Del Pezzo quintics as equivariant compactifications of vector groups

DEL PEZZO QUINTICS AS EQUIVARIANT COMPACTIFICATIONS OF VECTOR GROUPS ADRIEN DUBOULOZ, TAKASHI KISHIMOTO, AND PEDRO MONTERO Abstract. We study faithful actions with a dense orbit of abelian unipotent groups on quintic del Pezzo varieties over a field of characteristic zero. Such varieties are forms of linear sections of the Grassmannian of planes in a 5-dimensional vector space. We characterize which smooth forms admit these types of actions and show that in case of existence, the action is unique up to equivalence by automorphisms. We also give a similar classification for mildly singular quintic del Pezzo threefolds and surfaces. arXiv:2209.04152v1 [math.AG] 9 Sep 2022 Introduction Vector group varieties are defined by analogy to toric varieties as being varieties X endowed with an effective action of an abelian unipotent group U ∼ = Gna with a Zariski dense open orbit. For varieties defined over a field of characteristic zero, the group U then embeds equivariantly as the open orbit, making X into a partial equivariant completion of U. The study of such equivariant completions which are Fano varieties was initiated by Hassett and Tschinkel in [21] with views towards Manin’s conjecture on the asymptotic distribution of rational points of bounded height over number fields (see e.g. [9, 10]). Over algebraically closed fields of characteristic zero, besides projective spaces of every dimension, many families of Fano varieties and other Mori fibers spaces including for instance smooth projective quadrics, Grassmannians and flag varieties are known to be vector group varieties, see e.g. [1, 3, 17, 18, 22, 31, 37]. Of particular interest in this context is the question of classification of possible equivalence classes of structures of vector group variety up to isomorphisms on a given variety. Indeed, it was observed by Hassett and Tschinkel [21] that in contrast to toric structures, vector group variety structures on PnC , n ≥ 2, are not unique and that for n ≥ 6 there are even infinitely many equivalence classes of such structures. In contrast, it is known that over algebraically closed fields of characteristic zero, Grassmannians other than projective spaces [3] and smooth quadrics [37] admit a unique equivalence class of structure of vector group variety. For Fano varieties of Picard rank one over the field of complex numbers, Fu and Hwang [15] gave a uniform characterization of uniqueness of vector group variety structures in terms in the smoothness of the Variety of Minimal Rational Tangents (VMRT) of a dominating family of minimal rational curves. In this article, we consider the problem of existence and uniqueness of vector group variety structures on smooth and mildly singular quintic del Pezzo varieties over an arbitrary field k of characteristic zero. By definition, a smooth quintic del Pezzo variety is a smooth geometrically connected projective k-variety X whose base extension Xk̄ = X ×Spec(k) Spec(k̄) to an algebraic closure k̄ of k has an ample invertible sheaf ∨ ∼ ⊗(n−1) L of degree 5 such that ωX k̄ =L . If n = 2 then Xk̄ is a del Pezzo surface of degree 5. On the other hand, by [19], a smooth quintic del Pezzo k-variety X exists if and only if n ≤ 6 and furthermore, for each n = 3, 4, 5, 6, Xk̄ is unique up to isomorphism, isomorphic to any smooth linear section of the Grassmannian G(2, 5) ⊂ P9k̄ of 2-dimensional vector subspaces of k̄ ⊕5 . For n ≤ 3, the automorphism group of Xk̄ is either finite if n = 2 or isomorphic to PGL2 (k̄) if n = 3 (see e.g. [7, Proposition 7.1.10]). In particular, it cannot contain any abelian unipotent subgroup defining a vector group variety structure on X. Over the field of complex numbers, existence and uniqueness of vector group variety structures on the remaining varieties, which are all Fano of Picard rank 1, has been settled affirmatively by Fu and Hwang as a consequence of a series of articles [16, 17] devoted to the broader study using VMRT techniques of so-called Euler-symmetric projective varieties. Here, since we work over arbitrary fields k of characteristic zero, possibly non-closed, the actual question becomes to determine and classify k-forms of vector group variety structures on del Pezzo quintics. Of course, the non-existence of such structures after base extension to an algebraic closure is a clear obstruction for existence of these structures over the given base field, But on the other hand, neither the existence nor its combination with uniqueness up to equivalence after base extension is enough in general to conclude, say by Galois descent arguments, the existence of such structures defined over the base field. For instance, a smooth n-dimensional quadric in Pn+1 Q without Q-rational point does not admit any vector group variety structure defined over Q, even though its base extension to Q̄ admits infinitely many such structures, which, as a consequence of [37], are all equivalent. Our approach is thus by necessity different from that using 2010 Mathematics Subject Classification. 