|34)
Let us denote these six configurations by ABCDEF . The action of S 6 on them induces a morphism S 6 → S 6 , and a direct examination shows that it sends the transposition [START_REF]Quartic surfaces with icosahedral symmetry[END_REF] to the permutation (AB)(CD)(EF ). So it has to correspond to the outer automorphism of S 6 , and our final claim follows.

Question. Is there an interpretation in terms of the root system E 7 ? In fact the Lie algebra e 7 admits a Z 3 -grading of the form

e 7 = sl 3 × sl 6 ⊕ (C 3 ⊗ ∧ 2 C 6 ) ⊕ (C 3 ⊗ ∧ 2 C 6 ) ∨ ,
and roots of e 7 defined by weights of C 3 ⊗ ∧ 2 C 6 can be interpreted as triples of pairs [START_REF] Manivel | Configurations of lines and models of Lie algebras[END_REF]. Note that roots of e 7 are classically connected with the 28 bitangents of a plane quartic.

The Fano fourfold

Recall that our main character

X 4 ⊂ G(2, V 4 ) × G(3, V 5 ) is defined by θ a general element in V ∨ 4 ⊗ ∧ 2 V ∨ 5
, considered as a general section of the vector bundle U ∨ ⊠ ∧ 2 V ∨ . Here U denotes the tautological rank two bundle on G(2, V 4 ), while V denotes the tautological rank three bundle on G(3, V 5 ).

In this section we describe the geometry of X 4 by blowups and contractions.

4.1. The main invariants. We start by computing the main numerical invariants of X 4 , including its Hodge numbers. Proof. The first assertion is an immediate consequence of the adjunction formula.

The Hodge numbers and invariants can computed using exact sequences, along the lines explained in [START_REF] Bernardara | Nested varieties of K3 type[END_REF]. (They could also be deduced from the geometric descriptions that will follow.) Since 172 = 4 × 43 is not divisible by any fourth power, the index must be one.

Note that h

0 (-K X4 ) = 40 < dim(∧ 2 V 4 ⊗ ∧ 3 V 5 ) = 60, which means that X 4 is linearly degenerate inside G := G(2, V 4 ) × G(3, V 5
). This can be checked by considering the twisted Koszul complex

0 -→ ∧ 6 E ∨ (1, 1) -→ • • • -→ E ∨ (1, 1) -→ O G (1, 1) -→ O X4 (1) -→ 0. Indeed H 0 (E ∨ (1, 1)) ≃ V 4 ⊗ V 5 has dimension 20, while it can be checked that H 0 (∧ k E ∨ (1, 1)) = 0 for k > 1.
We will describe in some details the two projections p 1 , p 2 :

X 4 p1 { { ✈ ✈ ✈ ✈ ✈ ✈ ✈ ✈ ✈ p2 $ $ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ G(2, V 4 ) G(3, V 5 ).
We start with the second one.

4.2.

The second projection and the Cremona-Richmond configuration.

We start with the projection to G(3, V 5 ), which is very similar to the resolution of singularities of the Segre cuic primal.

Proposition 4.2. The projection of X 4 to G(3, V 5 ) is a small resolution of a codimension two subvariety C 4 of degree 12, contracting ten planes to ten singular points in natural bijection with L 0 .

Proof. The fiber of p 2 :

X 4 -→ G(3, V 5 ) over a point [V ] ∈ G(3, V 5 ) is defined by the morphism θ V : ∧ 2 V → V ∨ 4 induced by θ.
In particular the fibers are non trivial when the rank is at most two, which happens in codimension two. We conclude that the image C 4 of X 4 is a determinantal fourfold. Its structure sheaf admits a resolution by the Lascoux complex [START_REF] Lascoux | Syzygies des variétés déterminantales[END_REF] (1)

0 -→ V ∨ (-3) -→ V 4 ⊗ O G(2,V5) (-2) -→ O G(2,V5) -→ O C4 -→ 0,
where V denotes the rank three tautological bundle. We deduce in particular that the class of C 4 in the Chow ring of the Grassmannian G(3, V 5 ) is 3σ 11 + 2σ 2 , so that its degree is 3

× 2 + 2 × 3 = 12.
The rank of θ V drops to one on the singular locus of C 4 , which must have codimension 6, hence be a finite set, over which the fibers are projective lines. The fact that θ V has a two dimensional kernel means that we can find a basis

v 1 , v 2 , v 3 of V such that θ i (v 1 , v 2 ) = θ i (v 1 , v 3 ) = 0 for all i.
Completing with two vectors v 4 , v 5 and taking the dual basis, we conclude that every θ i belongs to the space of forms generated by

v ∨ 1 ∧ v ∨ 4 , v ∨ 1 ∧ v ∨ 5 and ∧ 2 (v ⊥ 1 ). In particular θ 1 , θ 2 , θ 3 , θ 4 has to meet ∧ 2 (v ⊥ 1
) in dimension at least two, which means that V defines a pair of planes π p , π q in P 0 , whose intersection point is a line in V . Finally, V defines a hyperplane H pq of V 4 , and the corresponding fiber is the set G(2, H pq ) ≃ P 2 of planes in H pq .

Conversely, such a pair of planes being given, we can decompose θ is an adapted basis as

θ = e ∨ 1 ⊗ f ∨ 1 ∧ f ∨ 2 + e ∨ 2 ⊗ f ∨ 3 ∧ f ∨ 4 + e ∨ 3 ⊗ θ 3 + e ∨ 4 ⊗ θ 4 , and then the conditions θ 3 (f 5 , •) = θ 4 (f 5 , •) = 0 define a 3-plane V containing f 5 .
This exactly means that the singular locus of C 4 consists in ten points, in natural bijection with L 0 . Proof. By definition, a point [v] ∈ P(V 5 ) belongs to C 3 when the four linear forms θ i (v, •) on V 5 are linearly dependent. In the proof above, we have seen that a singular point in C 4 corresponds to a three-plane

V = v 1 , v 2 , v 3 in V 5 with θ(v 1 , v 2 ) = θ(v 1 , v 3 ) = 0. So for any v ∈ V , the linear forms θ i (v, •) vanish on v 1 ,
and also on v by skew-symmetry. When v and v 1 are independent, the four linear forms θ i (v, •) therefore belong to the three-dimensional space v, v 1 ⊥ ⊂ V ∨ 5 , so they must be linearly dependent. Hence P(V ) ⊂ C 3 .

That the projectivized kernels P(K j ) of the five singular skew forms θ j are contained in C 3 is obvious, since θ j (v, •) = 0 for v ∈ K j is a linear dependence relation between the θ i (v, •).

Note that we also have a special point [v 1 ] in each of the ten planes P(V ). Moreover the five planes P(K 1 ), . . . , P(K 5 ) meet pairwise at a single point. In particular, they provide one of the special subcollections of the Cremona-Richmond configuration described in Proposition 3.5. Also observe that a form ω which is as above in the span of

v ∨ 1 ∧ v ∨ 4 , v ∨ 1 ∧ v ∨ 5 and ∧ 2 (v ⊥ 1 ), but does not belong to ∧ 2 (v ⊥ 1 ), can be written as v ∨ 1 ∧ w ∨ + γ with γ ∈ ∧ 2 (v ⊥ 1
) and w ∨ a combination of v ∨ 4 and v ∨ 5 . It has rank two when γ has rank (at most) two and w ∨ ∧ γ = 0, which means if w ∨ = 0 that γ is divisible by w ∨ . But then ω itself is divisible by w ∨ , and since w ∨ is a combination of v ∨ 4 and v ∨ 5 this implies that ω(v 2 , v 3 ) = 0. In other words, the linear form that defines H pq ⊂ V 4 vanishes at the point that corresponds to ω. This exactly means that

p i ∈ H jk i = j, k.
We thus get in G(2, V 4 ) a collection of 5 + 10 planes, such that each plane of the second type meets exactly three planes of the first type. Hence a configuration (10 3 , 5 6 ). The condition that (jk) be disjoint from (lm), so that the two hyperplanes meet in p n , defines a copy of the Petersen graph.

Being a degeneracy locus of a morphim between vector bundles, C 4 admits two natural resolutions of singularities; X 4 is one of them. For the other one, we need to impose a rank one kernel in the source of the morphism

∧ 2 V → V ∨ 2 ; note that a rank one subspace of ∧ 2 V is always of the form ∧ 2 W for W ⊂ V a rank two subspace. But then the composition ∧ 2 W → V ∨
2 vanishes exactly when W defines a point in the del Pezzo surface X 2 ⊂ G(2, V 5 ). Our second resolution of singularities is thus simply P X2 (Q), the projectivisation of the quotient bundle of G(2, V 5 ), restricted to X 2 . The two resolutions are dominated by X4 , the set of triples (U 2 , V 3 ⊃ W 2 ) such that (U 2 , V 3 ) belongs to X 4 and W 2 belongs to X 2 . We get a diagram:

X4 α 6⑥ ⑥ ⑥ ⑥ ⑥ ⑥ ⑥ π β # # • • • • • • • • • X 4 p1 { { ✇ ✇ ✇ ✇ ✇ ✇ ✇ ✇ ✇ p2 ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ P X2 (Q) q2 { { ✇ ✇ ✇ ✇ ✇ ✇ ✇ ✇ ✇ q1 # # • • • • • • • • • G(2, V 4 ) C 4 X 2 .
Proposition 4.4. The morphism q 2 : P X2 (Q) -→ C 4 is a small resolution of singularities, contracting ten lines to the ten singular points of C 4 . These ten lines are mapped by q 1 to the ten lines in the del Pezzo surface X 2 . The morphism β is the blow-up of the ten exceptional lines of q 2 , as well as α is the blow-up of the ten exceptional planes of p 2 .

Finally, π is the blow-up of the ten singular points of C 4 , its exceptional divisor being the disjoint union of ten copies of P 2 × P 1 .

