Matteo Borga
email: borgamat@dei.unipd.it

Matteo Meneghini

Davide Benazzi

Eleonora Canato

Roland Püsche

Joff Derluyn

Idriss Abid

F Medjdoub

Gaudenzio Meneghesso

Enrico Zanoni

Buffer breakdown in GaN-on-Si HEMTs: a comprehensive study based on a sequential growth experiment

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

Introduction

Gallium Nitride (GaN) based transistors are promising devices in the voltage range between 200V and 1200V [START_REF] Amano | The 2018 GaN power electronics roadmap[END_REF], being suitable for the next generation of switching power converters. Thanks to the high breakdown field of GaN, which is greater than 3 MV/cm [START_REF] Kaminski | SiC and GaN devices -wide bandgap is not all the same[END_REF], the devices can withstand high voltages, showing an excellent robustness toward OFF-state condition. Moreover, growing the GaN material system over a silicon substrate, improve the process yield and reduces the costs of the final device.

The optimization of the GaN-on-Silicon technology is crucial to improve the efficiency of power High Electron Mobility Transistors (HEMTs), by reducing the leakage losses between drain and substrate when the devices are kept in a high-voltage OFF-state condition. Different approaches have been exploited in order to improve the vertical robustness of the devices, such as a partial removal of the silicon substrate in the drain-gate region [START_REF] Srivastava | Si Trench Around Drain (STAD) technology of GaN-DHFETs on Si substrate for boosting power performance[END_REF] [START_REF] Herbecq | V, 1.6 mΩ cm 2 AlN/GaN-on-Si power devices realized by local substrate removal[END_REF], or a local doping of the silicon substrate below the drain contact [START_REF] Umeda | Blockingvoltage boosting technology for GaN transistors by widening depletion layer in Si substrates[END_REF]. Nevertheless, we recently presented a study [START_REF] Borga | Evidence of Time-Dependent Vertical Breakdown in GaN-on-Si HEMTs[END_REF] on the failure statistics, demonstrating that the failure of a complete GaN-on-Si structure show a time dependent breakdown, with Weibull distributed time to failure. Previous works in literature studied the conduction mechanisms though the vertical stack, showing that the vertical leakage current is determined by a Space Charge Limited conduction through trap states within the materials [START_REF] Zhou | Vertical leakage/breakdown mechanisms in AlGaN/GaN-on-Si structures[END_REF] [START_REF] Moens | Impact of buffer leakage on intrinsic reliability of 650V AlGaN/GaN HEMTs[END_REF]. In addition, Li at all [START_REF] Li | Investigation on Carrier Transport Through AlN Nucleation Layer From Differently Doped Si(111) Substrates[END_REF] presented and extensive analysis on the conduction mechanisms through the nucleation layer grown on silicon, showing that the defects, deep levels, and interfacial states, play a key role in the current conduction through a defective AlN layer.

The aim of this paper is to investigate the breakdown mechanisms in the vertical buffer used for the fabrication of GaN-on-Si HEMTs. To this aim, we carried out a buffer-decomposition experiment: three different wafers were fabricated, where the epitaxial growth was interrupted at different stages. In this way we were able to separately evaluate the role of the AlN nucleation layer, of the AlGaN buffer, and of the vertical stack up to the C-doped layer on the conduction and breakdown processes. We also show the temperature dependence of the current through the tested structures. Moreover, focusing on the Silicon-AlN layer, we evaluate for the first time the breakdown field of the AlN nucleation layer grown on a silicon substrate, and its dependency on the ambient temperature. Finally, a set of constant voltage stresses suggest that the failures of the AlN layer are Weibull-distributed.

Experimental details

In this work three different samples were tested and studied. The first sample consists of a 200nm thick layer of AlN epitaxially grown on a conductive 1•10 19 cm -3 boron-doped silicon substrate; in the next paragraphs this will be referred as Sample A. The second sample, Sample B, has a similar structure to sample A, but over the thin AlN layer, an additional 350nm thick Al 70 Ga 30 N layer is grown. Lastly, sample C is made up of a p-doped silicon substrate, a 200nm AlN layer, a 2350nm thick multi-layer AlGaN backbarrier, and a 2450nm thick carbon doped GaN layer; the carbon doping concentration is 2•10 19 cm -3 . The ohmic Ti(40nm)/Au(20nm) contacts were defined by photolithography; 95x95 µm patterns have been used for the electrical measurements. The structures details are summarized in Figure 1, while a schematic representation of the band diagrams at zero-bias of the three described structures are shown in Figure 2.

The voltage-current characterizations were carried out with a high voltage parameter analyzer Keysight B1505A, whose high voltage (HV) Source Measurement Unit (SMU) is limited at ±3kV, 4mA. The breakdown voltage is extrapolated with a failure-current level of 8mA.

The constant current stresses were performed by means of a Keithley 2410, a high voltage source meter limited at 1.1kV, 20mA. In order to determine the time to failure of a device, a threshold current level of 1µA was set.

The high temperature measurements were carried out heating the samples with a ceramic heater controlled by a Thorlabs TC 200 heater controller.

Results and discussion

The discussion is split in two sections. The first one addresses the comparison of the three samples, while the second one concerned the sample A, and is focused on the failure statistics of the AlN layer.

Current-Voltage characterization

Several devices on each sample were submitted to a voltage sweep on the ohmic pad, while the substrate was grounded. These measurements have been carried out at 30°C, 100°C, and 170°C. The applied voltage was swept from 0V up to the failure of the device, so that both the leakage current behavior and the failure voltage were monitored on a single measurement as a function of the ambient temperature.

Sample A consists of just a thin AlN layer Due to the conductive nature of the silicon substrate, we can suppose that all the applied voltage drops on the AlN layer. Moreover, the AlN can be considered as an insulating material, meaning that the electric field through it is almost constant within the whole thickness. These two approximations allow us to estimate the maximum electric filed that cause the breakdown of the AlN layer by calculating the ratio between the failure voltage and the thickness of the AlN layer (Figure 3(b)). At 30°C the mean breakdown electric field of the AlN layer results 3.2 MV/cm, and it drops to 2.78 MV/cm at 170°C. These breakdown filed values are compatible with previous works in literature [START_REF] Yacoub | The effect of AlN nucleation growth conditions on the inversion channel formation at the AlN/silicon interface[END_REF] [START_REF] Yacoub | The effect of the inversion channel at the AlN/Si interface on the vertical breakdown characteristics of GaN-based devices[END_REF]. The sample B has a thicker structure with respect to the sample A, and over the AlN nucleation layer, a first strain relief 350nm thick AlGaN layer was grown. Usually, in a GaN-on-Si stack, several AlGaN layers are interposed between the nucleation layer and the GaN layer; the aim of these layers is to generate compressive stress to compensate for the thermal-mismatch-induced tensile stress that is generated during cool-down. At the same time, the strain fields at the layer transitions cause some of the threading dislocations to bend and thereby limit the density of the dislocations that propagate from the defective nucleation layer toward the subsequent layers. This guarantees a reduced defect and trap densities close to the active region of the final devices.