14J45, 14J50, 14L30, 14M15, 14M20. 1  DEL PEZZO QUINTICS AS EQUIVARIANT COMPACTIFICATIONS OF VECTOR GROUPS 2 VMRT techniques in [16, 17, 18], which, in particular, depend on the Cartan-Fubini analytic extension theorem [24] that has no arguably straightforward counterpart over arbitrary fields. Instead we build on elementary birational geometry of the Grassmannian G(2, 5), its linear Schubert sub-varieties and their associated rational projections, a material which, over algebraically closed fields, goes back to a classical article of Todd [41]. Our first main result is a classification of k-forms of vector group variety structures on smooth del Pezzo quintics of dimension ≥ 4 which can be summarized as follows: Theorem 1. For a smooth quintic del Pezzo k-variety Xn of dimension n ∈ {4, 5, 6}, the following hold: 1) If n = 6 then X6 admits a vector group variety structure if and only if it has a k-rational point. If so, X6 is isomorphic to G(2, 5) ⊂ P9k and it admits a unique vector group variety structure up to equivalence. 2) If n = 4, 5 then Xn is unique up to isomorphism, isomorphic to any smooth section of G(2, 5) ⊂ P9k by a linear subspace of codimension 6 − n. Furthermore, it admits precisely one vector group variety structure up to equivalence. In a second step, we apply the same methods to k-forms of vector group variety structures on mildly singular del Pezzo quintic threefolds and surfaces. The case of quintic del Pezzo surfaces with canonical singularities has already been fully settled by Derenthal and Loughran [12] who studied vector group variety structures on del Pezzo surfaces with canonical singularities over arbitrary fields of characteristic zero. We therefore mainly focus on the case of quintic del Pezzo threefolds with terminal singularities. We obtain the following characterization, which says in particular that in contrast to smooth del Pezzo quintics of dimension 4 and 5 and canonical del Pezzo quintic surfaces which have no non-trivial k-forms for any field k, trinodal del Pezzo quintics threefolds do in general have non-trivial k-forms: Theorem 2. A quintic del Pezzo threefold X3 with terminal singularities admits a vector group variety structure if and only if its base extension to k̄ has precisely three ordinary double points. In this case, the vector group structure is unique up to isomorphism. Furthermore, isomorphism classes of such threefolds are in one-to-one correspondence with PGL2 (k)-orbits of smooth 0-dimensional sub-schemes of P1k of length three. The article is organized as follows. In the first section we collect standard facts on Grassmannians and basic properties of vector groups and their actions on varieties. Section two is devoted to a review of certain classes of linear Schubert sub-varieties of the Grassmannian G(2, 5) and its smooth linear sections, and of their associated rational linear projections. These preliminary results are then applied in the third and fourth sections to derive the proofs of Theorem 1 and Theorem 2, respectively. Acknowledgements. This research was initiated during the stay of the first and the third authors at Saitama University in March 2020 at the occasion of the last in-person workshop “Affine and Birational Geometry” held before the Covid-19 pandemic. The first author was partially supported by the French ANR project "FIBALGA" ANR-18-CE40-0003 and acknowledge the support of the EIPHI Graduate School ANR-17-EURE-0002 to the Institute of Mathematics of Burgundy. The second author was partially funded by JSPS KAKENHI Grant Number 19K03395. The third author was partially supported by Fondecyt ANID projects 11190323 and 1200502. 1. Preliminaries We work over a field k of characteristic zero, with a fixed algebraic closure k̄ and associated Galois group Γ = Gal(k̄/k). We consider Γ as the profinite group lim Gal(k ′ /k) endowed with the profinite topology, the ← limit being taken over the directed set of finite Galois extension k ⊂ k ′ of k, each group Gal(k ′ /k) being endowed with the discrete topology. A k-variety is an integral k-scheme of finite type. 1.1. Notation and conventions. For vector bundles and projective bundles, we follow [20]. Namely, given a quasi-coherent sheaf F on a scheme X, we let SymF be the symmetric algebra of F and we denote by p : VX (F ) = SpecX (Sym· F ) → X the “vector bundle” over X associated to F and by π : PX (F ) = Proj (SymF ) ∼ X = (VX (F ) \ 0X )/Gm,X → X, where 0X is the zero section of p, its associated “projective bundle”. We denote by OPX (F ) (1) the canonical coherent invertible sheaf of OPX (F ) -modules associated to SymF viewed as graded sheaf of modules over itself with the degree shifted by 1. When E is a coherent locally free sheaf, VX (E) is a usual Zariski locally trivial vector bundle of finite rank and PX (E) is its associated