Remark. Contrary to X 4 , the fourfold X ′ 4 = P X2 (Q) is not Fano but only weak Fano. Indeed, the canonical bundle of X 2 is det(Q ∨ ), so the canonical bundle of X ′ 4 is O X ′ 4 (-3). The quotient bundle Q is obviously not ample on G(2, V 5 ), and neither is it when restricted to X 2 since the morphism defined by O X ′ 4 (1) is precisely q 2 and has non trivial fibers. But of course Q is obviously nef, and it is also big since

X ′ 4 O X ′ 4 (1) 4 = X2 s 2 (Q) = G(2,V5) s 2 (Q)σ 4 1 = 2 > 0.
Note also the striking similarity with the two main projective resolutions of the Segre cubic, which can be encapsulated in a similar diagram

Z3 ⑦ ⑦ ⑦ ⑦ ⑦ ⑦ ⑦ ⑦ # # ❋ ❋ ❋ ❋ ❋ ❋ ❋ ❋ ❋ Z 3 | | ③ ③ ③ ③ ③ ③ ③ ③ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ P X2 (U ) { { ① ① ① ① ① ① ① ① ① # # • • • • • • • • • P(V 4 ) C 3 X 2 ,
where Z 3 is the blowup of P(V 4 ) = P 3 at five points. Two important differences: Z 3 , contrary to X 4 , is only weak Fano; Z 3 and Z ′ 3 = P X2 (U ), contrary to X 4 and X ′ 4 , are related by flops and therefore derived-equivalent. Instead of that, we have: Proposition 4.5. The birational map q 2 • p -1 2 : X 4 X ′ 4 is a flip. Proof. Since X 4 is Fano, we need only to check that the canonical bundle of X ′ 4 is nef on the non trivial fibers of the projection to C 4 . But we have seen that

K X ′ 4 = O X ′ 4 (-3
), the fiber of O X ′ 4 (-1) at a point defined by a flag U 2 ⊂ U 3 being U 3 /U 2 . On a fiber F of the projection to C 4 , by definition U 3 is fixed, so O X ′ 4 (-1) |F is base point free, hence also K X ′ 4 |F . According to the Bondal-Orlov conjecture, there should therefore exist a fully faithful functor D b (X ′ 4 ) -→ D b (X 4 ) that would be interesting to describe explicitely.

4.3. Pencils of skew-forms and the first projection. In order to describe the projection to G(2, V 4 ), we first note that a plane in V 4 defines through θ a pencil of skew-symmetric forms in five variables, and that such pencils have been classified. In fact, for a two dimensional vector space V 2 , the action of GL(V 2 ) × GL(V 5 ) on V ∨ 2 ⊗ ∧ 2 V ∨ 5 has finitely many orbits, which are described in [START_REF] Kraskiewicz | Geometry of orbit closures for the representations associated to gradings of lie algebras of type e 6 , f 4 and g 2[END_REF]. Let us only mention that there are exactly eight orbits: the open orbit O 7 , an orbit O 6 of codimension two and another O 5 of codimension four, and then all the other orbits have bigger codimension.

The orbit O 5 (or rather its closure) is characterized as consisting of tensors of rank at most four, in the sense that they belong to

V ∨ 2 ⊗ ∧ 2 V 4 for some hyperplane V 4 ⊂ V ∨ 5 .
The orbit O 6 (or rather its closure) is characterized as consisting of those pencils in ∧ 2 V ∨ 5 admiting a rank two element. So the open orbit O 7 parametrizes pencils of forms of constant rank four. By [START_REF] Manivel | On linear spaces of skew-symmetric matrices of constant rank[END_REF]Proposition 2], given such a pencil one can find a basis of V 5 for which the two skew-forms

ω 1 = f ∨ 1 ∧ f ∨ 3 + f ∨ 2 ∧ f ∨ 4 , ω 2 = f ∨ 1 ∧ f ∨ 4 + f ∨ 2 ∧ f ∨ 5 are generators. The projective line f ∨ 1 , f ∨ 2
is the pivot of the pencil. Now, observe that if a three-plane V ⊂ V 5 is isotropic with respect to any skew-form sω 1 + tω 2 of the pencil, it has to contain its kernel, which is generated by s 2 f 3 -stf 4 + t 2 f 5 . So necessarily V = f 3 , f 4 , f 5 , the orthogonal to the pivot. Proposition 4.6. The projection of X 4 to G(2, V 4 ) is birational. The exceptional locus in G(2, V 4 ) is the union of five planes, intersecting in the ten points of L 0 , whose fibers are quadratic surfaces.

Proof. The fiber of the projection p 1 :

X 4 -→ G(3, V 5 ) over the point [U ] ∈ G(2, V 4 ) is defined by the morphism θ U : U → ∧ 2 V ∨ 5 .
This morphism is injective and we thus get a pencil of skew-symmetric forms. If this pencil is generic, which means that it has constant rank, then we have just seen that there is a unique three-plane in V 5 which is isotropic with respect to any skew-form in the pencil. This threeplane is the image of the induced map θ

(2)

U : S 2 U → ∧ 4 V ∨ 5 ≃ V 5 .
In particular, p 1 is birational.

Special fibers will occur when the pencil Im(θ U ) becomes special in some way. By the usual arguments for orbital degeneracy locus [START_REF] Benedetti | Orbital degeneracy loci and applications[END_REF], we need to take into account, in the space of pencils, only those orbits of codimension smaller than five, which apart from the open orbit are the orbits O 5 and O 6 we have described above.

Pencils in O 5 contain two skew-forms of rank two. In our case, they must be two of the skew-forms ω 1 , . . . , ω 5 , say θ 1 and θ 2 . Choose an adapted basis such that

θ 1 = f ∨ 1 ∧ f ∨ 2 and θ 2 = f ∨ 3 ∧ f ∨ 4 , so that θ U = e ∨ 1 ⊗ f ∨ 1 ∧ f ∨ 2 + e ∨ 2 ⊗ f ∨ 3 ∧ f ∨ 4 .
It is straightforward to check that the three-planes that are isotropic with respect to any skew-form in the pencil are those generated by f 5 , a vector in f 1 , f 2 , and a vector in f 3 , f 4 . We thus get for fiber a copy of P 1 × P 1 .

Finally, pencils in O 6 contain exactly one skew-form of rank two, say θ 1 . To describe the corresponding fiber we must understand the 3-planes isotropic with respect to both the generic form θ 2 and the degenerate form

θ 1 = f ∨ 1 ∧ f ∨ 2 .
Such a 3-plane must contain the kernel of θ 2 ; let us choose a generator f 5 and a hyperplane H 4 in V 5 not containing f 5 . We may suppose that f ∨ 2 vanishes on f 5 . The 3-planes we are looking for are in correspondence with the 2-planes H = h, h ′ in H 4 such that ω 2 (h, h ′ ) = 0 and f ∨ 1 (h) = f ∨ 1 (h ′ ) = 0. Such a 2-plane must be contained in the kernel K 3 of f ∨ 1 , and it has to contain the kernel K 1 of the restriction of ω 2 to K 3 . We finally get for fiber a pencil of planes.

To summarize, the exceptional locus is the union of five planes π 1 , . . . , π 5 in G(2, V 4 ), where π i parametrizes the planes in V 4 containing ω i . Any two of these five planes meet at a single point, over which the fiber of p 1 is a quadratic surface.

If U 2 does not belong to any of the five exceptional planes, we have seen that U 3 is the span of the kernels of the two-forms θ(v), for v ∈ U 2 . Since this kernel can be computed as θ(v) ∧ θ(v), there is a natural associated conic bundle over G(2, V 4 ) minus the five exceptional planes. This also stresses the analogy with the construction of the Segre primal C 3 as the image of a rational map P(V 4 ) P(V 5 ) defined by θ. Here we get C 4 as the image of a rational map G(2, V 4 ) G(2, V 5 ) also defined by θ. We will put its equations in simple form in the next section. 4.4. Blow-up and contract. Proposition 4.6 suggests to construct X 4 by first blowing-up G(2, V 4 ) along the 10 points of L 0 , then the strict transforms of the 5 planes, which are Del Pezzo surfaces of degree five. The first blow-up Bl 0 : G 0 -→ G(2, V 4 ) gives 10 exceptional divisors E ij ≃ P 3 for 1 ≤ i < j ≤ 5, each with a pair of skew lines ℓ i , ℓ j coming from the two planes π i and π j intersecting at p ij . The second blow-up Bl P : G 1 -→ G 0 produces five other exceptional divisors F k for 1 ≤ k ≤ 5, while the strict transform of Ẽij of E ij is the blowup of E ij along ℓ i ∪ ℓ j . Since the blowup of P 3 along two skew lines is the total space of P(O(-1, 0) ⊕ O(0, -1)) over P 1 × P 1 , we deduce that the rational map to X 4 is a morphism. More precisely, it has to coincide with the blowup Bl Q : G 1 -→ X 4 of the ten quadratic surfaces S ij = p -1 1 (p ij ) in X 4 . This explains in particular why the Picard number is equal to 6.

G(2, V 4 ) X 4 G 0 Bl 0 G 1 Bl P Bl Q p 1 Let F = F 1 + • • • + F 5 ,
and let E be the sum of the ten divisors Ẽij in G 1 . from the identity

K G1 = -4H 1 + 3E + F = K X4 + E = -H 1 -H 2 + E we deduce the relation 3H 1 = H 2 + 2E + F .
The exceptional locus of p 2 defines a collection of 10 planes in X 4 , contracted to the ten singular points of C 4 , and that we can identify with their isomorphic images in G(2, V 4 ). Recall that in this Grassmannian we have the five planes π 1 , . . . , π 5 .

Proposition 4.7. The resulting collection of 10 + 5 planes in G(2, V 4 ) is in natural correspondance with the Cremona-Richmond configuration.

4.5.

Incidences with the Segre cubic. Now we relate the two varieties X 3 and X 4 by considering the incidence correspondence

I = {(A 1 , B 1 ), (U 2 , U 3 ) ∈ X 3 × X 4 , A 1 ⊂ U 2 , B 1 ⊂ U 3 }.
Recall that by definition, B 1 is (contained in) the kernel of θ(v) for v ∈ A 1 , while U 3 is the linear span of the kernels of the two forms θ(u) for u ∈ U 2 ; this kernel depends quadratically on u since it is given by θ(u) ∧ θ(u). This implies that I is (generically) a P 1 -bundle over X 4 , and (generically) a P 2 -bundle over X 3 . We have a commutative diagram I x x r r r r r r r r r r r r

' ' ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ X 3 F l(1, 2, V 4 ) y y r r r r r r r r r r & & ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ X 4 P(V 4 ) G(2, V 4 ).
One easily checks that:

Proposition 4.8. The base locus of the birational projection I -→ F l(1, 2, V 4 ) is the union of five disjoint planes.

Proof. Denote by ψ i the plane G(2, V 4 ) parametrizing the lines in P(V 4 ) that pass through p i , and by Ψ i its lift in F l(1, 2, V 4 ). The preimage in I of a point in ψ i is given by a flag B 1 ⊂ U 3 ⊂ V 5 such that U 1 is contained in the kernel K i of ω i and U 3 contains 4.6. Projective duality. We have seen that X 4 is birationally equivalent to the projective bundle P X2 (Q) over the del Pezzo surface X 2 . Since Q ∨ has no section, we would rather write it as P = P X2 (∧ 2 Q ∨ ), in which case the relative tautological bundle O P (-1) sends P to P(∧ 2 V ∨ 5 ) ≃ P(∧ 3 V 5 ), the image being C 4 ⊂ G(3, V 5 ). We are then in the context of Homological Projective Duality for projective bundles, according to which P → P(∧ 2 V ∨ 5 ) is dual to P * → P(∧ 2 V 5 ), with P * the projective bundle P X2 (W ∧ V 5 ), where W denotes the rank to tautological bundle. Proposition 4.9. The image of P * → P(∧ 2 V 5 ) is an octic hypersurface in P(∧ 2 V 5 ), containing the Grassmannian G(2, V 5 ) in its singular locus.