By comparing the current-voltage characteristics of the devices grown on sample B (Figure 4(a)) with those grown on sample A, it is clear the role of the AlGaN layer: the current-voltage curves of sample B results more stable, and the device-to-device variability is considerably reduced. Moreover, due to the higher total thickness of the epitaxial layers, the breakdown voltage of the devices on sample B is considerably higher with respect to the failure voltage of the devices of sample A. In the case of sample B, the average electric field at the breakdown was calculated as the ratio of the failure voltage and the total thickness of the AlN and the AlGaN layers, and it results 2.58 MV/cm at 30°C. This calculation is a strong approximation, since a non-uniform partitioning of the applied voltage is present. The failure electric field of sample B is lower than the one calculated in sample A, meaning that the electric field is not uniformly distributed within the AlN/AlGaN stack. This can be ascribed either to the different dielectric constant and the different conductivity of the two materials; moreover, also the piezoelectric charge may play a role in the voltage partitioning within the epitaxial layers. As on sample A, the higher ambient temperature results in a lowered breakdown voltage (Figure 4(b)). The third tested sample, sample C, consists of the AlN nucleation layer grown on a silicon substrate, a complete stack of AlGaN strain relief layers, which prevents the dislocation to propagate through the epitaxial stack, and a 2450nm thick carbon doped GaN layer. The aim of the C-doped layer is to prevent punch-through effect on the final devices, and to reduce the intrinsic conductivity of the GaN. Several works in literature show that this layer plays a crucial role not only in the vertical robustness but also in the stability of the devices under dynamic bias conditions.

The current-voltage characteristics of the sample C are shown in Figure 5(a). Thanks to the thick epitaxial layer and thanks to the high material quality of the carbon doped GaN layer, Figure 5(b) shows that the breakdown voltage is considerably improved (higher than 700V), and there is a remarkably low variability among the tested devices. The ambient temperature has a clear impact on the current through the structure, causing this latter to be higher at higher temperatures. This can be explained either by the improved thermal generation of free charges within the semiconductors, and by the enhanced conductivity of the defective material under high temperature and high electric field conditions.

Constant voltage stress SAMPLE A

This section focuses on the failure mechanisms of the AlN layer grown on a silicon substrate. The leakage characterizations at different ambient temperatures show a very unstable behavior of the current with the applied voltage, thus indicating the high defectiveness of the material. Studying the failure statistic of this sample allow us to better understand if the defects play a role in the breakdown of the devices.

A set of constant voltage stresses were carried out by biasing at 50V the ohmic contact on top of the AlN layer and by keeping the silicon substrate grounded. Due to the high conductivity of the silicon substrate, all the applied voltage drops on the AlN layer, thus allowing the study of its failure statistic. During these voltage stresses the ambient temperature was 30°C.

In Figure 6(a) one can observe the noisy behavior of the current through the AlN layer over the time; this can be ascribed to the defectiveness of the material, which results in a dynamic formation and disruption of leakage paths. Once a set of defects is aligned, a conductive path gets formed between the ohmic contact and the silicon substrate, the device shows a catastrophic breakdown. This failure mechanisms was previously defined as percolation [START_REF] Degraeve | New insights in the relation between electron trap generation and the statistical properties of oxide breakdown[END_REF], and it is typical of the breakdown of the dielectric materials.

By plotting the time to failure of a set of 10 devices grown on sample A on a Weibull plot (Figure 6(b)), it can be noticed that the points perfectly align, meaning that the failure are Weibull distributed. The beta parameter value of the distribution extracted with 99% confidential level is 1.05, meaning that the failures are caused by a wearout process that degrades the devices. The alpha parameter, which represent the time at which the 63.2% of the devices is expected to be failed, is 189 s for the adopted stress condition. The Weibull distribution of the failures agrees with the percolation model, confirming the hypothesis made on the failure mechanism of the AlN layer.

Conclusions

In this work we studied the electrical behavior of three different structures obtained by stopping the epitaxial growth of a standard GaN-on-Silicon process, showing that both the thickness and the composition of the epitaxial stack, beside enhancing the breakdown voltage, improve the material quality by limiting the propagation of defects and dislocations. Furthermore, for the first time we evaluated the critical electric of the AlN nucleation layer grown on a silicon substrate, which is equal to 3.2MV/cm. In addition, focusing on sample A, we showed that the nucleation layer shows a wear-out process when submitted to a constant voltage stress, with Weibull-distributed times to failure, thus suggesting that a percolation process is the failure mechanism of the aluminum nitride layer.

Figure 1 :

 1 Figure 1: schematic representation of the three tested samples. Sample A (left) consists of 200nm thick layer of AlN grown on a conductive (1•10 19 cm -3 boron-doped) silicon substrate. Sample B (center) has an additional 350nm thick AlGaN layer. Sample C (right), over the AlN nucleation layer, has a 2350nm thick strain-relief layer and a 2450nm carbon doped (2•10 19 cm -3 ) GaN layer.

Figure 2 :

 2 Figure 2: schematic equilibrium band diagram of the studied samples: (top-left) sample A, (top-right) sample B and (bottom) sample C

  a conductive p-doped silicon substrate. The AlN layer, beside avoiding the Ga-Si eutectic reaction, act as a base layer for the growth of the subsequent (Al)GaN layers. Due to the remarkably high lattice mismatch between the Aluminum Nitride and the Silicon as well as the low surface mobility of Al species during the epitaxial growth, the AlN layer has a high defectivity. This results in both an unstable current-voltage characteristic of the devices and a high device-todevice variability (Figure3(a)). The current behavior shows a low dependency on the ambient temperature, since the current mostly depends on the local defectivity of the material.

Figure 3 :

 3 Figure 3: (a) current-voltage characteristics of the devices on sample A at different ambient temperature (i.e. 30°C, 100°C and 170°C). (b) box chart of the failure voltages of

Figure 4 :

 4 Figure 4: (a) current-voltage characteristics of the devices on sample B at different ambient temperature (i.e. 30°C, 100°C and 170°C). (b) box chart of the failure voltages of the tested device of sample B. Right y-axis shows the corresponding electric field.

Figure 5 :

 5 Figure 5: (a) current-voltage characteristics of the devices

Figure 6 :

 6 Figure 6: (a) current behavior during the constant voltage stress at 50V of sample A. (b) Weibull plot of the failed devices submitted to the constant voltage stress at 50V.

Acknowledgement

This work was partially supported by the project InRel-NPower (Innovative Reliable Nitride based Power Devices and Applications). This project has received funding from the European Union's Horizon 2020 research and innovation program under grant agreement No. 720527.


Boris Pasquier

Laurent Manivel
email: manivel@math.cnrs.fr

HOROSPHERICAL TWO-ORBIT VARIETIES AS ZERO LOCI

Keywords: Mathematics Subject Classification. 14J45, 14M17, 14M27, 14N15? 14N35, 20G41 Fano variety, horospherical variety, exceptional Lie group, blow-up, vector bundle, Chow ring, quantum cohomology

We present geometric realizations of horospherical two-orbit varieties, by showing that their blow-up along the unique closed invariant orbit is the zero locus of a general section of a homogeneous vector bundle over some auxiliary variety. As an application, we compute the cohomology ring of the G2-variety, including its quantum version. We also consider the Spin7-variety, which deserves a different treatment.

Introduction

Homogeneous varieties play an important role in the classification of complex Fano manifolds, one of the main building blocks in the classification of complex projective varieties. Already in dimension three, the Fano-Iskovskih classification of Fano threefolds of Picard number one and index one reveals that many of them (those of genus between 6 and 10, to be precise) can be realized as complete intersections in certain homogeneous spaces [START_REF] Iskovskikh | Fano varieties[END_REF]. In genus 12, one has to consider an equivariant bundle over a Grassmannian in order to realize the Fano threefolds with this genus as zero loci of global sections. If other approaches are also possible, this vector bundle method was applied systematically by Mukai and many others for Fano threefolds and K3 surfaces. It has the great advantage of allowing an easy access to the geometry of these varieties [START_REF]New developments in Fano manifold theory related to the vector bundle method and moduli problems[END_REF]. Very recently, the vector bundle method was used in order to cover the whole of Mori-Mukai's classification of Fano threefolds [START_REF] De Biase | Fano 3-folds from homogeneous vector bundles over grassmannians[END_REF].