projective bundle of lines. Given a quasi-coherent sheaf F on a scheme X and an integer d ≥ 0, we denote by ρ : GX (F , d) → X the Grassmann bundle whose T -points, where f : T → X is any X-scheme, are equivalence classes of  DEL PEZZO QUINTICS AS EQUIVARIANT COMPACTIFICATIONS OF VECTOR GROUPS 3 coherent locally free quotients E of f ∗ F of constant rank d, two such quotients being called equivalent if the corresponding surjections q : f ∗ F → E and q ′ : f ∗ F → E ′ have the same kernel, see e.g. [26]. We denote by ρ∗ F → Q the universal coherent locally free quotient of constant rank d and refer the kernel S of this surjection to as the universal subsheaf of ρ∗ F so that we have the universal exact sequence 0 → S → ρ∗ F → Q → 0. For a coherent locally free sheaf F of constant rank n ≥ d + 1, there is a canonical isomorphism GX (F , d) → GX (F ∨ , n − d) given on T -points by mapping a quotient q : f ∗ F → E with kernel K to the quotient f ∗ F ∨ → K∨ . For d = 1, GX (F , d) ∼= PX (F ) and the universal exact sequence coincides under this isomorphism with the relative Euler exact sequence of PX (F ) 0 → Ω1PX (F ) (1) := Ω1PX (F ) ⊗ OPX (F ) (1) → π ∗ F → OPX (F ) (1) → 0. 1.2. Grassmannians. We summarize basic properties of Grassmannian varieties, see e.g. [28] and [43, Chapters 3 and 4] for the details. For a k-vector space V of dimension n ≥ 2 and an integer 1 ≤ d ≤ n − 1, the Grassmann bundle ρ : Gk (V ∨ , d) → Spec(k) is the d(n − d)-dimensional Grassmannian whose geometric points correspond to equivalence classes of d-dimensional quotients of Vk̄∨ , equivalently to d-dimensional k̄-vector subspaces E of Vk̄ . 1.2.1. Tautological sheaves. We put VG∨k (V ∨ ,d) = V ∨ ⊗k OGk (V ∨ ,d) and we write 0 → S = SGk (V ∨ ,d) → VG∨k (V ∨ ,d) → Q = QGk (V ∨ ,d) → 0 for the universal sequence of coherent locally free sheaves on Gk (V ∨ , d). The sheaf of Kähler differential Ω1Gk (V ∨ ,d)/k is canonically isomorphic to Hom(Q, S) ∼= S ⊗ Q∨ and its determinant ωGk (V ∨ ,d) is canoni- ⊗n−d ∨ ⊗d ∼ cally isomorphic to (det S) ⊗ (det Q ) = (det Q∨ )⊗n . The k-vector spaces H 0 (Gk (V ∨ , d), Q) and H (Gk (V , d), S ) are canonically isomorphic to V ∨ and V respectively, H 1 (Gk (V ∨ , d), Ω1Gk (V ∨ ,d)/k ) ∼ 0 ∨ ∨ = k ∨ ∨ 1 and all other cohomology spaces of Q, Q , S, S and ΩGk (V ∨ ,d)/k are zero. 1.2.2. Plücker embedding and automorphisms. We denote by jP : Gk (V ∨ , d) → Pk (Λd V ∨ ) the Plücker em- bedding, that is, the closed immersion determined by the surjection Λd VG∨k (V ∨ ,d) → det Q induced by the universal quotient homomorphism. Letting Autk (Pk (Λd V ∨ ), Gk (V ∨ , d)) be the stabilizer of jP (Gk (V ∨ , d)) in Autk (Pk (Λd V ∨ )), it follows from [8] that the composition of the homomorphism of k-group schemes (1.1) PGLk (V ∨ ) = Autk (Pk (V ∨ )) → Autk (Pk (Λd V ∨ ), Gk (V ∨ , d)), ϕ 7→ Λd ϕ with the restriction homomorphism Autk (Pk (Λd V ∨ ), Gk (V ∨ , d)) → Autk (Gk (V ∨ , d)) is a closed immersion, which is an isomorphism when n 6= 2d (otherwise, its image is a k-subgroup scheme of index 2). 1.3. Vector groups and vector group varieties. 1.3.1. Vector groups. A vector k-group is an abelian unipotent algebraic k-group scheme. By [11, IV.2.4], there is an equivalence between the category of finite dimensional k-vector spaces and the category of vec- tor k-groups, given by the map associating to a finite dimensional k-vector space V the k-group scheme (Vk (V ∨ ), +), where the comorphism of the k-group scheme structure + is induced by the diagonal homo- morphism V ∨ → V ∨ ⊕ V ∨ , and to a k-linear homomorphism f : W → V the k-group homomorphism (Vk (W ∨ ), +) → (Vk (V ∨ ), +) induced by t f : V ∨ → W ∨ . The choice of a k-basis of V determines an isomorphism of k-groups schemes (Vk (V ∨ ), +) ∼ = Gna,k . We will repeatedly use the following elementary results. Lemma 3. With the notation above, the following hold: (1) Every k-subgroup and quotient k-group of a vector k-group is a vector k-group. ∼ = (2) Every extension 0 → U′ → U → U′′ → 0 of vector k-groups has a splitting h : U → U′ × U′′ . U (3) The set M of U-invariants of a rational U-module M of finite positive dimension is nonzero. Recall [29, I.3] (see also [23, 4.2]) that a quasi-coherent G-sheaf of OX -modules on a k-scheme X with an action µ : G × X → X of an algebraic k-group G is a pair (F , θ) consisting of a coherent sheaf of OX -modules ∼= F and an isomorphism θ : µ∗ F → p∗2 F of coherent sheaves of OG×X -modules, called a G-linearization of F , that satisfies the cocycle relation (mG × idX )∗ θ = p∗23 θ ◦ (idG × µ)∗ θ on G × G × X, where mG : G × G → G is the group law on G and where p23 : G × G × X → G × X is the projection onto the last two factors. In particular, when G acts trivially on X, a G-linearization of F is the same as a homomorphism of G into the group AutX (F ) of OX -module automorphisms of F . Two G-linearizations θ and θ′ of F are called equivalent if there exists an OX -module automorphism ϕ of F such that p∗2 ϕ ◦ θ′ = θ ◦ µ∗ ϕ. Lemma 4. Let X be a normal k-variety endowed with an action of a vector k-group U. Then every coherent invertible sheaf of OX -modules L admits a U-linearization θL unique up to equivalence.  