Proof. First consider the full projective bundle P G(2,V5) (W ∧ V 5 ) and its projection to P(∧ 2 V 5 ). The generic fiber is a copy of Q 3 (while the special fibers, that occur over G(2, V 5 ), are codimension two Schubert cycles). When we restrict to X 2 , we cut the fibers by linear spaces of codimension four. Generically, they meet the span of the fiber at one point; in codimension one, this point will be on the fiber itself. This implies that P * → P(∧ 2 V 5 ) is birational onto its image, which must be a hypersurface. As usual, we compute the degree of this hypersurface as

P * O P * (1) 8 = X2 s 2 (W ∧ V 5 ) = G(2,V5) (2σ 2 + σ 11 )σ 4 1 = 8.
Here we used exact sequences to compute the Segre class s(W

∧V 5 ) = c(Q) 5 c(S 2 U ), with c(Q) = 1 + σ 1 + σ 2 + σ 3 and c(U ) = 1 -σ 1 + σ 11 .
Over a point W 0 of the Grassmannian, the fiber of P G(2,V5) (W ∧V 5 ) → P(∧ 2 V 5 ) is the Schubert cycle of planes W meeting W 0 along at least a line. It is desingularized by a P 3 -bundle over P(W 0 ). If we fix a line L ⊂ W 0 , there exists a plane W ⊃ L in X 2 if and only if the four linear forms θ(L, •) on V 5 /L are linearly dependent. This defines a section of ∧ 4 (Q ∨ (1)) = O(3) over P 1 , and we conclude that the general fiber of P X2 (W ∧ V 5 ) → P(∧ 2 V 5 ) over G(2, V 5 ) consists in three points. Since this morphism is birational onto its image, Zariski's main theorem implies that G(2, V 5 ) is contained in the singular locus.

Symmetries

The symmetries of the Segre cubic primal must be reflected in X 4 . In this section we describe the symmetries of X 4 in some detail. In particular we will prove: Proposition 5.1. The generic stabilizer of the action of P GL(V 4 ) × P GL(V 5 ) on

P(V ∨ 4 ⊗ ∧ 2 V ∨ 5 )
is the symmetric group S 5 . What is classically known, as we mentionned in the introduction, is that the action of P GL(V 4 ) × P GL(V 5 ) on V ∨ 4 ⊗ ∧ 2 V ∨ 5 is prehomogeneous. The representative of the open orbit given in [START_REF] Ozeki | On the microlocal structure of the regular prehomogeneous vector space associated with SL(5) × GL(4)[END_REF] is

θ = e ∨ 1 ⊗ (f 25 -f 34 ) + e ∨ 2 ⊗ (f 15 -f 24 ) + e ∨ 3 ⊗ (f 23 -f 14 ) + e ∨ 4 ⊗ (f 45 -f 12 ), with the notation f ij = f ∨ i ∧ f ∨ j .
The corresponding points in P(V 4 ) and rank two forms are easy to identify; we get

p 1 = e 2 + ie 4 , ω 1 = (f 1 + if 4 ) ∧ (f 2 + if 5 ), p 2 = e 2 -ie 4 , ω 2 = (f 1 -if 4 ) ∧ (f 2 -if 5 ), p 3 = e 1 + e 3 + e 4 , ω 3 = (f 2 + f 4 ) ∧ (f 1 + f 3 + f 5 ), p 4 = e 1 + je 3 + j 2 e 4 , ω 4 = (f 2 + j 2 f 4 ) ∧ (f 1 + j 2 f 3 + jf 5 ), p 5 = e 1 + j 2 e 3 + je 4 , ω 5 = (f 2 + jf 4 ) ∧ (f 1 + jf 3 + j 2 f 5 ).
Here j and i are primitive fourth and third roots of unity. Each pair ω p , ω q defines two planes in V ∨ 5 whose common orthogonal is a line [e pq ]. Then the planes of the Cremona-Richmond configuration are obtained as follows: P pq is generated by the three points e ij , e jk , e ik for ijk distinct from pq; and P p is generated by the four points e ip for i = p. Explicitely, the ten vectors e pq can be chosen as follows: e 12 = (0, 0, 1, 0, 0) e 24 = (i, -j 2 , -2ij, 1, ij 2 ) e 13 = (1, -i, -2, i, 1) e 25 = (i, -j, -2ij 2 , 1, ij) e 14 = (-i, -j 2 , 2ij, 1, -ij 2 ) e 34 = (1, 0, j 2 , 0, j) e 15 = (-i, -j, 2ij 2 , 1, -ij) e 35 = (1, 0, j, 0, j 2 ) e 23 = (1, i, -2, -i, 1) e 45 = (1, 0, 1, 0, 1).

Each ω i defines a plane π i in V ∨ 5 , from which we can deduce a collection of hyperplanes π ij = π i + π j and points p ijk = π i ∩ (π j + π k ). Proposition 5.2. For any permutation i, j, k, l, m of 1, . . . , 5, p ijk = p ilm .

Proof. Explicit check.

We have no convincing explanation of this coincidence, but as a consequence, we don't get thirty but only fifteen points in P(V ∨ 5 ). Obviously, p ijk belongs to π i , hence to any of the four hyperplanes π il , l = i. Conversely, π ij contains the three points p iab plus the three points p jcd .

Proposition 5.3. The fifteen points π ijk and the ten hyperplanes π ij in P(V 4 ) form a configuration (15 4 , 10 6 ).

We thus recover the abstract configuration classically defined by the Segre primal. In particular the fifteen points π ijk should be in natural correspondence with planes in the Segre primal.

Automorphisms in P GL(V 4 ) × P GL(V 5 ) that fix θ are in bijective correspondence with elements of P GL(V 5 ) fixing the four-plane generated by the ω i 's. Automatically such an automorphism will preserve the set of five planes π 1 , . . . , π 5 , hence the collection of the thirty points p ijk .

In order to show that any permutation of the five planes can be lifted to P GL(V 5 ), it is enough to lift two generators of S 5 , say a transposition and a complete cycle. By sending f i to ǫ i f i with ǫ i = 1 for i odd and ǫ i = -1 for i even, we exchange π 1 and π 2 and let the three other planes fixed. So let us turn to a maximal cycle. We claim that the cycle (12345) ∈ S 5 can be lifted to the tranformation of GL(V 5 ) given by

f 1 → j 3 f 1 -2ijf 2 + j 3 f 3 -ijf 4 + 4j 3 f 5 , f 2 → -2i 3 f 1 -f 2 + i 3 f 3 + i 3 f 5 , f 3 → 4j 2 3 f 1 + 4ij 2 f 2 -2j 2 3 f 3 -4ij 2 f 4 + 4j 2 3 f 5 , f 4 → -ij 3 f 1 -ij 3 f 3 -jf 4 + 2ij 3 f 5 , f 5 → 4 3 f 1 + if 2 + 1 3 f 3 + 2if 4 + 1 3 f 5 .
Corollary 5.4. The automorphism group of the Fano fourfold X 4 is Aut(X 4 ) = S 5 .

Proof. An automorphism of X 4 is induced by a linear transformation in P GL(V 4 )× P GL(V 5 ) preserving θ. Considered as a homomorphism from V 4 to ∧ 2 V ∨ 5 , θ defines a codimension four linear section of G(2, V 5 ), that is a degree five del Pezzo surface S 5 . This implies that Stab(θ) embeds into Aut(S 5 ), which is well-known to be S 5 . Since we know by the previous computations that Stab(θ) contains S 5 , we are done.

Once we identify S 5 with the stabilizer of θ in SL(V 4 )× SL(V 5 ), we get actions of S 5 on V 4 and V 5 , clearly irreducible. Up to the sign representation there is a unique irreducible representation of S 5 of dimension 4, and a unique one of dimension 5. The complex [START_REF] Avilov | Automorphisms of singular three-dimensional cubic hypersurfaces[END_REF] shows that C 4 is cut out by a family of quadrics on G(2, V 5 ) parametrized by V 4 , hence a S 5 -invariant copy of V 4 inside S 22 V 5 . We will show later on that this copy is unique. (This point of view from finite group representation theory is typically used in [START_REF]Quartic surfaces with icosahedral symmetry[END_REF]. Something with the same flavour has been done in [START_REF] Bauer | s 5 -equivariant syzygies for the del Pezzo surface of degree 5[END_REF] for the quintic del Pezzo surface.)

We use the character table of S 5 (see for example [START_REF] Dolgachev | Geometry of the Wiman-Edge pencil, I: algebrogeometric aspects[END_REF]) to compute some plethysm and tensor product representations. Recall that S 5 has irreducible representations of dimension 1, 1, 4, 4, 5, 5, 6 that we denote by U 1 , U - 1 , U 4 , U - 4 , U 5 , U - 5 , U 6 . All these representations are self-dual, being defined over the real numbers. Concretely,

U 1 is the trivial representation, U - 1 is the sign representation, U 4 is the natural representation, U - 4 = U 4 ⊗ U - 1 and U 6 = ∧ 2 U 4 . One computes that S 2 U 4 = U 5 ⊕ U 4 ⊕ U 1 , ∧ 2 U 5 = U - 4 ⊕ U 6 .
The last decomposition implies in particular that (U - 4 ) ∨ ⊗ ∧ 2 U ∨ 5 contains a unique S 5 -invariant tensor θ S5 , up to scalars.

At this point it could therefore be reasonable to reverse the whole process and start from the representation theory of S 5 . One should be able to check directly that θ S5 is generic, and then we should get θ S5 -invariant descriptions of all the objects we have been studying.

Note that S 2 U 4 = U 5 ⊕ U 4 ⊕ U 1 allows to construct U 5 from U 4 , as the space of quadrics which are apolar to the obvious invariant cubic. In coordinates x 1 , . . . , x 5 permuted by S 5 , the representation U 4 is the hyperplane

x 1 + • • • + x 5 = 0, the invariant cubic is x 3 1 + • • • + x 3 5
and the apolar quadrics are of the form i =j a ij x i x j with a ij = a ji ∀i = j, i =k a ik = 0 ∀k.

We get ten indeterminates and five independent relations, consistently with the fact that these quadrics should span a copy of V 5 . Inside the space V 5 of apolar quadrics to the invariant cubic, note that we have q ij,kl = (x i -x j )(x k -x l ) for i, j, k, l distinct integers. These quadrics are subject to the Plücker type relations q ij,kl -q ik,jl + q il,jk = 0. This suggests to define the following elements of ∧ 2 V 5 :

Q 1 = q 23,
45 ∧ q 24,35 , Q 2 = q 13,45 ∧ q 14,53 , Q 3 = q 12,45 ∧ q 14,25 , Q 4 = q 12,35 ∧ q 13,52 , Q 5 = q 12,34 ∧ q 13,24 .