In higher dimensions, complete intersections and, more generally, zero loci of sections of homogeneous vector bundles on homogeneous varieties also allow to construct lots of interesting varieties (in dimension four, see [START_REF] Küchle | On Fano 4-fold of index 1 and homogeneous vector bundles over Grassmannians[END_REF][START_REF] Benedetti | Manifolds of low dimension with trivial canonical bundle in Grassmannians[END_REF] for a sample of these techniques). Nevertheless, it is certainly important to enlarge the class of ambient manifolds on which one could use the vector bundle method. Close to homogeneous varieties, one can consider quasi-homogeneous varieties (those varieties whose automorphism group acts with a dense orbit), especially those that have been classified by combinatorial data, such as spherical varieties, or even more special ones, such as symmetric varieties or horospherical varieties. Under the hypothesis that the Picard group is cyclic (which implies that these varieties are Fano), symmetric varieties were classified by Ruzzi [START_REF] Ruzzi | Smooth projective symmetric varieties with Picard number one[END_REF]. They are in fact homogeneous, or hyperplane sections of homogeneous varieties, up to two exceptions. It is remarkable that these two exceptional varieties can both be realized geometrically by the vector bundle method; this was used in [START_REF] Manivel | The Cayley Grassmannian[END_REF][START_REF]The double Cayley Grassmannian[END_REF] in order to study their geometries and compute their (quantum) intersection rings.

In this paper we consider the case of horospherical varieties. Before stating our main result, let us recall the classification of smooth projective horospherical varieties with Picard group Z.

Theorem 1. [Pas09, Th. 0.1] Let G be a connected reductive algebraic group. Let X be a smooth projective horospherical G-variety with Picard group Z, which is not homogeneous. Then X is horospherical of rank one, and its automorphism group is a connected nonreductive linear algebraic group, strictly containing G, acting with exactly two orbits.

Moreover, X is uniquely determined by its two closed G-orbits Y and Z, isomorphic to G/P Y and G/P Z respectively. With the convention that Z is fixed by Aut(X), the possible triples (G, P Y , P Z ) are the following:

(1) (B m , P (̟ m-1 ),

P (̟ m )) with m ≥ 3 (2) (B 3 , P (̟ 1 ), P (̟ 3 )) (3) (C m , P (̟ i+1 ), P (̟ i )) with m ≥ 2 and i ∈ {1, . . . , m -1} (4) (F 4 , P (̟ 2 ), P (̟ 3 )) (5) (G 2 , P (̟ 2 ), P (̟ 1 ))
We denoted by P (̟ i ) the maximal parabolic subgroup of G corresponding to the dominant weight ̟ i , with the notations of Bourbaki [START_REF] Bourbaki | Groupes et algèbres de Lie. Chapitre IV: Groupes de Coxeter et systèmes de Tits. Chapitre V: Groupes engendrés par des réflexions. Chapitre VI: systèmes de racines[END_REF]. We will also denote it P i for simplicity.

Note that for each group G there is at most one variety in this list, that we will call the G-variety. We refer to [START_REF] Gonzales | Geometry of horospherical varieties of Picard rank one[END_REF] for more geometric information on these varieties (in particular their dimesion and their index). It turns out that the Spin 7 -variety has the special feature of being a (generic) hyperplane section of the spinorial variety Spin 10 /P 5 ⊂ P(∆), where ∆ denotes any of the half-spin representations of Spin 10 . Since its dimension is 9 and its index is 7, this follows from Mukai's classification of Fano varieties of coindex three [START_REF] Mukai | Biregular classification of Fano 3-folds and Fano manifolds of coindex 3[END_REF]).

Our main result provides geometric models of the remaining varieties.

Theorem 2. Let X be a smooth projective horospherical G-variety with Picard group Z, which is not homogeneous, and with G = Spin 7 . Let q : X → X be the blow-up of X along the closed orbit Z fixed by Aut(X). Then X can be realized as the zero locus of a general section of a vector bundle over some homogeneous space. More precisely, there exist a fundamental G-module V , and a positive integer k , such that X coincides with the zero locus of a general section s of the vector bundle

E = Q ⊠ U * over G/P k+1 × G(k + 1, V ⊕ C), where Q is the tautological quotient bundle over G/P k+1 ⊂ G(k + 1, V ), and U the tautological bundle over G(k + 1, V ⊕ C).
Finally, the projection p to G/P k+1 realizes X as the projective bundle P(C ⊕ V * ), if V denotes the tautological rank k + 1 bundle over G/P k+1 .

We can illustrate the theorem by the following diagram.

Q ⊠ U * Q G/P k+1 × G(k + 1, V ⊕ C) s general [ [ u u ❦ ❦ ❦ ❦ ❦ ❦ ❦ ❦ ❦ ❦ ❦ ❦ ❦ ❦ ❦ * * ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ U * G(k + 1, V ) G/P k+1 ? _ o o X ? O O p u u ❦ ❦ ❦ ❦ ❦ ❦ ❦ ❦ ❦ ❦ ❦ ❦ ❦ ❦ ❦ ❦ ❦ ❦ q * * ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ G(k + 1, V ⊕ C) G/P k+1 X ? O O
In the second part of the paper, we use these simple geometric models to improve our understanding of the cohomology (or Chow) ring of the horospherical varieties. The Chevalley formulas for those varieties have been obtained in [START_REF] Gonzales | Geometry of horospherical varieties of Picard rank one[END_REF], including the quantum version, which was recently used to prove that Galkin's Conjecture O does hold for these varieties [START_REF] Bones | Conjecture O holds for some horospherical varieties of picard rank 1[END_REF]. We give a complete treatment of the G 2 -variety in Section 3. We also discuss the cohomology ring of the Spin 7 -variety in Section 4 and extend these results to quantum cohomology. Unfortunately, although the Chow ring of the F 4variety could in principle be determined following the same approach, the computational complexity seems too big for this case to be accessible.

Proof of Theorem 2

Let G be a simple algebraic group over C. Fix a Borel subgroup B of G containing a maximal torus T .

We will show that to almost any horospherical two-orbit G-variety, we can associate a horospherical G-variety of rank one, which is naturally embedded in a homogeneous G × GL d+1 -variety. Then we will prove that this variety is the zero locus of a general section of a vector bundle over this homogeneous space.

2.1. The general construction. Our main construction will involve a certain fundamental G-module V , whose dimension will be denoted by d. Let k be a positive integer, with k < d. Denote e 1 , . . . , e k+1 T -semi-invariant linearly independent vectors of maximal weights in V . We order them starting from the highest weight, in some compatible way with the partial dominance order. In particular

v k := e 1 ∧• • •∧e k and v k+1 := e 1 ∧• • •∧e k+1 are B-semi-invariant vectors of k V and k+1 V , respectively. Denote by V k (respectively V k+1 ) the sub-G-module of k V (resp. k+1 V ) generated by v k (resp. v k+1 ).
We will suppose that the weights of v k and v k+1 are fundamental weights ̟ i and ̟ j of (G, B, T ), in particular V k ≃ V (̟ i ) and V k+1 ≃ V (̟ j ). This is a quite restrictive hypothesis, which is sensible only when V is a fundamental module associated to an end of the Dynkin of diagram of G; in this situation the closed orbit in P(V k ) can naturally be realized as a subvariety of G(k, V ), as discussed in [START_REF] Landsberg | On the projective geometry of rational homogeneous varieties[END_REF] (see in particular Proposition 4.15). Finally, fix a non-zero element e 0 in the trivial G-module C. Definition 3. With the notations above, we define X as the G-orbit closure:

X := G • [v k+1 ⊗ (v k ∧ (e k+1 + e 0 ))] ⊂ P(V k+1 ⊗ k+1 (V ⊕ C)).
Denote by P k and P k+1 the maximal parabolic subgroups containing B associated to ̟ i and ̟ j respectively. Note that P k and P k+1 are the stabilizers in G of the lines Cv k and Cv k+1 , respectively. Proposition 4. The variety X is a horospherical G-variety. Moreover, there exists a horospherical G-variety X of Picard group Z such that X is obtained by blowing-up X along a closed G-orbit.