DEL PEZZO QUINTICS AS EQUIVARIANT COMPACTIFICATIONS OF VECTOR GROUPS 4 Proof. This follows from [5, Lemma 2.13] and the fact that since U ∼ = Ank , for every normal k-variety X, the ∗ i ∗ i ∗ pullback homomorphisms pX : H (X, OX ) → H (X × U, OX×U ), i = 0, 1, are isomorphisms.  1.3.2. Vector group structures and vector group varieties. Definition 5. A vector group variety is a k-variety X endowed with an effective action µ : U × X → X of a vector k-group U which has a Zariski dense open orbit UX . The action µ is said to define a vector group structure on X. Two vector group structures µ : U × X → X and µ′ : U′ × X → X on X are said to have the same equivalence class if there exists an isomorphism of k-groups α : U′ → U and a k-automorphism ϕ of X such that ϕ ◦ µ′ = µ ◦ (α × ϕ). Lemma 6. Let X be a vector group variety with open orbit UX . Then UX is a trivial U-torsor. In particular, UX contains a k-point of X. Proof. Since the action of U is effective and UX is Zariski dense, the morphism (µ, p2 ) : U ×k UX → UX × UX is an isomorphism, i.e., UX endowed with the induced action of U is a U-torsor. The conclusion then follows from the additive form of Hilbert Theorem 90 which asserts that every such torsor is trivial.  Example 7. Since all orbits of unipotent group actions on a quasi-affine k-variety are closed [36, Theorem 2], Lemma 6 implies that a quasi-affine vector group variety X is a trivial U-torsor. In particular, Ank is the unique quasi-affine k-variety with a Gna,k -structure and this structure is unique up to isomorphism. On the other hand, by Sumihiro’s equivariant completion [38, Theorem 3], every normal vector group k-variety X admits a U-equivariant open immersion j : X ֒→ X̄ into a complete vector group k-variety X̄. When X is smooth, the existence of equivariant resolution of singularities [27, Theorem 3.36 and Proposition 3.9.1] implies in addition that X̄ can be chosen to be smooth and such that X̄ \ j(X) is the support of a U-stable smooth normal crossing divisor on X̄. Proposition 8. Let X be a k-variety endowed with a vector group structure µ : U × X → X, let f : X → Y be a proper morphism to a k-variety Y such that f∗ OX = OY and let i : F ֒→ X be the scheme-theoretic fiber of f over a k-point y0 of Y in the image by f of the open U-orbit UX . Then there exists an extension a b of vector k-groups 0 → U′ → U → Ū → 0 such that the following hold: (1) The variety Y is endowed with a vector group structure µY : Ū ×k Y → Y with open Ū-orbit UY and f : X → Y is equivariant with respect to the homomorphism b : U → Ū. (2) The scheme F is a k-variety endowed with a vector group structure µF : U′ ×k F → F and the closed immersion i : F ֒→ X is equivariant with respect to the homomorphism a : U′ → U. (3) Given any section c : Ū → U of b : U → Ū, the morphism ∼ = j : µ ◦ (c × i) ◦ (µ−1 Y (·, y0 ) × idF ) : UY × F → Ū × F → U × X → X is a U′ × Ū-equivariant open immersion with image f −1 (UY ). Proof. By Blanchard’s lemma [6, Theorem 7.2.1], there exists a unique action ν : U × Y → Y such that f is U-equivariant. Let U′ ⊂ U be the stabilizer of y0 and let Ū = U/U′ . Since y0 belongs to f (UX ), the U-orbit of y0 is a constructible set which is not contained in any proper closed subset of Y . It thus contains a Zariski dense open subset of Y , hence, being homogeneous under the action of U, is a Zariski dense open subset of Y . This implies that U′ acts trivially on Y and that the induced action µY : Ū × Y → Y of Ū is a vector group structure on Y with the property that f : X → Y is equivariant with respect to the k-group homomorphism b : U → Ū. This proves (1). The closed subscheme F is U′ -stable, with U′ -action µF : U′ × F → F induced by µ. Given a section c : Ū → U of b, we have the following cartesian square of Ū-equivariant morphisms µ◦(c×i) Ū × F /X idŪ ×f f  µY (·,y0 )  Ū × {y0 } /Y where Ū acts on Ū × F and Ū × {y0 } by translations on the first factor and on X by µ ◦ (c × idX ). For every v ∈ Ū(k), µ ◦ (c(v) × idX ) is an automorphism of X which maps F isomorphically onto the scheme-theoretic fiber of f over the point µY (v, y0 ). Since µY (·, y0 ) : Ū × {y0 } → Y is an open immersion with image UY , it follows that µ ◦ (c × i) is an open immersion with image f −1 (UY ). Furthermore, µ ◦ (c × i) is equivariant for the isomorphism h = (c, a) : Ū × U′ → U with respect to the product action of Ū × U′ on Ū × F and the U-action on X. Assertion (3) follows. Finally, assertion (2) follows from the observation that the intersection of the inverse image of UX ⊂ f −1 (UY ) by µ ◦ (c × i) with {0} × F is a Zariski dense U′ -stable open subset UF of F which is a principal homogeneous space of U′ .   