Obviously, for any permutation σ ∈ S 5 one must have σ(Q i ) = ±Q σ(i) . We also let, for a pair i = j with complement p, q, r in 1 . . . 5, Q i,j = q ip,qr ∧ q jp,qr + q iq,rp ∧ q jq,rp + q ir,pq ∧ q jr,pq .

Proposition 5.5. The action of S 5 on Q 1 , . . . , Q 5 gives a copy of the representation U - 4 in ∧ 2 U 5 . Similarly, the action of S 5 on Q i,j , 1 ≤ i < j ≤ 5 gives a copy of the representation U 6 .

What have we gained in doing all that? First, we get a better, more symmetric normal form for the generic θ than that of Ozeki, as

θ S5 = e 1 ⊗ Q 1 + e 2 ⊗ Q 2 + e 3 ⊗ Q 3 + e 4 ⊗ Q 4 + e 5 ⊗ Q 1 , with e 1 + • • • + e 5 = 0.
Also, we can make explicit the quadratic equations of C 4 . A character computation yields: Lemma 5.6. The multiplicity of U - 4 inside S 2 (∧ 2 U 5 ) is equal to one. So the space of quadratic equations we are looking for is uniquely defined in terms of the S 5 -action. Moreover, recall that ∧ 2 U 5 = U - 4 ⊕ U 6 . Another character computation shows that the copy of U - 4 that we are looking for inside

S 2 (∧ 2 U 5 ) is in fact contained inside U - 4 ⊗ U 6 = U - 4 ⊗ ∧ 2 (U - 4 ) ⊂ U - 4 ⊗ End(U - 4 ) (recall that U -
4 is self-dual). So there is an obvious map to U - 4 , and dually, this says that the space of quadrics we are looking for is generated by the five quadrics

CQ i = j =i Q i,j Q j , 1 ≤ i ≤ 5.
Remark. Since Aut(C 3 ) = S 6 , certain automorphisms of the Segre primal do not lift to X 4 . would it be possible that S 6 act on X 4 by birational transformations?

6. The Chow ring of X 4

In this section we completely determine the Chow ring of X 4 , with its structure of S 5 -module. Let us start with the Picard group.

From the relation 3H 1 = H 2 + 2E + F that we found on G 1 , we compute that

H 4 1 = 2, H 3 1 H 2 = 6, H 2 1 H 2 2 = 13, H 1 H 3 2 = 14, H 4 2 = 12.
The Picard group is generated by H 1 , H 2 and the five components of F , which are permuted by S 5 . We deduce: Proposition 6.1. The Chow ring of X 4 is generated by A * (G) and the five divisors F 1 , . . . , F 5 . As a representation of S 5 , the Picard group decomposes as

P ic(X 4 ) ⊗ Z C = 2U 0 ⊕ U 4 .
We know by Proposition 4.1 that the middle dimensional Chow group A 2 (X 4 ) has dimension 17, and we expect that the invariant part has dimension four, with two classes coming from G(2, V 4 ) and two other classes from G(2, V 5 ). We will show that are all come (at least over Q) from products of divisor classes.

We compute the multiplicative structure of the Chow ring by embedding it in the Chow ring of G 1 , that we shall now describe. First, the Chow ring of G 0 is generated by the Chow ring of G = G(2, V 4 ) and the ten exceptional divisors E 0 pq of the blow-up b 0 = Bl 0 , such that (E 0 pq ) 4 = -1, E 0 pq E 0 p ′ q ′ = 0 for {p, q} = {p ′ , q ′ }, E 0 pq .b * 0 C = 0 for any class C ∈ A * (G) of positive degree. After this first blow-up, the five planes π 1 , . . . , π 5 give five disjoint surfaces Σ 1 , . . . , Σ 5 , each one being a plane blow-up in five points, that is a del Pezzo surface of degree 5. We denote the four exceptional lines in Σ p by ℓ q p , whose image in G is the point π pq , for q = p. The second blow-up b 1 = Bl P is the blow-up of these five surfaces. We denote by F 1 p the five exceptional divisors, and by E 1 pq the strict transforms of the divisors E 0 pq . Since F 1 p = P(N p ), for N p the normal bundle of Σ p inside G 0 , we need to describe this normal bundle. Recall that when one blows up one point in a smooth variety X, creating an exceptional divisor E inside the blow-up Y π → X, the tangent exact sequence is 0 → T Y → π * T X → i * T E → 0, where i : E → Y denotes the inclusion. Since the normal bundle of π p inside the Grassmannian G is the quotient bundle Q, we get the following diagram:

0 0 0       0 ----→ T Σ p ----→ T G 0|Σp ----→ N p ----→ 0       0 ----→ b * 0 T π p ----→ b * 0 T G |πp ----→ b * 0 Q ----→ 0       0 ----→ ⊕ q =p T ℓ q p ----→ ⊕ q =p T E 0 pq ----→ ⊕ q =p N q p ----→ 0       0
0 0 Here we denoted by N q p the normal bundle to ℓ q p ≃ P 1 inside E 0 pq ≃ P 3 , which is just O ℓ q p (1) ⊕ O ℓ q p (1). We deduce the Segre class

s(N p ) = s(b * 0 Q) q =p c(O ℓ q p (1)) 2 ∈ A * (Σ p ).
One the one hand, the Segre class s(Q) equals the Chern class of the tautological bundle on G, that is s(Q) = 1-H 1 +σ 11 , and the Schubert class σ 11 restricts to zero This gives F 4 = F 4 1 + • • • + F 4 5 = 60, hence F 4 p = 12. (But note that this is not equal to (c * F p ) 4 = -4, as a consequence of the fact that F p contains four of the quadratic surfaces blown-up by c.)

The other intersection numbers can be computed by pulling-back by c and using Lemma 6.2. Proposition 6.6. The square map S 2 A 1 (X 4 ) -→ A 2 (X 4 ) is surjective. As a consequence, the S 5 -module structure of A 2 (X 4 ) is

A 2 (X 4 ) = 4U 0 ⊕ 2U 4 ⊕ U 5 .
Proof. The decomposition of the S 5 -module S 2 A 1 (X 4 ) is 4U 0 ⊕ 3U 4 ⊕ U 5 , the sum of three isotypic components, and the kernel of the square map must decompose accordingly.

First consider the four invariant classes H 2 1 , H 1 F, F (2) , F (11) , where

F (2) = p F 2 p , F (11) = p<q F p F q .
Suppose that there is a relation aH 2 1 + bH 1 F + cF (2) + dF (11) = 0. Multiplying successively by H 2 1 , H 1 F p , F 2 p , F p F q and using the results of Proposition 6.5, we deduce that 2a

-5c = 0, b + c = 0, a + b + 16c -8d = 0, 4c -d = 0, hence a = b = c = d = 0.
Now consider the possibility that U 5 be contained in the kernel of the square map. We claim that U 5 is embedded inside S 2 A 1 (X 4 ) as the space of linear combinations p =q a pq F p F q with a pq = a qp and r a pr = 0 for all p, q. Indeed, this defines an invariant five-dimensional subspace of S 2 A 1 (X 4 ), not containing any invariant class, so it must be U 5 . A typical element is

3F p F q -(F p + F q ) r =p,q F r + s,t =p,q F s F t .
If this was zero in A 2 (X 4 ), multiplying by F p F q would imply that the intersection number F 2 p F 2 q = 0, which is not the case. We can conclude that the kernel of the square map must be contained in the isotypical component 3U 4 of S 2 A 1 (X 4 ), which is generated by the three copies of U 4 respectively obtained as the linear combinations p a p H 1 F p , p a p F F p and p a p F 2 p for p a p = 0. A copy of U 4 in the kernel corresponds to a relation of the form

uH 1 F p + vF F p + wF 2 p = I ∀p,
for I an invariant class. Since I is invariant, multiplying by H 1 F p and H 1 F q must then give the same intersection number, which gives the relation -u -v -w = 0. Similarly, multiplying by F 2 p or F 2 q must give the same result, that is -u+4v+12w = -v + w. Finally, multiplying by F p F q or F q F r with q, r distinct from p must also give the same result, that is -v -2w = 0. These three equations are linearly dependent and reduce to u = w and v = -2w, which proves that there is a unique copy of U 4 in the kernel of the square map. This concludes the proof.

Threefolds. Consider the two families of divisors in X 4 given by sections of H 1 and H 2 , respectively. Since a general hyperplane section in G(2, V 4 ) ≃ Q 4 will avoid the ten points π pq , the first ones are just blowups of five disjoint lines in Q 3 . For the same reason, the second ones, say Z 3 , are isomorphic with their images in G(3, V 5 )∩ H 2 , which are codimension two degeneraci loci defined by the condition that the morphism ∧ 2 V → V ∨ 4 has rank exactly two. Its image is then the pullback from G(2, V 4 ) of the dual quotient bundle Q ∨ . In particular we get an exact sequence

0 → O(1, -2) → p * 2 (∧ 2 V ) → p * 1 Q ∨ → 0 on Z 3 .
This shows in particular that O(-1, 2), the restriction of 2H 2 -H 1 = 5H 1 -2F , is generated by sections on Z 3 . The image in Z 3 is the closure of the planes U ⊂ V 4 such that the image of S 2 U → ∧ 2 V 5 ≃ V 5 is isotropic with respect to some three-form on V 5 . This defines a section of ∧ 3 (S 2 U ) ∨ = det(U ∨ ) 3 , so that the image of Z 3 in G(2, V 4 ) is a singular cubic hypersurface. K3 surfaces. By taking sections of H 1 ⊕ H 2 in X 4 , we get a family of smooth K3 surfaces S in X 4 . We denote by h 1 , h 2 , f 1 , . . . , f 5 the restriction to S of the divisors H 1 , H 2 , F 1 , . . . F 5 . Proposition 6.7. The intersection numbers of these divisors in S are

h 2 1 = 6, h 2 2 = 14, h 1 h 2 = 13, h 1 f i = 1, h 2 f i = 5, f i f j = -2δ ij .
Proof. This is an immediate consequence of the computations above, since for two divisors A, B on X restricting to a, b on S, we have ab = ABH 1 H 2 .

An obvious consequence is that h 1 , f 1 , . . . , f 5 are linearly independent. Moreover, the curves C i = F i ∩ S are (-2)-curves on S, mapping to lines on G(2, V 4 ) and to rational quintics in G(2, V 5 ). The divisor 5h 1 -h 2 = 2h 1 + f should contract these five (-2)-curves to the five singular points of a surface S. Note that this is a divisor of degree 34, so S could be a degeneration of a smooth K3 surface of genus 18. Mukai described the generic such K3 surface as the zero locus in OG(3, 9) of five sections of the rank two spinor bundle. What is the connection? Note that we have a family of surfaces of dimension 5 + 9 = 14 = 19 -5, which is coherent with the expectation that imposing 5 nodes on a K3 surface of genus 18 should give five independent conditions.