To prove the proposition we will use the general theory of horospherical varieties, in particular, the classification in terms of colored fans, the description of divisors and the ampleness criterion. For a survey on this theory, see for example [START_REF]Variétés horosphériques de Fano[END_REF] or [START_REF]A survey on the singularities of spherical varieties[END_REF].

Proof. We start by studying the open G-orbit Ω of X more closely. We have Ω ≃ G/H where H = Stab G [v k+1 ⊗(v k ∧(e k+1 +e 0 )] is the kernel of ̟ j -̟ i in the parabolic subgroup P = P k ∩ P k+1 that stabilizes both Cv k and Cv k+1 . In particular Ω is a horospherical homogeneous space of rank 1 with the following spherical data:

(1) the weight lattice M = Z(̟ j -̟ i ) ≃ Z; M

0 ̟ i ̟ j • • • • Q Q ′ • • Figure 1. Two choices of moment polytopes for G/H
According to the Luna-Vust classification of G/H-embeddings in terms of colored fans, there exist four complete G/H-embeddings, obtained by picking or not picking each of the two colors. Each of these embeddings has three G-orbits: Ω and two closed orbits isomorphic to either G/P k , G/P k+1 or G/P (which is a divisor).

The Picard group and the ample divisors of each of these G/H-embeddings can easily be described. In particular, they are all projective and locally factorial, and their Picard number can be 1, 2 or 3. Also, in order to realize these embeddings, we can choose a (small) ample divisor (which is automatically very ample because G/H has rank one [Pas06, Th. 0.3]), and by computing its global sections, we can describe the corresponding embedding into the projective space of the dual G-module of global sections (here we need to suppose that G is simply connected, so that our line bundle can be G-linearized). In another point of view, the projective G/H-embeddings are classified and can be described by moment polytopes, see [Pas15, Section 2.3].

For example, to get the G/H-embedding X obtained by picking the two colors of D, we can choose D so that the corresponding moment polytope is Q = [̟ i , ̟ j ], and then

X = G • [v k + v k+1 ] ⊂ P(V k ⊕ V k+1 ),
which is of Picard number one; and to get the G/H-embedding X ′ obtained by picking only the color D k , we can choose D so that the corresponding moment polytope is

Q ′ = [̟ i + ̟ j , 2̟ j ],
and then

X ′ = G • [v k+1 ⊗ v k + v k+1 ⊗ v k+1 ] ⊂ P((V k+1 ⊗ V k ) ⊕ (V k+1 ⊗ V k+1 )),
which is of Picard number two. To get the descritpion of X ′ , we also use that

V (̟ i +̟ j ) ⊂ V (̟ j ) ⊗ V (̟ i ) ≃ V k+1 ⊗ V k and V (2̟ j ) ⊂ V (̟ j ) ⊗ V (̟ j ) ≃ V k+1 ⊗ V k+1 .
We illustrate the choice of the moment polytope in Figure 1.

Remark that X ′ can also be obtained by blowing-up X along the closed G-orbit isomorphic to G/P k . Indeed, blowing-up one of the closed G-orbit in X we obtain another projective G/H-embedding where the closed G-orbit of X has been replaced by a G-stable divisor, which has to be a closed G-orbit isomorphic to G/P . In terms of colored fans, we have deleted a color. Now, observe that

(V k+1 ⊗V k )⊕(V k+1 ⊗V k+1 ) = V k+1 ⊗(V k ⊕V k+1 ) ⊂ V k+1 ⊗( k V ⊕ k+1 V ) ≃ V k+1 ⊗ k+1 (V ⊕C),
and that by this G-equivariant isomorphism,

X ′ ≃ G • [v k+1 ⊗ (v k ∧ (e k+1 + e 0 ))] ⊂ P(V k+1 ⊗ k+1 (V ⊕ C)).
This exactly means that X ′ is G-equivariantly isomorphic to X.

Remark 5. With the same G-equivariant isomorphism as above, we also have the following G-equivariant emmbedding

X ≃ G • [v k ∧ (e k+1 + e 0 )] ⊂ P( k+1 (V ⊕ C)).
Now, X is by construction a closed subvariety of the homogeneous G × GL d+1 -space

G/P k+1 × G(k + 1, V ⊕ C) ⊂ P(V k+1 ) × P( k+1 (V ⊕ C)) ⊂ P(V k+1 ⊗ k+1 (V ⊕ C)).
Moreover, the immersion X -→ G/P k+1 × G(k + 1, V ⊕ C) is G-equivariant so that composing with the projection to the second factor we get a proper G-equivariant morphism

π : X -→ G(k +1, V ⊕C). The image of π is the closure of the image of G•[v k ∧(e k+1 +e 0 )] in P( k+1 (V ⊕ C))
, which is nothing else than X. This means that the projection to

G(k + 1, V ⊕ C) induces the blow-up X ′ ≃ X → X. Note also that the closed orbit G • [v k+1 ⊗ (v k ∧ e k+1 )] ≃ G/P k+1 maps by π to Z := G • [v k ∧ e k+1 ] ≃ G/P k+1 . Moreover G • [v k+1 ⊗ (v k ∧ e 0 )] ≃ G/(P k ∩ P k+1 ), which is the exceptional divisor of the blow-up, maps by to Y := G • [v k ∧ e 0 ] ≃ G/P k .
2.2. The variety X as the zero locus of a general section. We begin by the following description of X. Recall that G/P k+1 is embedded in

G(k + 1, V ). Proposition 6. Let (A, B) ∈ G/P k+1 × G(k + 1, V ⊕ C), in particular A and B are subspaces of dimension k + 1 of V and V ⊕ C, respectively. Then (A, B) is a point of X if and only if the projection of B to V is contained in A, or equivalently B ⊂ A ⊕ C. Proof. The points of the G-orbit G • [v k+1 ⊗ (v k ∧ (e k+1 + e 0 ))] are of the form [(x 1 ∧ • • • ∧ x k+1 ) ⊗ (x 1 ∧ • • • ∧ (x k+1 + e 0 ))],
and then correspond to a pair (A, B) ∈ G/P k+1 × G(k + 1, V ⊕ C) where A is generated by x 1 , . . . , x k+1 , and B is generated by x 1 , . . . , x k and x k+1 + e 0 . In particular, the projection of B in V is contained in A. Since this is a closed condition, it remains true on the closure of G

• [v k+1 ⊗ (v k ∧ (e k+1 + e 0 ))].
Conversely, if the projection of B in V is contained in A, there exists a basis (x 1 , . . . , x k+1 ) of A such that (x 1 , . . . , x k , x k+1 + e 0 ) or (x 1 , . . . , x k+1 ) or (x 1 , . . . , x k , e 0 ) is a basis of B.