DEL PEZZO QUINTICS AS EQUIVARIANT COMPACTIFICATIONS OF VECTOR GROUPS 5 Recall that a coherent locally free sheaf E on a k-variety X is called simple if its only endomorphisms are scalar homothethies. Lemma 4 and Proposition 8 imply the following result. Corollary 9. For a simple coherent locally free sheaf E of rank r ≥ 2 on a normal k-variety X, the total space of the projective bundle π : PX (E) → X does not admit a vector group structure. Proof. Assume that PX (E) admits a vector group structure given by the action of vector group U. By Lemma 4, the invertible sheaf OPX (E) (1) is canonically U-linearized. By Proposition 8, π : PX (E) → X is U-equivariant for a uniquely determined U-action on X, which factors through an effective action of a non- trivial quotient Ū = U/U′ defining a vector group structure on X. Since π is U-equivariant, E ∼ = π∗ OPX (E) (1) is endowed with an induced U-linearization for the action on X, hence with a linearization for the trivial action of the positive dimensional vector group U′ . The latter is determined by some group homomorphism U′ → AutX (E) which is injective by Proposition 8. But this is impossible since AutX (E) ∼ = Gm,X by hypothesis.  2. Linear sections of G(2, 5) and their rational linear projections We collect results concerning linear Schubert sub-varieties of dimension ≥ 2 of the Grassmannian G(2, 5) of 2-dimensional k-vector subspaces of k ⊕5 and of its smooth linear sections in the Plücker embedding. We then review the description of the rational maps given by projections with respect to these linear Schubert sub-varieties. Over algebraically closed fields, all this material is classical, see e.g. [13, 19, 33, 41]. 2.1. Linear Schubert subvarieties of G(2, 5) and its hyperplane sections. We put G = Gk (V ∨ , 2) ∼ = G(2, 5) for some fixed 5-dimensional k-vector space V . We denote by 0 → S → VG∨ → Q →0 the universal sequence on G and by jP : G ֒→ Pk (Λ2 V ∨ ) the Plücker embedding. For any algebraic extension k ′ of k, we interpret k ′ -points of Gk′ either as 2-dimensional k ′ -vector subspaces E ⊂ Vk′ or as their corresponding lines Pk′ (E ∨ ) in Pk′ (Vk∨′ ). For concrete examples, we fix the following coordinate convention: Notation 10. For a chosen basis e1 , . . . , e5 of V with dual basis e∨ ∨ ∨ 4 1 , . . . , e5 , we identify Pk (V ) with Pk and G(2, 5) with the closed subvariety of Pk (Λ V ) = Pk endowed with the Plücker coordinates wij = ei ∧ e∨ 2 ∨ 9 ∨ j, 1 ≤ i < j ≤ 5, defined by the equations    w12 w34 − w13 w24 + w14 w23 = 0  w12 w35 − w13 w25 + w15 w23 = 0   w12 w45 − w14 w25 + w15 w24 = 0 w13 w45 − w14 w35 + w15 w34 = 0     w23 w45 − w24 w35 + w25 w34 = 0  2.1.1. Solids and planes in G(2, 5). We consider the following linear Schubert sub-varieties of G: Definition 11. Let {V1 ⊂ V3 ⊂ V4 } be a partial flag of k-vector subspaces of V , with dimk Vi = i: • The σ3,0 -solid σ3,0 (V1 ) ∼= Pk ((V /V1 )∨ ) associated to V1 is the zero scheme of the homomorphism V1,G ֒→ VG → S ∨ . Its intersection σ3,1 (V1 ⊂ V4 ) ∼= Pk ((V4 /V1 )∨ ) with the zero scheme of the homomorphism (V /V4 )∨ G ֒→ V ∨ G → Q is called the σ3,1 -plane associated to {V1 ⊂ V4 }. • The σ2,2 -plane σ2,2 (V3 ) = Gk (V3∨ , 2) ∼ P (V = k 3 ) associated to V3 is the zero scheme of the homomorphism (V /V3 )∨ G ֒→ V ∨ G → Q. The above sub-schemes are linear subspaces of G in the Plücker embedding G ⊂ Pk (Λ2 V ∨ ), given respec- tively as the intersections G ∩ Pk (Λ2 V ∨ /Λ2 (V /V1 )∨ ), σ3,0 (V1 ) ∩ Pk (Λ2 V4∨ ) and G ∩ Pk (Λ2 V3∨ ). It follows from [41] that they are the only linear k-subspaces of dimension ≥ 2 contained in G.1 Geometrically, the closed points of σ3,0 (V1 )k̄ and σ3,1 (V1 ⊂ V4 )k̄ correspond respectively to lines in Pk̄ (Vk̄∨ ) passing through the point Pk̄ (V1,∨k̄ ) and its subset of those which are contained in the subspace Pk (V4∨ ). The closed points of σ2,2 (V2 )k̄ correspond to lines contained in the subspace Pk̄ (V3,∨k̄ ) of Pk̄ (Vk̄∨ ). Remark 12. The conormal sheaves in G of a solid Π = σ3,0 (V1 ) and of a plane Ξ = σ2,2 (V3 ) are canonically isomorphic to Ω1Π (1) ⊗ V1,Π and to Ω1Ξ (1) ⊗ (V /V3 )∨ Ξ , respectively. Indeed, since Π is the zero scheme of V1,G → S ∨ , its conormal sheaf CΠ/G is isomorphic to (S ⊗ V1,G )|Π ∼ = S|Π ⊗ V1,Π and the canonical 1The two types of planes in G are called planes of the first and second system in [41], and in [19], planes of vertex type and of non-vertex type, respectively.  