The Igusa quartic and the Coble fourfold

Given a linear form h on V 5 , there is an associated quadratic form Q on V 4 :

Q h (v) = h ∧ θ(v) ∧ θ(v) ∈ ∧ 5 V ∨ 5 ≃ C. Proposition 7.1.
The quartic det(Q h ) = 0 is the Igusa quartic in P 4 = P(V ∨ 5 ). Proof. Recall that the generic point of the Segre cubic C 3 ⊂ P(V 5 ) is the kernel of one of the two-forms θ(v), and that we can get this kernel as the line generated by θ(v) ∧ θ(v) ∈ ∧ 4 V ∨ 5 ≃ V 5 . At this generic point, the affine tangent space to the Segre cubic is therefore the hyperplane of V 5 generated by the vectors of the form θ(v) ∧ θ(w) ∈ ∧ 4 V ∨ 5 ≃ V 5 . This hyperplane is defined by a linear form h v ∈ V ∨ 5 that vanishes on these vectors, which exactly means that h v ∧ θ(v) ∧ θ(w) = 0 for any w ∈ V 5 . In other words, Q hv (θ(v), θ(w)) = 0 for all w ∈ V 5 , which means that θ(v) belongs to the kernel of the quadratic form Q hv . In particular the latter is degenerate.

We have thus proved that the generic point of the projective dual variety of the Segre cubic is contained in the quartic hypersurface det(Q h ) = 0. But this projective dual is well-known to be the Igusa quartic in P(V ∨ 5 ), and these two quartics have to coincide. This yields a simple determinantal representation of the Igusa quartic. Using Ozeki's representative we get

Q h =     2h 1 -h 2 -h 3 -h 5 -h 2 2h 3 -h 4 0 -h 3 -h 4 2h 5 -h 1 -h 5 0 -h 1 2h 3     ,
whose determinant is readily computed to be -det(Q h ) = 4h One can consider inside P(V ∨ 5 ) × G(2, V 4 ) the locus J 5 of pairs ([h], U ) such that U is isotropic with respect to Q h . Recall that OG Q (2, 4) = P 1 ∪ P 1 is the disjoint union of two smooth conics when Q is non degenerate. When Q is a quadratic form of corank one on V 4 , the corresponding orthogonal Grassmannian OG Q (2, V 4 ) is a single conic (while if Q has corank two, OG Q V 4 ) is the union of two planes meeting at one point, defined by the kernel). This means that the Stein factorization of the projection of J 5 to P(V ∨ 5 ) is J 5 → Cob 4 → P(V ∨ 5 ) where Cob 4 is the double cover of P(V 5 ) branched over the Igusa quartic: that is, the Coble fourfold [START_REF] Cheltsov | Coble fourfold, s 6 -invariant quartic threefolds, and Wiman-Edge sextics[END_REF].

On the other hand, denote by Q X4 the pull-back to X 4 of the rank two quotient bundle on G(3, V 5 ). The P 1 -bundle P(Q X4 ) over X 4 has a natural map to G(2, V 4 )× P(V ∨ 5 ) and we claim that its image is precisely J 5 . Indeed, a generic element (U, V ⊂ H) of G(2, V 4 ) × F l(3, 4, V 5 ) belongs to P(Q ∨ X4 ) when V , hence H, contains the kernels of all the two-forms θ(v), v ∈ U . But if h is a linear form defining H, the condition that H contains the kernel of θ(v) exactly means that h ∧ θ(v) ∧ θ(v) = 0, hence our claim. Moreover the projection map P(Q ∨ X4 ) → J 5 is birational, being clearly bijective outside the five special planes in G(2, V 4 ). So we get a diagram

P(Q ∨ X4 ) bir { { ① ① ① ① ① ① ① ① ① P 1 # # • • • • • • • • J 5 conic # # • • • • • • • • • X 4 Cob 4
where the south-west arrow is a conic bundle, at least generically. But the picture does not seem to recover the small resolutions of Cob 4 described in [START_REF] Cheltsov | Coble fourfold, s 6 -invariant quartic threefolds, and Wiman-Edge sextics[END_REF].

Local rigidity

Since V ∨ 4 ⊗ ∧ 2 V ∨ 5 is prehomogeneous, we expect that X 4 has strong rigidity properties. What we can prove is the following statement. Proposition 8.1. X 4 is locally rigid and has finite automorphism group.

Proof. Local rigidity is equivalent to the vanishing of H 1 (T X 4 ). In order to check this, as usual we rely on the normal exact sequence, which yields an exact sequence of cohomology groups

H 0 (T G |X4 ) -→ H 0 (E |X4 ) -→ H 1 (T X 4 ) -→ H 1 (T G |X4 ).
So local rigidity will follow from the following statements, to be proved separately:

Proposition 4 . 1 .

 41 X 4 is a rational Fano fourfold of index one. Its cohomology is pure, with h 1,1 = 6 and h 2,2 = 17. Moreover h 0 (-K X4 ) = 40 and K 4 X4 = 172.

Proposition 4 . 3 .

 43 Each singular point of C 4 defines a plane in the Segre cubic primal C 3 . The five remaining planes are the projectivized kernels of the five singular form ω 1 , . . . , ω 5 .

  4 3 + 4h2 3 (3h1 h 5 -h 2 h 4 )-4h 3 (h 3 1 + h 3 5 + h 1 h 2 4 + h 2 2 h 5 )+ (h 1 h 2 -h 4 h 5 ) 2 .

on π p . On the other hand, on the del Pezzo surface Σ p we have O ℓ q p (1) = O(-ℓ q p ) |ℓ q p , from which we get the Segre class s(O ℓ q p (1)) = 1 + ℓ q p + 2(ℓ q p ) 2 . Finally,

We can deduce several intersection numbers, since for any class C 3-k of degree 3-k on G 0 , we have the classical formulas

Lemma 6.2.

Note also that F 1 p does not meet E 1 rq for r, q = p, but it meets E 1 pq transverselly along the surface S q p = b -1

Applying the previous formula to C 3-k = (E 0 pq ) 3-k we get:

On the other hand, E 0 pq gets blown-up along the two-skew lines ℓ q p and ℓ p q , and its strict transform E 1 pq is contracted to the quadratic surface ℓ q p × ℓ p q in X 4 . This surface is also the intersection of F p and F q in X 4 , in particular it is contained in F p . We deduce, denoting Bl Q by c, that

Summing up over p, we get the relation c

) by the linear system |I π (3H 1 )| of cubics vanishing along the union π of the five planes π 1 , . . . , π 5 .

We have enough information to describe the full intersection product on X 4 .

Proposition 6.5. The nonzero intersection numbers among the divisor classes H 1 , F 1 , . . . , F 5 are the following, for 1 ≤ p = q ≤ 5:

Moreover we always have H 1 F p F q = 0 for p = q and F p F q F r = 0 for p = q = r = p.

Proof. The values of F 3

p H 1 and F 2 p H 2 1 can be computed directly by restricting to a general hyperplane or a general codimension two section of G; then we avoid the points π qr and we are reduced to compute the self-intersection of the exceptional divisor for the blow-up of a line in a three-dimensional quadric, or a point in a surface. Then we can deduce the value of F 4 p by computing the self-intersection of

The third statement follows from the fact that P(V ∨ 4 ⊗ ∧ 2 V ∨ 5 ) is prehomogeneous under P GL(V 4 )×P GL(V 5 ), more precisely from the fact that the orbit of [θ] is open, since this implies that the image of the natural differential sl 4 × sl 5 = H 0 (T G) -→ V ∨ 4 ⊗ ∧ 2 V ∨ 5 / θ sending X to X(θ) mod θ is surjective; since this morphism can also be defined by restricting first to X 4 and then composing with the morphism we are interested in, the latter must also be surjective.

In order to prove the second statement we twist by E the Koszul complex resolving the structure sheaf of X 4 . By standard cohomological arguments, it is enough to check that H 0 (G, End 0 (E)) = 0 and H i (G, E ⊗ ∧ i+1 E ∨ ) = 0 for any i > 0. For the first claim, observe that

is in fact acyclic. For the second claim, check that E ⊗ ∧ i+1 E ∨ is also acyclic for any i > 0. Similarly, in order to prove the first statement we need to check that H i+1 (G, T G⊗∧ i E ∨ ) = 0 for any i ≥ 0, which is again a straightforward application of Bott's theorem.

By the same type of arguments (or using a computer to check that χ(T X 4 ) = 0), we deduce that H 0 (T X 4 ) = 0, which implies that the automorphism group is discrete, hence finite since X 4 is Fano.

The question remains open, whether X 4 is also globally rigid, which would be remarkable for a Fano fourfold with such a big Picard number. The first thing to check is whether X 4 remains smooth when we degenerate θ to the codimension one orbit. If yes, we would get a similar situation to the case of codimension two linear sections of the spinor tenfold (which has Picard number one, though).

Another question one may ask is whether the quotient bundle restricted to X 4 is rigid. In other words, is the morphism to G(3, V 5 ) uniquely defined?

Higher dimensions

Let us briefly describe the higher dimensional models. Proposition 9.1. X 6 is a rational Fano sixfold of index one and Picard rank two.

The projection of X 6 to G(3, V 5 ) is birational, with non trivial fibers isomorphic to P 1 over the smooth locus of C 4 , and to P 2 over its ten singular points.

The projection to P(V 4 ) is a Q 3 -bundle outside P 0 , with five four-dimensional fibers over P 0 .

From this description and that of X 4 , we deduce that in the Grothendieck ring of varieties one has the relation [

]. This yields the Poincaré polynomial of X 6 , P X6 (t) = 1 + 2t + 8t 2 + 9t 3 + 8t 4 + 2t 5 + t 6 . Proposition 9.2. X 8 is a Fano eightfold of pseudo-index three, while X ′ 8 is Fano eigthfold of index three.

Proposition 9.3. The projections of X 8 , X ′ 8 to G(2, V 5 ) are dual P 2 -fibrations over the complement of a del Pezzo surface of degree five, the exceptional fibers being isomorphic to P(V 4 ) and G(2, V 4 ) respectively.

We can readily deduce that X 8 and X ′ 8 have pure cohomology, with Poincaré polynomials P X8 (t) = 1 + 2t + 4t 2 + 6t 3 + 11t 4 + 6t 5 + 4t 6 + 2t 7 + t 8 , P X ′ 8 (t) = 1 + 2t + 5t 2 + 11t 3 + 13t 4 + 11t 5 + 5t 6 + 2t 7 + t 8 .

Of course X 6 , X 8 , X ′ 8 inherit the same symmetries as X 4 . E. Fatighenti and F. Tanturri checked the necessary vanishing conditions to establish, as for X 4 , that they are also infinitesimally rigid.