In the first case, (A, B) is a point of the open G-orbit G • [v k+1 ⊗ (v k ∧ (e k+1 + e 0 ))]. In the two other cases, (A, B) is a point in one of the two closed G-orbits G • [v k+1 ⊗ v k+1 ] and G • [v k+1 ⊗ (v k ∧ e 0 )].
We will deduce the following proposition, which completes the proof of Theorem 2. Since G/P k+1 is embedded in the Grassmannian G(k + 1, V ), it admits a vector bundle Q obtained by restricting the tautological quotient vector bundle, of rank d -k -1. We also denote by U the tautological vector bundle, of rank k + 1, over

G(k + 1, V ⊕ C).
Proposition 7. The variety X is the zero locus of a general section of the vector bundle

E = Q ⊠ U * over G/P k+1 × G(k + 1, V ⊕ C).
We first prove the following statement.

Lemma 8. The space of global sections of

E contains V ⊗ (V ⊕ C) * .
Proof of the lemma. By the Borel-Weil theorem on the Grassmannian,

H 0 (G(k + 1, V ⊕ C), U * ) = (V ⊕ C) * ,
so it suffices to prove that H 0 (G/P k+1 , Q) contains V . But by its very definition, Q is a quotient of the trivial bundle with fiber V , so there is a non trivial equivariant map V → H 0 (G/P k+1 , Q). Since V is irreducible, it must be injective.

We can then conclude with the following general statement. Suppose given a globally generated rank r vector bundle F over some variety Z, and denote by F its space of global sections, of dimension d. Consider a vector space W of dimension w ≥ d, some positive integer ℓ < w, and denote by U the tautological rank ℓ bundle on the Grassmannian G(ℓ, W ). The space of global sections of

F ⊠ U * on Z × G(ℓ, W ) is then H 0 (Z × G(ℓ, W ), F ⊠ U * ) = F ⊗ W * ≃ Hom(W, F ).
Since F is globally generated, there is an exact sequence

0 -→ M -→ F ⊗ O Z -→ F -→ 0,
where M is a vector bundle of rank d -r.

Proposition 9. Let s be a section of F ⊠ U * , defined by σ ∈ Hom(W, F ). Then:

(1) the zero locus

Z(s) ⊂ Z × G(ℓ, W ) is the set of pairs (x, U ) such that σ(U ) ⊂ M x ⊂ F ;
(2) if σ is surjective, Z(s) is smooth and has the structure of a G(ℓ, d -r)-bundle over Z. More precisely, if K ⊂ W denotes the kernel of σ,

Z(s) ≃ G(ℓ, K ⊗ O Z ⊕ M).
Proof. The first claim is clear, since the evaluation of s at (x, U ) is the vector in Hom(U, F x ) obtained by restricting σ to U ⊂ W and projecting from F to its quotient F x . The condition can be rewritten U ⊂ σ -1 (M x ). When σ is surjective, σ -1 (M x ) has constant dimension e -r. This implies that the projection to Z is locally trivial, and the second claim follows since σ -1 (M x ) can be identified with K ⊕ M x .

Conclusion of the proof of Proposition 7.

Apply Proposition 9 (1) to the canonical section of E given by σ ∈ Hom(V ⊕ C, V ), the projection to V . By Proposition 7, its zero locus coincides with X.

By Proposition 9 (2), the projection of X to G/P k+1 is the fiber-bundle G(k + 1, C ⊕ V), if V denotes the tautological bundle, restricted from G(k + 1, V ). Since C ⊕ V has rank k + 2, this is just the hyperplane bundle P(C ⊕ V * ), of relative dimension k + 1. This concludes the proof of the main Theorem.

Remark 10. The stabilizer of σ in G×GL d+1 is (G×C * )⋉V . Indeed, (g 1 , g 2 ) ∈ G×GL d+1 acts on Hom(

V ⊕ C, V ) ≃ M d,d+1 (C) by (g 1 , g 2 ) • M = g 1 M g -1 2 ; then (g 1 , g 2 ) • (I d , 0) = (I d , 0) if and only if g 2 = g 1 0 t x y ,
with x ∈ V and y ∈ C * . As a consequence, the G × GL d+1 -orbit of σ is always dense, and the zero-loci of the corresponding sections are all isomorphic to X.

̟ m = 1 2 (ǫ 1 + • • • + ǫ m ) and let e 2 be a T -semi-invariant vector of weight ̟ m -α m = 1 2 (ǫ 1 +• • •+ǫ m-1 -ǫ m ). Then e 1 ∧e 2 is of weight ̟ m-1 = ǫ 1 +• • •+ǫ m-1 .
• Case 3. (Sp 2m , P (̟ k+1 ), P (̟ k )) with m ≥ 2. The G-module V is the minimal representation, of dimension d = 2m, and k is any positive integer smaller than m. The T -weights of V are the ±ǫ h with 1 ≤ h ≤ m, all of multiplicity one. For any h ∈ {1, . . . , k + 1}, let e h be a T -semi-invariant vector of weight ̟

1 -α 1 -• • • -α h-1 = ǫ h . Then e 1 ∧ • • • ∧ e k and e 1 ∧ • • • ∧ e k+1 are of weights ̟ k = ǫ 1 + • • • + ǫ k and ̟ k+1 = ǫ 1 + • • • + ǫ k+1 respectively.
• Case 4. (F 4 , P (̟ 2 ), P (̟ 3 )).

The G-module V is the minimal representation, of dimension d = 26, and k = 2. The highest weight of V is ̟ 4 = ǫ 1 . Consider T -semi-invariant vectors e 1 of weight ̟ 4 , e 2 of weight ̟ 4 -α 4 = 1 2 (ǫ 1 + ǫ 2 + ǫ 3 + ǫ 4 ), and e 3 of weight ̟ 4 -α 4 -α 3 = 1 2 (ǫ 1 + ǫ 2 + ǫ 3 -ǫ 4 ). Then e 1 ∧ e 2 is of weight ̟ 3 = 1 2 (3ǫ 1 + ǫ 2 + ǫ 3 + ǫ 4 ), while e 1 ∧ e 2 ∧ e 3 is of weight

̟ 2 = 2ǫ 1 + ǫ 2 + ǫ 3 . • Case 5. (G 2 , P (̟ 2 ), P (̟ 1 )).
The G-module V is the minimal representation, of dimension d = 7, and k = 1. The highest weight of V is ̟ 1 . Consider T -semi-invariant vectors e 1 of weight ̟ 1 , and e 2 of

̟ 1 -α 1 . Then e 1 ∧ e 2 is of weight 2̟ 1 -α 1 = ̟ 2 .
3. Cohomology of the G 2 -variety 3.1. The Hasse diagram. Recall that the Chow ring of the horospherical variety X of type G 2 has two natural basis, made of classes coming from the two closed G 2 -orbits [START_REF] Gonzales | Geometry of horospherical varieties of Picard rank one[END_REF]. The latter are Z = G 2 /P 1 ≃ Q 5 , the only closed Aut(X)-orbit, and the adjoint variety Y = G 2 /P 2 , also of dimension 5. Both are homologically rational projective homogeneous spaces, in the sense that their Hodge numbers are the same as those of P 5 . We deduce that the Chow ring of X is free of rank 6 + 6 = 12. Let us choose the basis (τ ′ i , σ j ), made of classes indexed by their degrees, where the τ ′ i are induced from Z = G 2 /P 1 in degree 0 to 5, while the σ j are induced from Y = G 2 /P 2 in degree 2 to 7. More precisely, the τ ′ i are the classes of the closures in X of C 2 -bundles over Schubert varieties of G 2 /P 1 (locally X is a C 2 -bundle over G 2 /P 1 ), and the σ j are the classes of the Schubert varieties of G 2 /P 2 .