DEL PEZZO QUINTICS AS EQUIVARIANT COMPACTIFICATIONS OF VECTOR GROUPS 6 isomorphism S|Π ∼ = Ω1Π (1) is given by the following commutative diagram 0 0   0 / Ω1 (1) / (V /V1 )∨ / OΠ (1) /0 Π Π   0 / S|Π / ∨ / Q|Π /0 ▼▼▼ 0 VΠ ▼▼▼ &   ∨ ∨ V1,Π V1,Π   0 0 Similarly, since Ξ is a codimension 4 local complete intersection equal to the zero-scheme of the homo- morphism (V /V3 )∨ ∨ ∨ ∼ G → Q, the conormal sheaf CΞ/G is canonically isomorphic to (Q ⊗ (V /V3 )G )|Ξ = ∨ ∨ ∼ ∨ (Q |Ξ ) ⊗ (V /V3 )Ξ , and since Q|Ξ equals the universal quotient sheaf on Ξ = Gk (V3 , 2), the canonical = Pk (V3 ) gives the identification Q∨ |Ξ ∼ isomorphism Gk (V3∨ , 2) ∼ = Ω1Ξ (1). Notation 13. Given a d-dimensional k-vector subspace Vd ⊂ V , 1 ≤ d ≤ 4, we denote by ∆Vd the stabilizer in GLk (V ∨ ) of the subspace (V /Vd )∨ ⊂ V ∨ . The choice of a splitting V ∼ = Vd ⊕ (V /Vd ) identifies this k-group scheme with that consisting of block-matrices of the form   A5−d U (2.1) M (A5−d , Ad , U ) = ∈ GLk ((V /Vd )∨ ⊕ Vd∨ ) 0 Ad with A5−d ∈ GLk ((V /Vd )∨ ), Ad ∈ GLk (Vd∨ ) and U ∈ Homk (Vd∨ , (V /Vd )∨ ). The associated subgroups P∆Vd of PGLk (V ∨ ) correspond under the isomorphism PGLk (V ∨ ) ∼ = Autk (G) of (1.1) to the stabilizer subgroups of the solid σ3,0 (V1 ) if d = 1 and of the plane σ2,2 (V3 ) if d = 3. 2.1.2. Smooth hyperplane sections. By a hyperplane section of G, we mean the zero scheme Zhsi of a non- zero global section s ∈ H 0 (G, Λ2 Q) = Λ2 V ∨ , equivalently the intersection of G in the Plücker embedding with the hyperplane Pk (Λ2 V ∨ /hsi) of Pk (Λ2 V ∨ ). Denote by s̃ : V → V ∨ the k-linear homomorphism corresponding to the form s under the canonical isomorphism Homk (V ⊗ V, k) ∼ = Homk (V, V ∨ ). The fivefold ∨ Zhsi is the isotropic Grassmannian Is Gk (V , 2): a closed point E ⊂ Vk̄ of Gk̄ belongs to (Zhsi )k̄ if and only if the homomorphism s̃k̄ |E : E → Vk̄ → Vk̄∨ has image contained in (Vk̄ /E)∨ , hence if and only if E is sk̄ -isotropic. Considering the conormal sequence d 0 → CZhsi /G = Λ2 Q∨ |Zhsi ⊗OZhsi hsiZhsi → Ω1G/k |Zhsi = Hom(Q|Zhsi , S|Zhsi ) → Ω1Zhsi /k → 0 we see that Zhsi,k̄ is smooth at E ⊂ Vk̄ if and only if the map d|E : Λ2 E ⊂ Homk (E ∨ ⊗ E ∨ , k) ∼ = Homk (E ∨ , E) → Homk̄ (E ∨ , (Vk̄ /E)∨ ), f 7→ s̃k̄ |E ◦ f is nonzero, hence if and only if E 6⊂ Kers̃k̄ . In particular, Zhsi is smooth if and only if s̃ has rank 4. This yields a functorial correspondence between k-points hsi∨ of Pk (Λ2 V ) \ Gk (V, 2) and smooth hyperplane sections Zhsi of G from which we get in particular that the action of Autk (G)(k) ∼ = PGL5 (k) on the set of smooth hyperplane sections Zhsi of G is transitive. Given a smooth hyperplane section Zhsi , the s-orthogonal V ⊥ = Kers̃ of V has dimension 1. We put V̄ = V /V ⊥ and denote by s̄ ∈ Λ2 V̄ ∨ the symplectic form on V̄ induced by s. Letting Ḡ = Gk (V̄ ∨ , 2) with universal sequence 0 → SḠ → V̄Ḡ∨ → QḠ → 0, the zero scheme Qhs̄i ⊂ Ḡ of s̄ is the Lagrangian Grassmannian Ls̄ Gk (V̄ ∨ , 2) whose k-points are the maximal s̄-isotropic subspaces of V̄ . The Plücker embedding Ḡ ֒→ Pk (Λ2 V̄ ∨ ) induces a closed immersion of Qhs̄i in Pk (Λ2 V̄ ∨ /hs̄i) as the zero scheme of the non-degenerate quadratic form q̄ associated to the symmetric bi-linear form b̄ ∈ Sym2 (Λ2 V̄ ∨ /hs̄i) induced by the bi-linear form b : Λ2 V̄ ⊗ Λ2 V̄ → k, ū1 ∧ v̄1 ⊗ ū2 ∧ v̄2 7→ (s̄ ∧ s̄)(ū1 ∧ v̄1 ∧ ū2 ∧ v̄2 ). Lemma 14. With the notation above, the following hold: (1) Every smooth hyperplane section Zhsi of G contains a unique solid Πhsi = σ3,0 (V ⊥ ) and conversely every solid of G is contained in a (non-unique) smooth hyperplane section of G. (2) A plane σ2,2 (V3 ) of G is contained in a smooth hyperplane section Zhsi if and only if V ⊥ ⊂ V3 and V3 /V ⊥ is s̄-isotropic. In other words, the σ2,2 -planes in Zhsi are in one-to-one correspondence with the k-points of Qhs̄i and they intersect the unique solid Πhsi of Zhsi along lines. Proof. A solid σ3,0 (V1 ) of G is contained in Zhsi if and only if V = V1⊥ hence, since by hypothesis V ⊥ is 1-dimensional, if and only if V1 = V ⊥ . Conversely, for every 1-dimensional k-vector subspace V1 of V ,  DEL PEZZO QUINTICS AS EQUIVARIANT COMPACTIFICATIONS OF VECTOR GROUPS 7 any choice of symplectic form s̄ ∈ Λ2 (V /V1 )∨ on the 4-dimensional k-space V /V1 determines through the inclusion Λ2 (V /V1 )∨ ⊂ Λ2 V ∨ a skew-symmetric form s ∈ Λ2 V ∨ with Kers̃ = V1 , hence a corresponding smooth hyperplane section Zhsi of G containing σ3,0 (V1 ). A plane σ2,2 (V3 ) of G is contained in Zhsi if and only if every 2-dimensional k-vector subspace E ⊂ V3 is s-isotropic, which holds if and only if V3 is s-isotropic. This property is equivalent to the fact that V3 contains V ⊥ and V̄3 = V3 /V ⊥ is an s̄-isotropic subspace of V̄ . Every σ2,2 -plane with this property intersects Πhsi ∼ = Pk (V̄ ∨ ) along the line Pk (V̄3∨ ).  Remark 15. For a smooth hyperplane section Z = Zhsi of G, the exact sequence s̄ 0 → CZ/G |Π ∼ = Λ2 Q∨ |Π → CΠ/G ∼ = Ω1Π (1) → CΠ/Z → 0 identifies the conormal sheaf CΠ/Z of the unique solid Π = Πhsi ∼ = Pk (V̄ ∨ ) of Zhsi with the null-correllation rank 2 locally sheaf Nhs̄i associated to the symplectic form s̄ ∈ Λ2 V̄ ∨ , that is, the sheaf whose fiber over a closed point ℓ ⊂ V̄k̄ of Pk̄ (V̄k̄∨ ) is the quotient ℓ⊥ /ℓ, where ℓ⊥ is the s̄k̄ -orthogonal of ℓ. 2.1.3. Smooth linear sections of codimension 2. A codimension 2 linear section of G is the zero scheme WL of a homomorphism LG → Λ2 Q determined by a 2-dimensional k-vector subspace L ⊂ H 0 (G, Λ2 Q) = Λ2 V ∨ , equivalently the intersection of G in the Plücker embedding with the codimension 2 