Vincente Cardona

Viviana Lago

THE EFFECT OF RAREFACTION LEVEL ON SHOCK/SHOCK INTERFERENCES APPLIED TO ATMOSPHERIC RE-ENTRY: EXPERIMENTAL STUDY IN A SUPERSONIC RAREFIED FLOW

Keywords: Aerodynamics, Atmospheric re-entry, Supersonic rarefied flow, Shock/shock interferences, Wind tunnel

In the case of proximal bodies flying in Earth's atmosphere, shock/shock interferences can occur. According the altitude, interacting bodies suffer different rarefaction levels which impact their aerodynamics and trajectories. To better understand the effect of rarefaction level, the six types of shock/shock interferences were studied with two iso-Mach number nozzles of the MARHy wind tunnel with different Reynolds and Knudsen numbers: M4 -2.67 Pa and M4 -8 Pa nozzles. Models used for the experiments are a couple of spheres of same diameter, representative of 1-m diameter blunt bodies interacting at 76 and 69 km in altitude. Results are obtained with three different diagnostics allowing to explore the influence of the viscosity on shock wave interferences. They enlighten differences in terms of shock wave shapes, pressure distribution on the following sphere, and values of forces and wall pressures.

INTRODUCTION

The studies of proximal bodies flying in Earth's atmosphere can be associated to a multitude of applications whether in the case of a spacecraft launch or the re-entry of space debris. In the case of spacecraft launch, shock/shock interferences (SSI) of a Two-Stage-To-Orbit were studied to propose a new booster model that enable lower heat transfer rate [START_REF] Hanai | Two-stageto-orbit booster configuration for reducing aerodynamic heating at hypersonic speed[END_REF] [START_REF] Jia | Aerodynamic interactions of a reusable launch vehicle model with different nose configurations[END_REF], or to better understand the stage separation [START_REF] Bordelon | Stage separation wind tunnel tests of a generic tsto launch vehicle[END_REF] that occurs at 70 to 90 km. In the case of space debris or meteoroid re-entry, the fragmentation occurring around 80 km in altitude also results in SSI. Proximal bodies interaction have often been studied [START_REF] Laurence | Proximal bodies in hypersonic flow[END_REF]. Most of these studies were conducted in the continuum regime, numerically by using Computational Fluid Dynamic (CFD) and experimentally. However, as the flight altitude increases, the rarefaction level impacts the aerodynamics due to viscous effects. Some numerical studies took into account rarefaction Fig. 1. MARHy wind tunnel effects [START_REF] White | The effect of increasing rarefaction on the edney type iv shock interaction problem[END_REF], but few are those who experimented SSI in a rarefied flow [START_REF] V V Riabov | Shock interference in hypersonic rarefied-gas flows near a cylinder[END_REF]. The present study proposes to better understand the effect of the rarefaction level on SSI occuring at high altitudes. This work is focused on the interaction of bow shock waves produced by two spheres at two different levels of rarefaction but same velocity. Results will help deepen the physics of SSI in a rarefied flow, and will serve as a database for the completion of numerical method.

EXPERIMENTAL SETUP

Test facility

The experiments were carried out in the MARHy wind tunnel (Fig. 1), at ICARE, CNRS, Orléans. MARHy is a low density facility that produces flows with Mach numbers from 0.8 to 20 thanks to a large set of nozzles. The powerful pumping group, composed with 16 pumps, allows to obtain steady operating conditions with no limit in time. Regardless the nozzle used, the flow is laminar and homogeneous in the isotropic core (which diameter and length vary according the nozzle used). For this study, the M4 -2.67 Pa (Nozzle a) and M4 -8 Pa (Nozzle b) nozzles presented in Table 1 were used to investigate the impact of the rarefaction level on shock/shock interferences. The stagnation conditions (subscript o ) refer to the settling chamber, and the free-stream conditions (subscript ∞ ) are those in the test chamber. The two chambers are separated by a nozzle which imposes flow conditions in terms of freestream pressure and Mach number, due to its specific design. The free-stream Mach number only depends on the stagnation and free-stream pressures. The stagnation pressure is accurately regulated with the opening of a micro-valve at the gas inlet. The free-stream pressure is set by commissioning the needed number of pumps, and then regulated by adjusting the opening of a butterfly valve placed between the diffuser and the pumping group. The stagnation and free-stream pressures are respectively monitored with a 10 Torr and a 0.1 Torr MKS absolute pressure sensors accurate at 0.15% and 0.12% of reading.

Models

As for many studies on proximal bodies, the experimented models are spheres. The two spheres are equal in size (16 mm in diameter) and are made of polyoxymethylene (POM). They are covered with a thin layer of black spray paint for visual reasons.

The Tsien number (1), also called the viscous parameter, is a dynamic similitude parameter that includes velocity and viscosity effects.

σ = M a ∞ √ Re ∞ (1)
It enables to characterize rarefaction level and determine the altitude experimentally simulated. Prévereaud et.al [START_REF] Prevereaud | Debris aerodynamic interactions during uncontrolled atmospheric reentry[END_REF] calculated the velocity of a 1 m-diameter sphere, at different altitudes. By analogy, a 16-mm sphere flying in the free-stream conditions detailed in In order to study the effect of rarefaction level on the SSI, three types of measurement were realized. The glow discharge technique enables to visualize shock-waves, and two quantitative measurements gave the drag and lift forces (measured with an aerodynamic balance), and the wall pressures (measured with a differential pressure sensor). The general set-up and the measurement techniques will be explained in the following sections.

General set-up

Figure 2 shows the general experimental set-up used. The first sphere (S 1 ) is fixed and held by a vertical profiled support mounted on a pneumatic rotary actuator that enables to remove S 1 from the flow. The other sphere (S 2 ) is held by the rear with an adapted profiled support mounted on a tri-axial displacement robot so that S 2 can be displaced behind S 1 .

Both spheres are placed in the middle plane of the isotropic core in the ⃗ y-direction. The diameter of the core is 7.5 cm for Nozzle a and 8 cm for Nozzle b. Due to the small diameter of the nozzles S 1 is placed below the horizontal plane of the nozzle, in order to enlarge the displacement area of S 2 . The placement of S 1 in the isotropic core does not impact much its shock-wave, as explained in a previous work [START_REF] Cardona | Shock/shock interferences in a supersonic rarefied flow: experimental investigation[END_REF]. S 2 is placed 24 mm behind S 1 is the ⃗ x-direction, and its altitude (in the ⃗ z-direction) varies to obtain the different types of SSI.

Glow-discharge technique

The visualization of the flow field is the first step that enables to determine the positions of the six types of SSI. In a rarefied flow, the density is so low that no optical technique based on

Aerodynamic balance

The aerodynamic balance used for the measurement of drag and lift forces was designed, tested and experimented by Noubel and Lago [START_REF] Noubel | Experimental analysis of waverider lift-to-drag ratio measurements in rarefied and supersonic regime[END_REF]. This device was also confronted to a pendulum method [START_REF] Cardona | Aerodynamic forces of interacting spheres representative of space debris re-entry: Experiments in a supersonic rarefied wind-tunnel[END_REF] that allows to measure drag forces, and the results of the two methods showed a good agreement. The balance is composed of two modules made of thin slates that deform according the perceived forces. The slats are equipped with strain gauges that send an electrical signal previously calibrated to allow its conversion into a force. The slats need to be positioned perpendicularly to the vertical axis (⃗ z) not to suffer gravity forces. Consequently, the balance measures the drag force in the ⃗ x-direction, and lift force in the ⃗ y-direction. So, contrarily to the set-up used for the visualization of shock waves, the axis of displacement of S 2 will not be vertical (in the ⃗ z-direction), but transversal (in the ⃗ y-direction). As a consequence, S 1 and S 2 are in the horizontal plane of the core middle, and S 1 is displaced in the ⃗ -y-direction to ensure a sufficiently large area of displacement for S 2 . This does not change the flow seen by S 2 while the two spheres are contained in the core since it is isotropic. As shown in Fig. 4, two measurements are realized. One measurement records, during 10 s, the forces perceived by S 2 behind S 1 (see the top image). As the pumping group constantly vacuum the test chamber, the measured forces correspond to the forces from the flow (the one perceived naturally by S 2 ) plus the suction forces from the pumping group. In order to remove suction forces from the measured values, an other measurement is recorded during 10 s, this time with a plate in front of S 2 (see the bottom image), so that S 2 and the balance do not see any force from the flow. The drag and lift forces are calculated by averaging each set and subtracting the suction forces from the global forces perceived by S 2 . For the accuracy of the measurements, these two recordings are realized ten times. Drag and lift forces are given with an accuracy of 0.1 mN.

Wall pressure measurement

For the wall pressure measurement, the general configuration and the sphere placement are the same as for the visualization (S 2 moves in the ⃗ z-direction).

Figure 5 shows the wall pressure set-up viewed from the side (top schemes) and from the top (bottom schemes). S 2 is drilled in two directions: in the ⃗ x-direction, the hole is 1.2 mm-diameter and serves as entrance for the air molecules; in the ⃗ y-direction, the 3 mm-diameter hole serves for the maintaining of S 2 (a tube is hermetically inserted in it), for its rotation (the tube is placed in a rotary system placed on the displacement robot), and for connection to the pressure sensor (via a flexible hose). Consequently, S 2 can be displaced behind S 2 , and rotated about the ⃗ y-direction. The pressure at its surface can be measured on the vertical line of the sphere for different types of SSI. A differential pressure sensor is made of two cavities separated by a sensitive deforming membrane. In order not to deform this membrane with gravity or vibrations, the pressure sensors are placed vertically on a vibration absorbing support. On one side of the membrane, the pressure is that at the surface of the sphere, and on the other side, the pressure is the free-stream pressure. Since the free-stream pressure is monitored by an MKS absolute pressure as already described, it is known. In consequence the wall pressure can be calculated by subtracting free-stream pressure from the measured pressure. The two sensors were calibrated on and beyond their own range and showed a linear deformation of the membrane until 150 Pa for the 0.0125 psi, and 350 Pa for the 0.02 psi, which includes our measured values.

For each position of S 2 towards S 1 , the wall pressure is measured for angles from 90°to -90°. At the end of each position (-90°), S 2 is rotated to its initial angle (90°) where a second measurement is realized to verify the accuracy of measurements.

RESULTS AND DISCUSSION

In this section, the two experimental conditions are compared using the results of the three types of measurement.

Shock/shock interferences

The shock/shock interferences were first analyzed qualitatively thanks to the shock wave detection. In order to compare the two experimented flow conditions, the middles of shock wave were superposed for each type of SSI as shown in Fig. 6, where plots are zoomed on the interference area. Whatever the type of SSI, three features can be observed:

• In both cases (Nozzle a and Nozzle b) the same patterns of shock/shock interference are observed:

-For SSI types I to III, there is a flattening of the shock wave below the intersection point.