From Proposition 1.14 of [START_REF] Gonzales | Geometry of horospherical varieties of Picard rank one[END_REF] we deduce that the Hasse diagram of X is the following:

• • • • • • • • • • • • ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ τ ′ 0 τ ′ 1 τ ′ 2 τ ′ 3 τ ′ 4 τ ′ 5 σ 2 σ 3 σ 4 σ 5 σ 6 σ 7
Recall that the edges in this diagram encode the multiplication by the hyperplane class h (which is nothing else than τ ′ 1 ). For example hτ ′ 2 = 2τ ′ 3 + σ 3 . In particular we can readily deduce the degrees of the classes (τ ′ i , σ j ), which are given in the following diagram:

• • • • • • • • • • • • ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅
56 56 38 10 4 1 18 18 6 3 1 1 3.2. Fundamental class. According to Theorem 2, the blow-up X of X along its closed orbit Z is the zero-locus of a general section of the vector bundle Q ⊠ U * over the product variety G 2 /P 2 × G, where G := G(2, V 7 ⊕ C). The Thom-Porteous formula implies that its fundamental class

[ X] = c 10 (Q ⊠ U * ) ∈ A 10 (G 2 /P 2 × G).
We can easily deduce the fundamental class of X ⊂ G. We will denote by σij the Schubert classes on G, for 0 ≤ j ≤ i ≤ 6, and by τkℓ , for 0 ≤ ℓ ≤ k ≤ 5, the Schubert classes on G(2, V 7 ).

Lemma 11. The fundamental class of X ⊂ G is

[X] = 2σ 41 + 2σ 32 ∈ A 5 (G). Proof. Decompose [ X] = k i α i k ⊗ β i 10-k , with α i k ∈ A k (G 2 /P 2 ) and β j ℓ ∈ A ℓ (G)
. Since the projection map p from X to G is birational on its image X, we deduce that

[X] = p * [ X] = i (p * α i 5 )β i 5 .
In order to compute this, let us denote by x 1 , . . . , x 5 the Chern roots of Q, and by y 1 , y 2 the Chern roots of U * . Recall that the m-th elementary symmetric function of x 1 , . . . , x 5 is the m-th Chern class of Q, which is nothing else than the Schubert class τm of G(2, V 7 ), restricted to G 2 /P 2 . We get

c 10 (Q ⊠ U * ) = i,j (y i + x j ) = 2 i=1 (y 5 i + y 4 i τ1 + y 3 i τ2 + y 2 i τ3 + y i τ4 + τ5 ).
The part of bidegree (5, 5) is

[ X] 5,5 = (y 5 1 + y 5 2 )τ 5 + (y 4 1 y 2 + y 1 y 4 2 )τ 4 τ1 + (y 3 1 y 2 2 + y 2 1 y 3 2 )τ 3 τ2 . In order to project this, we need to evaluate the classes τ5 , τ4 τ1 and τ3 τ2 on G 2 /P 2 . For this we need to recall that G 2 /P 2 ⊂ G(2, V 7 ) is the zero locus of a general section of the vector bundle Q * (1). In particular, its fundamental class is

[G 2 /P 2 ] = c 5 (Q * (1)) = 2τ 41 + 2τ 32 ∈ A 5 (G(2, V 7 )).
For any class α restricted from the Grassmannian, it is then straightforward to compute

G 2 /P 2 α = G(2,V 7 ) (2τ 41 + 2τ 32 )α.
In particular, we get the following evaluations:

G 2 /P 2 τ5 = 0, G 2 /P 2 τ4 τ1 = 2, G 2 /P 2 τ3 τ2 = 4.
Plugging in our formula for the fundamental class of X, we finally get

[X] = 2(y 4 1 y 2 + y 1 y 4 2 ) + 4(y 3 1 y 2 2 + y 2 1 y 3 2 ) = 2σ 2 (σ 3 1 -σ1 σ2 ) = 2σ 41 + 2σ 32 .

Generators and relations. Our next ingredient in order to compute the intersection product on X is

Lemma 12. The restriction map A * (G) Q → A * (X) Q is surjective. And τ ′
1 and σ 2 are the restrictions of the Schubert classes σ1 and σ11 of G, respectively.

Proof. The ring A * (X) Q is generated by τ ′ 1 and σ 2 . It is therefore enough to prove that these two classes are the restrictions of the Schubert classes σ1 and σ11 of G, respectively. Remark that τ ′ 1 is the hyperplane class in X while σ1 is the hyperplane class of G, so the claim is obvious for τ ′ 1 . Recall that σ 2 is the class of the closed G 2 -orbit Y of X, which is isomorphic to G 2 /P 2 . In a neighborhood of Y , more precisely on X\Z, X is a vector bundle of rank two over Y . More precisely, the map X\Z -→ Y is given by the restriction of the projection map

G = G(2, V 7 ⊕C) G(2, V 7 ). This projection is defined on G\{W ∈ G, | C ⊂ W }, which is a neighborhood of the Schubert variety G 11 := {W ∈ G | W ⊂ V 7 };
and it defines a vector bundle of rank two over G(2, V 7 ). It is now clear that the restriction of σ11 (the class of G 11 ) is σ 2 .

Remark 13. The ring A * (X) Q is also generated by τ ′ 1 and τ ′ 2 . We could also prove directly that τ ′ 2 is the restriction of the Schubert class σ2 , the class of {W ∈ G | W ∩ V 5 = {0}}, where V 5 is a 5-dimensional subspace of V 7 . Indeed, one can check that this Schubert variety intersects X transversely at general points, and that the intersection is the subvariety of X that defines τ ′ 2 . In order to simplify the notations we will denote by h, σ our two generators τ ′ 1 , σ 2 of the Chow ring of X. Given the Betti numbers of X, we deduce that the rational Chow ring

A * (X) Q = Q[h, σ]/ R 4 , R 6
for two relations R 4 of degree four and R 6 of degree six. Proposition 14. We can choose the relations to be

R 4 = 3σ 2 -h 2 σ and R 6 = 28h 4 σ -9h 6 .
Proof. The fact that R 4 = 0 in A * (X) follows from the observation that in the Chow ring of G(2, V 7 ), we have [X]σ 22 = 2σ 54 , while [X]σ 31 = 4σ 54 . Therefore the class 2σ 22 -σ31 = 3σ 2 11 -σ2 1 σ11 restricts to zero on X, and this restriction is 3σ 2 -h 2 σ. The fact that R 6 = 0 is even easier. Indeed A 6 (X) has rank one, so the classes h 6 and h 4 σ must be proportional. Since the degree of the latter is 18, while the degree of the former is 56, the claim follows immediately.

The multiplication table.