linear subspace Pk (Λ2 V ∨ /L) of Pk (Λ2 V ∨ ). A closed point E ⊂ Vk̄ of Gk̄ belongs to WL if and only if sk̄ |E = 0 for every skew-symmetric form s in L and, arguing as in § 2.1.2, we see from the conormal sequence d CWL /G = Λ2 Q∨ |WL ⊗ LWL → Ω1G/k |WL = Hom(Q|WL , S|WL ) → Ω1WL /k → 0 that WL,k̄ is smooth at a closed point E ⊂ Vk̄ if and only if E 6⊂ Kers̃k̄ for all s ∈ L \ {0}. In particular, WL is smooth if and only if the line Pk (L∨ ) ⊂ Pk (Λ2 V ) is contained in Pk (Λ2 V ) \ Gk (V, 2). This yields a natural correspondence between smooth codimension 2 linear sections WL of G and k-points of the open subset of Gk (Λ2 V, 2) parametrizing such lines, which is a homogeneous space under the action of PGL5 (k). Lemma 16. A smooth linear section WL of G does not contain any σ3,0 -solid of G. It contains a unique σ2,2 -plane ΞL = σ2,2 (V3,L ), where V3,L ⊂ V is the linear span of the kernels of the linear maps s̃, s ∈ L \ {0}. Proof. By adjunction formula, a solid Π contained in WL would have normal sheaf isomorphic to OΠ (−1), and hence OWL (Π) would be an invertible sheaf of degree 1 on WL . This is impossible since Pic(WL ) ∼ =Z is generated by an ample invertible sheaf of degree 5. A plane σ2,2 (V3 ) of G is contained in WL if and only if V3 contains the linear span of the kernels of the linear maps s̃ corresponding to the forms s ∈ L \ {0}. The image of Pk (L∨ ) ⊂ Pk (Λ2 V ) \ Gk (V, 2) by the morphism Pk (Λ2 V ) \ Gk (V, 2) → Gk (V, 4) given by the complete linear system of quadrics containing Gk (V, 2) is a smooth conic CL in Gk (V, 4) ∼ = Pk (V ∨ ) whose k-points are the kernels Kers̃ ⊂ V of the elements s ∈ L \ {0}. Letting V3,L ⊂ V be the unique 3-dimensional ∨ k-vector subspace such that Pk (V3,L ) contains CL , we conclude that σ2,2 (V3,L ) is the unique σ2,2 -plane in WL .  2.2. Projections from σ3,0 -solids. Let V1 ⊂ V be a 1-dimensional k-vector subspace, let p : V → V̄ = V /V1 be the quotient morphism and let Ḡ = Gk (V̄ ∨ , 2) with universal sequence 0 → SḠ → V̄Ḡ∨ → QḠ → 0. Let Zhsi ⊂ G be a smooth hyperplane section determined by the image s ∈ Λ2 V ∨ of a symplectic form s̄ ∈ Λ2 V̄ ∨ and let Qhs̄i ⊂ Ḡ be the zero scheme of s̄. Let Π = σ3,0 (V1 ) = G ∩ Pk (Λ2 V ∨ /Λ2 (V̄ )∨ ) be the solid of G contained in Zhsi determined by V1 . The projection of G from the solid Π is the dominant rational map (2.2) πΠ : G = Gk (V ∨ , 2) 99K Ḡ = Gk (V̄ ∨ , 2) given by the restriction to G of the linear projection Pk (Λ2 V ∨ ) 99K Pk (Λ2 V̄ ∨ ). The morphism πΠ |G\Π maps a k-point E ⊂ V of G not containing V1 to the k-point p(E) of Ḡ and conversely, the closure in G of a fiber of πΠ over a k-point Ē ⊂ V̄ of Ḡ is the plane σ2,2 (p−1 (Ē)) of G. The restriction πΠ : Zhsi 99K Qhsi of πΠ to Zhsi is called the projection of Zhsi from its solid Π = Πhsi . The restriction πΠ |Zhsi \Π maps a k-point of Zhsi represented by an s-isotropic k-point E ⊂ V of G not containing V1 to the k-point of Qhs̄i represented by the s̄-isotropic k-point p(E) of Ḡ. To state the next result, we put (X6 , Q4 , E6 ) = (G, Ḡ, Q∨ Ḡ ) and (X5 , Q3 , E5 ) = (Zhsi , Qhs̄i , S ), where S = Q∨ | Q Ḡ hs̄i is the spinor locally free sheaf of rank 2 on the quadric threefold Qhs̄i ⊂ Ḡ, see e.g. [32]. Proposition 17. For i = 5, 6, let YΠ ⊂ Xi × Qi−2 be the graph of πΠ with projections pXi : YΠ → Xi and let pQi−2 : YΠ → Qi−2 . Then we have the following Sarkisov link ∼ (2.3) PΠ (CΠ/Xi ) ∼ = PQi−2 (Ei )  / Y ❲❲❲ = / PQi−2 (Ei ⊕ OQi−2 ) pXi ❧❧❧ Π ❲❲❲❲p❲Qi−2 ❧❧❧ ❲❲❲❲❲❲   ❧❧❧ ❲  Π ❴ ❴ ❴ ❴ ❴ ❴ ❴ π❴Π ❴ ❴ ❴ ❴ ❴ ❴ ❴❲+/ Qi−2 / Xi u❧  DEL PEZZO QUINTICS AS EQUIVARIANT COMPACTIFICATIONS OF VECTOR GROUPS 8 where pXi : YΠ → Xi is the contraction of the sub-bundle PQi−2 (Ei ) ⊂ PQi−2 (Ei ⊕ OQi−2 ) onto Π. ∼ P (F ∨ ) → Ḡ, where Proof. The projection pḠ : YΠ → Ḡ is isomorphic to the projective bundle GḠ (F , 2) = Ḡ ∨ ∨ F is the cokernel of the injective homomorphism SḠ → V̄Ḡ → VḠ . The latter is locally free of rank 3, isomorphic to an extension of V1,∨Ḡ by QḠ , hence to QḠ ⊕ OḠ due to the vanishing of H 1 (Ḡ, QḠ ), and the projection pG : YΠ → G contracts the projective sub-bundle PḠ (Q∨ Ḡ ) ⊂ PḠ (Q∨ Ḡ ⊕OḠ ) to Π. This identifies in particular PḠ (Q∨ Ḡ ) with the exceptional divisor PΠ (CΠ/G ) of the blow-up of Π in G. The corresponding diagram for the smooth hyperplane section Zhsi follows immediately by restriction.  