-For SSI types V and VI, there is a flattening of the shock wave above the intersection point.

-For SSI types I and IV, there is a step-back area of the shock wave at the intersection point.

-From type I to VI, the lower part of the shock wave (below P i , the point of intersection between S 1 and S 2 middles of shock) progressively opens. A wider explanation is given in Cardona et.al [START_REF] Cardona | Shock/shock interferences in a supersonic rarefied flow: experimental investigation[END_REF].

• Whether S 2 is globally in the free-stream or globally in the wake of S 1 , the global level of rarefaction is lower for Nozzle a than for Nozzle b. As the level of rarefaction decreases, the middle of shock wave gets closer to the surface of S 2 . This result is in accordance with the work of Rembaut et.al [12] who experimentally demonstrated that the stand-off distance of a sphere in a free-stream flow decreases with the decrease of the Knudsen number.

• As the level of rarefaction decreases, the specificities of SSI are slightly accentuated, mostly for SSI types I to IV. Note that differences are not much pronounced because the levels of rarefaction of the two nozzles are close. Nevertheless, this phenomenon is due to a global increase in pressure values and to less diffuse shock waves, leading to a stronger impact between the shock waves.

Another feature concerns the thickness of the shock wave. Figure 7 presents the superposition of Nozzle a and Nozzle b shock waves in the intersection area for the three shock regions of the type III SSI. As can be observed, the boundary layers of the two cases are superposed, while the distance between Nozzle a and Nozzle b shock waves increases for the middles of shock and further more for the foots of shock. This means that the shock becomes thicker with a higher Knudsen number. Here, only the type III SSI is presented, but same results are observed for the five other types.

Drag and lift forces

For each type of SSI, one position of S 2 towards S 1 was chosen for the measurement of the aerodynamic forces of S 2 . This graph shows that the type of SSI where the drag force is maximal depends on the rarefaction level. Also, as the rarefaction level decreases, the impact of types IV and V SSI seems to diminish. Indeed, with lower local rarefaction, shock waves are denser but thinner. At P i , the intersection point, the incident shock begins to penetrates the boundary layer of S 2 , but more locally than for Nozzle a. It is possible that the impacted area at the surface of S 2 is small enough to induce a decrease in drag forces, even if parietal forces are locally higher.

The lift force values are not divided by the reference force because a single sphere has a null lift force. Nevertheless, it can be seen that the maximal value is reached for a type III SSI for Nozzle a while it is reached for a type II SSI for Nozzle b. Moreover, it seems that the lift force is positive for a wider range of S 2 positions when the rarefaction level increases. the displacement of S 2 in S 1 reference frame can locally be deduced:

• if F x < F x0 , S 2 approaches S 1 horizontally (⇐) • if F x > F x0 , S 2 moves away from S 1 horizontally (⇒) • if F z < F z0 , S 2 approaches S 1 vertically (⇓) • if F z > F z0 , S 2 moves away from S 1 vertically (⇑)
By combining the vector of the drag force with this of the lift force, the resulting force of S 2 (calculated in S 1 spatial referential) gives the local displacement of S 2 towards S 1 , as shown by the arrows in Fig. 9. For Nozzle a, S 2 is more repulsed by the wake of S 1 than for Nozzle b for which it is clearly seen that for types V and VI SSI, S 2 is strongly attracted by S 1 . For Nozzle b, this attraction can be the consequence of a denser S 1 shock wave that prevents S 2 to go through the incident shock, and potentially makes S 2 ricocheting on it.

Wall pressure

The pressure at the surface of S 2 were recorded for the same positions than those adopted for the measurement of aerodynamic forces. For each type of SSI, Fig. 10 gives the pressure distribution according Θ wp , the angle that makes the segment [P wp O S2 ] with the horizontal, where P wp is the point of the pressure measurement at the surface of S 2 and O S2 is the center of the sphere S 2 , as described in 5. Concerning Θ wp , -90°represents the bottom of S 2 , and 90°its top.

Wall pressure are normalized with p w0 , the wall pressure at the stagnation point of a single sphere in the free-stream. Respectively for Nozzle a and Nozzle b, (p w0 ) a = 54.17 Pa and (p w0 ) b = 164.04 Pa. The given data is dimensionless so that the analysis only highlights shock/shock interferences differences according the nozzle used, this way, only the effect of rarefaction is shown.

As can be seen, for each type of SSI, a specific pattern appears, and it is the same for the two nozzles. For the SSI types I, II and III, only the part below the intersection area sees a variation of wall pressure. As S 2 is manly outside of S 1 wake, the region above the intersection area is still in the free-stream so the upper surface of S 2 presents the same pressure distribution than for the reference case. For these first three types, in particular for the types I and III, it seems that wall pressure variation are more emphasized for Nozzle b. As this phenomenon is more marked for the type IV SSI, it will be explained in the next paragraph. For SSI type IV, the evolution of the curve presents three slopes. For Nozzle b, a first set of wall pressure were recorded (empty triangle), but as the pressure peak was not at the same angle than for Nozzle a, another set were recorded (solid triangle). As the measurements of wall pressure are time consuming, the second recording was only made around the pres- sure peak. It enables to observe that for the exact same position of pressure peak, the wall pressure of the two flow conditions present a slight difference in terms of distribution in the interference area. For Nozzle b, there is a pronounced inflection around 10°, where the incident shock wave probably does interfere with S 2 shock wave anymore. As the level of rarefaction increases, shock waves get thicker and the interference area is smoother and bigger. In consequence, as the incident shock wave penetrates S 2 shock wave it diffuses in its boundary layer, and so the wall pressures are a bit lower at the peak but the impacted region is bigger. For the types V and VI, there is not much of a difference between both rarefaction level. S 2 being mainly in the wake of S 1 , the local rarefaction level is higher than in the freestream, which results in a global decrease of the pressure values. Moreover, the interaction between the two shock waves is located at the top of S 2 or even higher, so its surface is not directly impacted, but the flow around it is. This results in an absence of pressure peak in a deformed area as observed for the other types of interference. Fig. 11 shows the values of maximal pressure in the interference area, where the deformation of the slopes are observed, according the angle of interference. It shows that whatever the SSI type, pressure peaks are greater for the least rarefied flow. Fig. 12 presents the area under the pressure curves (AUPC) presented in Fig. 10 according the angle of interference. The AUPC is representative of the pressure forces exercised at the front line of S 2 . As can be observed looking at the types III and IV, values are slightly higher for Nozzle a, where the flow is the most rarefied and the shock waves are thicker. From Fig. 11 and Fig. 12, it can be assumed that with an increase in rarefaction level, the lower pressure peaks are compensated by the fact that the pressure is distributed more widely over the surface of S 2 .

CONCLUSION

Trough the different experiments carried out in the MARHy wind tunnel, the impact of shock/shock interferences was studied for two levels of rarefaction. The analysis of the flow field images enabled to detect shock waves and, in particular, the different patterns of the six types of SSI. For the two experimented conditions, the SSI patterns are similar but two features were noticed: an increase of the rarefaction level results in the increase of the shock waves thickness, leading to a larger interference area and to a less pronounced pattern. The measurements of drag and lift forces showed that the maximal forces do not occur for the same interference angle and thus for different types of SSI. The maximal drag and lift forces are respectively obtained for SSI types IV and III for the highest rarefaction level and are higher than for the other experimental conditions, for which maximums are obtained for SSI types III and II. Observing wall pressure results, an increase of the rarefaction level also tends to show less pronounced SSI. The distribution of maximal wall pressure according the angle of interaction shows that a lower level of rarefaction implies higher pressure peak. The area under pressure curves shows the same results than the drag forces. Globally, interferences are less pronounced, and pressure peaks are lower for a higher rarefaction level. But the larger distribution of forces generate a greater impact of SSI for types III and IV.

Fig. 2 .

 2 Fig. 2. Models and their placement in the isotropic core

Fig. 3 .

 3 Fig. 3. Image post-processing. a) average image with models, b) average image without models, c) normalized image, and d) normalized image with shock wave detection.

Fig. 4 .

 4 Fig. 4. Set-up of the aerodynamic balance. Top image: Measurement of the forces applied on the sphere. Bottom image: Measurement of the suction forces of the roots blowers

Fig. 5 .

 5 Fig. 5. Schematics of the set-up of the wall pressure measurement -side view (top schemes) and top view (bottom schemes).

Fig. 6 .

 6 Fig. 6. Superposition of shock waves obtained with Nozzle a (blue circle) and Nozzle b (orange triangle) for each type of shock/shock interference.

Fig. 7 .

 7 Fig. 7. Superposition of the foots of shock (FS), middles of shock (MS) and boundary layers (BL) obtained with Nozzle a (blue circle) and Nozzle b (orange triangle) for the type III shock/shock interference.

Figure 8

 8 presents drag (F x ) and lift (F z ) forces of S 2 plotted according Θ i , the angle that makes the segment [P i O S2 ] with the horizontal, where P i is the point of intersection between S 1 and S 2 middles of shock and O S2 is the center of the sphere S 2 . Drag force values are divided by F x0 , the force of S 2 alone in the free-stream, in order to highlight the impact of the type of SSI on the drag force of S 2 . Respectively for Nozzle a and Nozzle b, (F x0 ) a = 8.1 mN and (F x0 ) b = 21.5 mN. As can be seen, the dimensionless drag forces (F x d ) of the Nozzle a and Nozzle b are equals for SSI types I, II and half of III. Then the slopes separate: for Nozzle b, F x d reaches a maximum for the SSI type III and decreases slowly until type V SSI, then decreases a bit faster; while for Nozzle a, F x d keep increasing until type IV SSI, and then decreases.

F

  x0 and F z0 are representative of the drag and lift forces of S 1 . Consequently, with the value of S 2 drag and lift forces,

Fig. 8 .

 8 Fig. 8. Dimensionless drag force (top graph) and lift force (bottom graph) according the interaction angle obtained with Nozzle a (blue circle) and Nozzle b (orange triangle).

Fig. 9 .

 9 Fig. 9. Calculated displacement of S 2 in S 1 reference frame according the shock/shock interaction type for Nozzle a (blue arrow) and Nozzle b (orange arrow).

Fig. 10 .

 10 Fig. 10. Pressure distribution for the six types of shock/shock interferences for Nozzle a (blue circle) and Nozzle b (orange triangle).

Fig. 11 .

 11 Fig. 11. Dimensionless pressure peak in the interference area according the interaction angle for Nozzle a (blue circle) and Nozzle b (orange triangle).

Table 1 .