Note that the information encoded in the Hasse diagram is already sufficient to express all the classes of X in terms of h and σ. We get:

τ ′ 2 = h 2 -σ, τ ′ 3 = h 3 2 -hσ, τ ′ 4 = h 4 2 - 4 3 h 2 σ, τ ′ 5 = h 5 2 - 3 2 h 3 σ, σ 3 = hσ, σ 4 = h 2 σ 3 , σ 5 = h 3 σ 6 , σ 6 = h 6 56 , σ 7 = h 7 56 .
Using the relations of Proposition 14, the multiplication table is then easily obtained:

(τ ′ 2 ) 2 = 2τ ′ 4 + 3σ 4 , τ ′ 2 σ 2 = 2σ 4 , σ 2 2 = σ 4 , τ ′ 3 σ 2 = σ 5 , σ 3 σ 2 = 2σ 5 , τ ′ 3 τ ′ 2 = τ ′ 5 + 2σ 5 , σ 3 τ ′ 2 = 4σ 5 , τ ′ 4 σ 2 = σ 6 , σ 4 σ 2 = 2σ 6 , τ ′ 4 τ ′ 2 = 3σ 6 , σ 4 τ ′ 2 = 4σ 6 , τ ′ 5 σ 2 = 0, σ 5 σ 2 = σ 7 , τ ′ 5 τ ′ 2 = σ 7 , σ 5 τ ′ 2 = 2σ 7 , (τ ′ 3 ) 2 = 2τ ′ 6 , τ ′ 3 σ 3 = 3σ 6 , σ 2 3 = 6σ 6 , τ ′ 4 σ 3 = σ 7 , σ 4 σ 3 = 2σ 7 , τ ′ 4 τ ′ 3 = σ 7 , σ 4 τ ′ 3 = σ 7 .
For completeness, we can also compute the Poincaré dual basis (which is not, as in classical Schubert calculus, a permutation of the original basis). Following the notations of [GPPS18, Proposition 1.10], we let σ ′ i = σ ∨ 7-i and τ j = (τ ′ 7-j ) ∨ . Then:

τ 6 = σ 6 , τ 5 = τ ′ 5 , σ ′ 5 = σ 5 -2τ ′ 5 , τ 4 = 2τ ′ 4 -σ 4 , σ ′ 4 = σ 4 -τ ′ 4 , τ 3 = 2τ ′ 3 -σ 3 , σ ′ 3 = σ 3 -τ ′ 3 , τ 2 = τ ′ 2 -2σ 2 , σ ′ 2 = σ 2 , σ ′ 1 = τ ′ 1 .
This is the other natural basis of the Chow ring of X, in terms of which the Hasse diagram becomes, in agreement with [GPPS18, Proposition 4.6]:

• • • • • • • • • • • • ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ σ ′ 0 σ ′ 1 σ ′ 2 σ ′ 3 σ ′ 4 σ ′ 5 τ 2 τ 3 τ 4 τ 5 τ 6 τ 7
In this basis, the multiplication table is the following:

(σ ′ 2 ) 2 = 2σ ′ 4 , σ ′ 2 τ 2 = 0, τ 2 2 = τ 4 , σ ′ 3 τ 2 = -τ 5 , τ 3 τ 2 = 2τ 5 , σ ′ 3 σ ′ 2 = σ ′ 5 + 2τ 5 , τ 3 σ ′ 2 = 0, σ ′ 4 τ 2 = -τ 6 , τ 4 τ 2 = 2τ 6 , σ ′ 4 σ ′ 2 = τ 6 , τ 4 σ ′ 2 = 0, σ ′ 5 τ 2 = -2τ 7 , τ 5 τ 2 = τ 7 , σ ′ 5 σ ′ 2 = τ 7 , τ 5 σ ′ 2 = 0, (σ ′ 3 ) 2 = 2τ 6 , σ ′ 3 τ 3 = -τ 6 , τ 2 3 = 2τ 6 , σ ′ 4 τ 3 = -τ 7 , τ 4 τ 3 = 2τ 7 , σ ′ 4 σ ′ 3 = τ 7 , τ 4 σ ′ 3 = -τ 7 .
Remarks.

(1) One important difference between the two multiplication tables is that the second one has some negative signs, while the first one has none. This is due to the fact that τ 2 is the class of G 2 /P 1 ⊂ X, which is the closed Aut(X)-orbit Z in X and is therefore not movable. On the contrary, σ is the class of a restricted Schubert cycle, and is therefore movable in X.

(2) The degree four relation R 4 can be expressed as σ 2 τ 2 = 0, and obviously follows from the fact that the two closed G-orbits of X do not meet. (3) More generally, and for any horospherical variety X with Picard number one, we have inside A * (X) two subalgebras A 1 (here generated by the σ i 's) and A 2 (here generated by the τ j 's) such that

hA 1 ⊂ A 1 , hA 2 ⊂ A 2 , A 1 A 2 = 0.
Formally we can even decompose A * (X) = A 1 ⊕ A ∨ 2 = A 2 ⊕ A ∨ 1 (where A ∨ 1 is the submodule generated by the Poincaré duals of the Schubert classes in A 1 ). Then

A ∨ 1 A 2 ⊂ A 2 and A ∨ 2 A 1 ⊂ A 1 .
If we add Poincaré duality and the Chevalley formula, do we get enough information to determine A * (X)? 3.5. Quantum cohomology. Recall that X has index four, so that the quantum parameter in its quantum cohomology ring QA * (X) = A * (X) Q [q] has degree four. By the general results of Siebert and Tian [START_REF] Siebert | On quantum cohomology rings of Fano manifolds and a formula of Vafa and Intriligator[END_REF], this quantum cohomology ring admits a presentation of the form QA * (X) Q = Q[h, σ, q]/ R 4 (q), R 6 (q) for two relations R 4 (q) of degree four and R 6 (q) of degree six, which are q-deformations of R 4 and R 6 . In particular we can write R 4 (q) = R 4 + r 4 q and R 6 (q) = R 6 + r 6 q,

where r 4 has degree zero (a rational number) and r 6 is a class of degree two. Note that since there is no term of degree bigger that one in q, these relations are determined by degree one Gromov-Witten invariants only. The quantum Chevalley formula has been computed in Proposition 4.6 of [START_REF] Gonzales | Geometry of horospherical varieties of Picard rank one[END_REF], in terms of our Poincaré dual basis. In the basis (σ i , τ ′ j ), this reads

h * h = σ 2 + τ ′ 2 , τ ′ 2 * h = 2τ ′ 3 + σ 3 , σ 2 * h = σ 3 , τ ′ 3 * h = τ ′ 4 + σ 4 + q, σ 3 * h = 3σ 4 + q, τ ′ 4 * h = τ ′ 5 + σ 5 + qh, σ 4 * h = 2σ 5 + qh, τ ′ 5 * h = σ 6 + qσ 2 , σ 5 * h = 3σ 6 + qτ ′ 2 , σ 6 * h = σ 7 + qτ ′ 3 , σ 7 * h = qτ ′ 4 + 2q 2 .
One immediately deduces the quantum Giambelli formulas (that is, how to determine each Schubert class in terms of the generators), and also the relation R 6 (q). Indeed, we get in quantum cohomology

τ ′ 2 * h 4 = 18τ ′ 6 + q(10σ 2 + 4τ ′ 2 ), h 6 = 56τ ′ 6 + 16q(2σ 2 + τ ′ 2 ),
and therefore R 6 (q) = R 6 + 8q(h 2 + 3σ). So the only ingredient missing is the computation of σ 2 in quantum cohomology, which is given by

σ 2 = σ 4 + I 1 (σ, σ, σ 7 )q.
Lemma 15. The Gromov-Witten invariant I 1 (σ, σ, σ 7 ) = 0.

Proof. Since the class τ is movable, the Gromov-Witten invariant I 1 (σ, σ, σ 7 ) is enumerative [GPPS18, Section 3.2]: it counts the number of lines ℓ in X that pass through a general point x (representing a plane P x in V 7 ⊕ C), and meet general Schubert cycles of class σ 11 , that is, two general sub-Grassmannians G(2, A 7 ) and G(2, B 7 ), for A 7 , B 7 two general hyperplanes of V 7 ⊕ C. But a projective line ℓ in G(2, V 7 ⊕ C) is made of planes containing a common line L 1 (and contained in a common three dimensional space L 3 ). We would thus get the inclusion L 1 ⊂ P x ∩ A 7 ∩ B 7 = 0, a contradiction.