 1 Flow conditions in the MARHy wind tunnel for Nozzle a (M4 -2.67 Pa) and Nozzle b (M4 -8 Pa)

		Nozzle a	Nozzle b
	Stagnation conditions
	gas	ambient air	ambient air
	p o (Pa)	404.79	1214.39
	T e o (K)	293.15	293.15
	ρ o (kg.m -3 ) 4.81 × 10 -3 1.44 × 10 -2
	Free-stream conditions
	gas	ambient air	ambient air
	p ∞ (Pa)	2.67	8.00
	T e ∞ (K)	69.80	69.80
	ρ ∞ (kg.m -3 ) 1.33 × 10 -4 3.99 × 10 -4
	µ ∞ (Pa.s)	4.84 × 10 -6 4.84 × 10 -6
	U ∞ (m.s -1 )	669.92	669.92
	Re u∞ (m -1 )	1.84 × 10 4	5.52 × 10 -4
	M a ∞	4	4
	λ

∞ (m) 2.23 × 10 -4 7.40 × 10 -5

Table 1

 1

	is representative of a 1-m blunt
	body flying at 76 km for Nozzle a, and 69 km for Nozzle
	b. The Knudsen numbers are respectively 1.39 × 10 -2 and
	4.64 × 10 -3 , representatives of slip regime.

2nd International Conference on Flight Vehicles, Aerothermodynamics and Re-entry Missions & Engineering (FAR)

2nd International Conference on Flight Vehicles, Aerothermodynamics and Re-entry Missions & Engineering (FAR) 19 -23 June 2022. Heilbronn, Germany


Evaluating the nD emission intensity

Simplification of the general expression of luminescence spectra

L(ω) = Cω 3 ∑ ke,kg
ρ ke (T )P (k g , k e )δ(E e,ke -E g,kg + ℏω)

First, we consider that C is an unknown constant as the absolute magnitude of the spectrum is difficult to obtain even experimentally, and is influenced by many measurespecific parameters.

Considering P (k g , k e ), it is by definition | ⟨k g | ⟨g| µ |e⟩ |k e ⟩ | 2 . Let's consider µ e,g = ⟨g| µ |e⟩.

The Condon approximation enables to write the following:

P (k g , k e ) = | ⟨k g | µ e,g |k e ⟩ | 2 = | ⟨k g |k e ⟩ µ e,g | 2 = | ⟨k g |k e ⟩ | 2 |µ e,g | 2 (2)
These considerations motivate the definition of I(ω) = 1 Cω 3 |µe,g| 2 with the next expression:

I(ω) = ∑ ke,kg ρ ke (T )| ⟨k g |k e ⟩ | 2 δ(E g,kg -E e,ke + ℏω) (3)
Now we consider that for a low enough temperature T , we can approximate ρ ke (T ) as follow:

ρ 0 (T ) = 1 ; ∀k e > 0, ρ ke (T ) = 0 (4)
Thus:

I(ω) = ∑ kg | ⟨k g |0⟩ | 2 δ(E e,0 -E g,kg + ℏω) (5)
| ⟨k g |0⟩ | 2 can be expressed exactly in the framework of the 1D harmonic oscillator as

e -S S k g kg!
where S is the Huang-Rhys factor. [START_REF] Henderson | Optical Spectroscopy of Inorganic Solids[END_REF] 2 Finally, in a 1D harmonic oscillator model:

E e,0 -E g,kg = E ZPL -ℏk g ω g (6)
Thus we have our simplified 1D expression:

I(ω) = ∑ kg e -S S k g k g ! δ(ℏω -E ZPL + ℏk g ω g ) (7)

Vibronic fine structure

Starting from the nested sum form:

I(ω) = ∑ k 1 ∈N . . . ∑ kn∈N ( n ∏ j=1 e -S j S j k j k j ! ) • δ ( ω - E ZPL ℏ + n ∑ j=1 k j ω j ) (8)
This formula is impossible to evaluate directly because of the infinite sums. A first approach would be to consider a certain value k max to limit the number of term in the sums.

However, the nested sum makes this strategy insufficient, as for a good value of k max needs to be significantly larger that S j , thus making the number of term in the sum k n still too large for any sizable system (n = 120 in the smallest supercell of our study).

Another approach described by Kubo and Toyozawa 2 is to rewrite the formula into a convolution product and then use the properties of the Fourier Transform to obtain the relevant expression.

First, we consider the following properties of the convolution product ⊛:

• δ(ω + ω 0 ) ⊛ δ(ω + ω 1 )
• For a and b constants:

a ⊛ b = a • b
From this we can see that:

( n ∏ j=1 e -S j S j k j k j ! ) • δ ( ω - E ZPL ℏ + n ∑ j=1 k j ω j ) = exp ( - n ∑ j=1 S j ) δ(ω - E ZPL ℏ ) ⊛ ( S 1 k 1 k 1 ! δ(ω + k 1 ω 1 ) ) ⊛ • • • ⊛ ( S n kn k n ! δ(ω + k n ω n ) ) (9)
Then considering the distributivity of the convolution product:

u(ω) ⊛ (v(ω) + w(ω)) = u(ω) ⊛ v(ω) + u(ω) ⊛ w(ω) (10)
We can find that:

∑ k 1 ∈N . . . ∑ kn∈N ( n ∏ j=1 e -S j S j k j k j ! ) • δ ( ω - E ZPL ℏ + n ∑ j=1 k j ω j ) = exp ( - n ∑ j=1 S j ) ∑ k 1 ∈N . . . ∑ kn∈N [ δ(ω - E ZPL ℏ ) ⊛ ( S 1 k 1 k 1 ! δ(ω + k 1 ω 1 ) ) ⊛ • • • ⊛ ( S n kn k n ! δ(ω + k n ω n ) )] = exp ( - n ∑ j=1 S j ) δ(ω - E ZPL ℏ ) ⊛ [ ∑ k∈N S 1 k k! δ(ω + kω 1 ) ] ⊛ . . . ⊛ [ ∑ k∈N S n k k! δ(ω + kω n ) ] (11)
We now introduce the Fourier transform:

F [u](ω) = ∫ R u(t)e -iωt dt (12)
The Dirac's distributions can be seen as Fourier transform of complex exponential:

I(ω) = exp ( - n ∑ j=1 S j ) δ(ω - E ZPL ℏ ) ⊛ [ ∑ k∈N S 1 k k! F [exp(-ikω 1 t)] ] ⊛ . . . ⊛ [ ∑ k∈N S n k k! F [exp(-ikω n t)] ] (13)
Then we apply the linearity of the Fourier transform to include the sums:

I(ω) = exp ( - n ∑ j=1 S j ) δ(ω - E ZPL ℏ ) ⊛ F [ ∑ k∈N S 1 k k! exp(-ikω 1 t) ] ⊛ . . . ⊛ F [ ∑ k∈N S n k k! exp(-ikω n t) ] (14)
The sums over k now have the shape of the exponential series:

I(ω) = exp ( - n ∑ j=1 S j ) δ(ω - E ZPL ℏ ) ⊛ F [exp (S 1 exp(-iω 1 t))] ⊛ . . . ⊛ F [exp (S n exp(-iω n t))] (15)
Considering the property of Fourier Transform over convolution product

F [u] ⊛ F [v] = F [u • v],
we can gather all the terms under the same Fourier Transform:

I(ω) = exp ( - n ∑ j=1 S j ) δ(ω - E ZPL ℏ ) ⊛ F [exp (S 1 exp(-iω 1 t)) . . . exp (S n exp(-iω n t))] = exp ( - n ∑ j=1 S j ) δ(ω - E ZPL ℏ ) ⊛ F [ exp ( n ∑ j=1 S j exp(-iω j t) )] (16)
We can now recognize that ∑ n j=1 S j = S and we can define s(t), the inverse Fourier Transform of the spectral function:

s(t) = n ∑ j=1 S j exp(-iω j t) (17)
And we are left with evaluating the following:

I(ω) = e -S δ(ω - E ZPL ℏ ) ⊛ F [e s(t) ] (18)
The δ(ω -E ZPL ℏ ) is the theoretical line shape at 0K and results in a high frequency oscillation in time space:

I(ω) = e -S F [e s(t)-i E ZPL ℏ t ] (19)
However, this term may be replaced with a Gaussian function or a Lorentzian function to take into account finite temperature.

Then, one computes the analytical inverse Fourier Transform, l(t), and numerically evaluates the following expression:

I(ω) = e -S F [l(t)e s(t) ] (20)

Determination of an effective frequency

The determination of the effective frequencies for the PES are required by the evaluation of the line width of the spectrum at 0 K, W (0). The effective frequency of the ground state is expressed as follow:

Ω g eff = ∑ i d FC,i ω i ∑ i d FC,i (21)
First, two approaches allow one to consider the broadening of the bands into the discrete formalism. The first one is to consider a single line width as previously introduced. The second is to consider that each oscillator contributes its own line width in the form of a Gaussian function g i (ω) centered around 0 and of FWHM at 0 K W i (0). This idea is to transform Equation 11 into Equation 22:

I(ω) = exp ( - n ∑ j=1 S j ) δ(ω- E ZPL ℏ )⊛ [ g 1 (ω) ⊛ ∑ k∈N S 1 k k! δ(ω + kω 1 ) ] ⊛. . .⊛ [ g n (ω) ⊛ ∑ k∈N S n k k! δ(ω + kω n ) ] (22)
Because the convolution product is associative, we can gather the Gaussian terms to the left.

I(ω) = g 1 (ω)⊛. . .⊛g n (ω)⊛exp ( - n ∑ j=1 S j ) δ(ω- E ZPL ℏ )⊛ [ ∑ k∈N S 1 k k! δ(ω + kω 1 ) ] ⊛. . .⊛ [ ∑ k∈N S n k k! δ(ω + kω n ) ] (23)
Now, the convolution product of Gaussian functions is also a Gaussian function and the square of its standard deviation is the sum of the square of the standard deviations of the components. As a consequence, we have a total line shape g tot defined by:

g tot = g 1 (ω) ⊛ . . . ⊛ g n (ω) (24)
with a FWHM at 0 K, W (0), defined by:

W (0) 2 = n ∑ i=1 W i (0) 2 (25)
Now considering that each W i (0) follows the 1D model exposed in main text, we have:

W i (0) 2 = 4 log (2)S i ω g i ω e i (ℏω g i ) 2 (26)
Which can be rewritten as:

W i (0) 2 = 4 log (2)d FC,i ω g i ω e i ℏω g i (27)
and gathered in a sum such as:

W (0) 2 = n ∑ i=1 W i (0) 2 = 4 log (2) n ∑ i=1 d FC,i ω g i ω e i ℏω g i ( 28
)
Now we consider that we are looking for some Ω g eff , and Ω e eff describing W (0) within a 1D model, namely:

W (0) 2 = 4 log (2)d FC Ω g eff Ω e eff ℏΩ g eff (29)
We are thus solving:

d FC Ω g eff Ω e eff ℏΩ g eff = n ∑ i=1 d FC,i ω g i ω e i ℏω g i (30)
Which, under the approximation that ∀i,

Figure S1 :Figure S2 :

 S1S2 Figures 73