Since there is no other quantum correction, we deduce:

Proposition 16. The quantum cohomology ring of X is

QA * (X) = Q[h, σ, q]/ 3σ 2 -h 2 σ + q, 28h 4 σ -9h 6 + 8q(h 2 + 3σ) .
Using the quantum Giambelli formulas

τ ′ 2 = h 2 -σ, τ ′ 3 = h 3 2 -hσ, τ ′ 4 = h 4 2 - 4 3 h 2 σ - 2 3 q, τ ′ 5 = h 5 2 - 3 2 h 3 σ -qh, σ 3 = hσ, σ 4 = h 2 σ 3 - 1 3 q, σ 5 = h 3 σ 6 - 2 3 qh, σ 6 = h 6 56 + 2 7 qσ - 4 7 qh 2 , σ 7 = h 7 56 + 9 7 qhσ - 15 14 qh 3 ,
it would then be easy to deduce the quantum multiplication table in our basis (or the dual one). Let us just mention that the quantum multiplication by σ is given by the following formulas:

τ ′ 2 σ = 2σ 4 + q, σ 2 σ = σ 4 , τ ′ 3 σ = σ 5 + qh, σ 3 σ = 2σ 5 , τ ′ 4 σ = σ 6 + qτ ′ 2 , σ 4 σ = 2σ 6 + qτ ′ 2 , τ ′ 5 σ = qτ ′ 3 , σ 5 σ = σ 7 + 2qτ ′ 3 , σ 6 σ = qτ ′ 4 + q 2 , σ 7 σ = qτ ′ 5 + q 2 h.
We could also check the generic semi-simplicity of QA * (X), which in [START_REF] Gonzales | Geometry of horospherical varieties of Picard rank one[END_REF] was directly deduced from the quantum Chevalley formula.

Cohomology of the Spin 7 -variety

Using the fact that the Spin 7 -variety X is a generic hyperplane section of the spinor variety S for Spin 10 , its cohomology is easily described. First observe that the two closed orbits in X are quadrics of dimensions 5 and 6, so that the topological Euler number is 6 + 8 = 14. If we use the Schubert basis (σ ′ i , τ j ), where the classes τ j are induced from the closed orbit Q 6 , we get the following Hasse diagram:

• • • • • • • • • • • • • • ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ σ ′ 0 σ ′ 1 σ ′ 2 σ ′ 3 σ ′ 4 σ ′ 5 τ + 6 τ 3 τ 4 τ 5 τ - 6 τ 7 τ 8 τ 9
We deduce that the Chow ring is generated by the hyperplane class h = σ ′ 1 and the degree three class τ = τ 3 . Moreover, from the Chevalley formula we get

σ ′ 2 = h 2 , σ ′ 3 = h 3 -τ 2 , τ 4 = hτ, σ ′ 4 = h 4 -3hτ 2 , τ 5 = h 2 τ, σ ′ 5 = h 5 -5h 2 τ 2 , τ + 6 = h 6 -5h 3 τ 2 , τ - 6 = 7h 3 τ -h 6 2 , τ 7 = h 7 12 , τ 8 = h 8 12 , τ 9 = h 9 12 .
There must be two relations between the generators, in degrees six and seven. For the latter we can choose R 7 = 6h 4 τ -h 7 . In order to find the former we use the fact that the restriction map from the Chow ring of S to the Chow ring of X is surjective. The Schubert classes in A * (S) will be denoted γ λ , for λ a strict partition with parts smaller than five. In degree three the Schubert classes γ 3 and γ 21 have degree seven and five, respectively. We deduce that τ is just the restriction of the difference γ 3 -γ 21 . Since A 6 (S) has rank two there must exist a linear relation between γ 2 3 , γ 21 γ 3 and γ 2 21 . By applying the Pieri formulas for the spinor variety we get γ 2 3 = 2γ 2 21 . By restricting to X we deduce the relation we are looking for, namely R 6 = τ 2 -6h 3 τ + h 6 .

(Note that it follows that hτ 2 = 0.) This provides enough information to write down the multiplication table. The multiplication by τ is given by σ ′ 1 τ = τ 4 , σ ′ 2 τ = τ 5 , σ ′ 3 τ = τ + 6 , τ 3 τ = τ - 6 -τ + 6 , τ 4 τ = 0, σ ′ 4 τ = τ 7 , τ 5 τ = 0, σ ′ 5 τ = τ 8 , τ + 6 τ = τ 9 , τ - 6 τ = -τ 9 .

And the missing products are the following:

(σ ′ 3 ) 2 = τ + 6 + τ - 6 , σ ′ 3 σ ′ 4 = τ 7 , σ ′ 3 τ 4 = τ 7 , σ ′ 3 σ ′ 5 = 0, σ ′ 3 τ 5 = τ 8 , σ ′ 3 τ + 6 = 0, σ ′ 3 τ - 6 = τ 9 , (σ ′ 4 ) 2 = 0, σ ′ 4 τ 4 = τ 8 , (τ 4 ) 2 = τ 8 , σ ′ 4 σ ′ 5 = -τ 9 , σ ′ 4 τ 5 = τ 9 , τ 4 σ ′ 5 = τ 9 , τ 4 τ 5 = 0.

In terms of the Poincaré dual basis we get the reversed Hasse diagram

• • • • • • • • • • • • • • ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ σ 0 σ 1 σ 2 σ - 3 σ 6 σ 5 σ 6 σ + 3 τ ′ 4 τ ′ 5 τ ′ 6 τ ′ 7 τ ′ 8 τ ′ 9
Since the index of X is seven, the quantum cohomology is very easy to deduce from the quantum Chevalley formula, computed in [START_REF] Gonzales | Geometry of horospherical varieties of Picard rank one[END_REF], Proposition 4.4: quantum corrections do appear only for hτ - 6 = τ 7 + q, hτ 7 = τ 8 + qh, hτ 8 = τ 9 + qσ ′ 2 and hτ 9 = qσ ′ 3 . In particular we get h(τ - 6 -τ + 6 ) = q, and since the the Giambelli type formulas above are valid in quantum cohomology up to degree six, we deduce that R 6 (q) = R 6 , R 7 (q) = 6h 4 τ -h 7 -q.

Moreover the Giambelli type formulas in degree bigger than six must be corrected as Finally, the quantum multiplication by the generator τ is given, in degree bigger than six, by the following formulas: τ 4 τ = q, σ ′ 4 τ = τ 7 -q, τ 5 τ = qh, σ ′ 5 τ = τ 8 -qh, τ + 6 τ = τ 9 , τ - 6 τ = -τ 9 + qh 2 , τ 7 τ = qσ ′ 3 , τ 8 τ = qσ ′ 4 , τ 9 τ = qσ ′ 5 .

τ

  (2) the set of colors D = {D k , D k+1 } corresponding to the set of inverse images by G/H -→ G/P of the two Schubert divisors in G/P ; (3) the images of D k and D k+1 in N := Hom Z (M, Z) ≃ Z, given respectively by α ∨ k|M (i.e. -1 ∈ Z) and α ∨ k+1|M (i.e. 1 ∈ Z).

2. 3 .

 3 More details on the module V . Here discuss what k and the module V are for each case of Theorem 1 (except the Spin 7 -variety). • Case 1. (Spin 2m+1 , P (̟ m-1 ), P (̟ m )) with m ≥ 3. The G-module V is the spinorial representation, of dimension d = 2 m , and k = 1. The T -weights of V are the 1 2 (±ǫ 1 + • • • + ±ǫ m ), all with multiplicity one. Let e 1 be a T -semiinvariant vector of weight