Example 18. With Notation 10, the kernel V ⊥ of the linear map s̃ associated to the skew-symmetric form s = e∨ ∨ ∨ ∨ 1 ∧ e3 − e2 ∧ e4 equals he5 i. The associated hyperplane section Zhsi = G ∩ {w13 − w24 = 0} is the smooth fivefold in Pk ⊂ P9k with coordinates wij , (i, j) 6= (2, 4), defined by the equations 8  2   w12 w34 − w13 + w14 w23 = 0  w12 w35 − w13 w25 + w15 w23 = 0   w12 w45 − w14 w25 + w13 w15 = 0 w13 w45 − w14 w35 + w15 w34 = 0     w23 w45 − w13 w35 + w25 w34 = 0  Putting V̄ = V /he5 i, the image of G by the projection Pk (Λ2 V ∨ ) = P9k 99K P5k = Pk (Λ2 V̄ ∨ ), [wij ]1≤i<j≤5 7→ [w12 : w13 : w14 : w23 : w24 : w34 ] from the solid Π = σ3,0 (V ⊥ ) = {w12 = w13 = w14 = w23 = w24 = w34 = 0} is the smooth quadric fourfold Ḡ = Gk (V̄ ∨ , 2) = {w12 w34 − w 13 w 24 + w 14 w 23 = 0} in P5k with Plücker coordinates w̄ij = ē∨ ∨ i ∧ ēj , 1 ≤ i < j ≤ 4, where ēi denotes the image of ei in V̄ . The image of Zhsi by this projection is the smooth quadric threefold Qhs̄i = {w12 w 34 − w213 + w 14 w 23 = 0} in P4k ⊂ P5k with coordinates w̄ij , (i, j) 6= (2, 4). With the notation above, let Spk (V̄ ∨ , s̄) be the symplectic group of the symplectic form s̄ ∈ Λ2 V̄ ∨ and let PSpk (V̄ ∨ , s̄) be its image in PGLk (V̄ ∨ ). We infer the following description: Corollary 19. There exists a split exact sequence of k-group schemes 0 → Autk (Zhsi , Πhsi )0 ∼ = G4a,k ⋊ Gm,k → Autk (Zhsi , Πhsi ) = Autk (Zhsi ) → Autk (Qhsi ) ∼ = PSpk (V̄ ∨ , s̄) → 0, where Autk (Zhsi , Πhsi )0 is the kernel of the restriction homomorphism Autk (Zhsi , Πhsi ) → Autk (Πhsi ). More- over, up to the choice of a splitting V ∼= V̄ ⊕V ⊥ , Autk (Zhsi ) is the image under the restriction homomorphism Autk (G, Zhsi ) → Autk (Zhsi ) of the subgroup Spk (V̄ ∨ , s̄) Homk ((V ⊥ )∨ , V̄ ∨ )   /{±Id} 0 Gm,k of PGLk (V ∨ ) under the isomorphism Φ : PGLk (V ∨ ) → Autk (G) of (1.1). Proof. Since, by Lemma 14, Π = Πhsi is the unique solid contained in Z = Zhsi , we have Autk (Z) = Autk (Z, Π). The action of Autk (Z, Π) lifts to the blow-up pZ : Y → Z of Π and since the fibers of the projection πΠ : Z 99K Qhs̄i of Z from Π over k-points of Qhs̄i meet Π along lines, Autk (Z, Π)0 equals the kernel of the homomorphism Autk (Z, Π) → Autk (Qhs̄i ) induced by pQhs̄i : Y → Qhs̄i . Let p : Autk (Z, Π) → B := Autk (Z, Π)/Autk (Z, Π)0 be the quotient morphism and let γ : B → Autk (Qhs̄i ) be the induced injective homomorphism. Let ∆V ⊥ ⊂ GLk (V ∨ ) be the stabilizer of V̄ ∨ ⊂ V ∨ , see Notation 13. Let ∆V ⊥ ,0 ∼ = G4a,k ⋊Gm,k be its normal subgroup consisting of matrices M (id4 , λ, U ) and let SV ⊥ be its subgroup consisting of matrices M (A4 , ±1, 0) with A4 ∈ Spk (V̄ ∨ , s̄). The image of P∆V ⊥ ⊂ PGLk (V ∨ ) by Φ : PGLk (V ∨ ) → Autk (G) is contained in the stabilizer Autk (G, (Z, Π)) of the pair (Z, Π). A direct verification shows that the homomorphism j : P∆V ⊥ → Autk (Z) obtained by composing with the restriction homomorphism Autk (G, Z) → Autk (Z) is injective and maps P∆V ⊥ ,0 ∼ = ∆V ⊥ ,0 isomorphically onto Autk (Z, Π)0 . Letting 2 2 ∨ q̄ ∈ Sym (Λ V̄ /hs̄i) be the quadratic form associated to the symplectic form s̄ (see § 2.1.2), the conclusion then follows from the fact that the restriction of the composition j p γ P∆V ⊥ → Autk (Z, Π) → B → Autk (Qhs̄i ) = POk (Λ2 V̄ ∨ /hsi, q̄) = SOk (Λ2 V̄ ∨ /hsi, q̄) to the subgroup PSV ⊥ ∼ = PSpk (V̄ ∨ , s̄) is an isomorphism onto its image SOk (Λ2 V̄ ∨ /hsi, q̄).   DEL PEZZO QUINTICS AS EQUIVARIANT COMPACTIFICATIONS OF VECTOR GROUPS 9 2.3. Projections from σ2,2 -planes. Let V3 ⊂ V be a 3-dimensional k-vector subspace, let K = Λ2 V /Λ2 V3 and let hsi ⊂ K ∨ and L ⊂ K ∨ be respectively a 1-dimensional and a 2-dimensional linear subspace of skew-symmetric bilinear forms on V whose non-zero elements all have maximal rank. These data determine respectively a plane Ξ = σ2,2 (V3 ) = G ∩ Pk (Λ2 V3∨ ) of G, a smooth hyperplane section Zhsi of G and a smooth codimension 2 linear section WL of G which both contain Ξ. • The projection of G from the plane Ξ is the birational map (2.4) πΞ : G = Gk (V ∨ , 2) 99K Pk (K ∨ ) induced by the restriction to G of the linear projection Pk (Λ2 V ∨ ) 99K Pk (K ∨ ). The morphism πΞ |G\Ξ maps a k-point E ⊂ V of G not contained in V3 to the image of Λ2 E ⊂ Λ2 V by the quotient homomorphism Λ2 V → K. Let ZΞ = G ∩ Pk (Λ2 V ∨ /Λ2 (V /V3 )∨ ) and HG =Pk (K ∨ /Λ2 (V /V3 )∨ ) be the hyperplane sections of G and Pk (K ∨ ) determined by the 1-dimensional k-vector subspace Λ2 (V /V3 )∨ of K ∨ ⊂ Λ2 V ∨ . Then the image SG of the rational map πΞ |ZΞ : ZΞ 99K HG equals that of the Segre embedding ∼ Pk (K ∨ /Λ2 (V /V3 )∨ ). s1,1 : Pk (V ∨ ) × Pk ((V /V3 )∨ ) ֒→ Pk (V ∨ ⊗k (V /V3 )∨ ) = 3 3 A k-point E ⊂ V of G belongs to ZΞ if and only if E ∩ V3 6= {0}, and the closure in G of the fiber of πΞ |ZΞ \Ξ over the image by s1,1 of a k-point (V1 ⊂ V3 , V4 /V3 ⊂ V /V3 ) of Pk (V3∨ ) × Pk ((V /V3 )∨ ) is the σ3,1 -plane σ3,1 (V1 ⊂ V4 ). • Since hsi ⊂ K ∨ , the projection of G from Ξ restricts on Zhsi to the birational map (2.5) πΞ : Zhsi 99K Pk (K ∨ /hsi) defined by the complete linear system of hyperplane sections of Zhsi containing Ξ, called the projection of Zhsi from the plane Ξ. Letting HZhsi = HG ∩ Pk (K ∨ /hsi), the image of the induced rational map πΞ |ZΞ ∩Zhsi : ZΞ ∩ Zhsi 99K HZhsi is the smooth cubic scroll SZhsi = SG ∩ Pk (K ∨ /hsi) in Pk (K ∨ /hsi). • Since L ⊂ K ∨ , the projection of G from Ξ restricts on WL to the birational map (2.6) πΞ : WL 99K Pk (K ∨ /L) defined by the complete linear system of hyperplane sections of WL containing Ξ, called the projection of WL from the plane Ξ. Putting HWL = HG ∩ Pk (K ∨ /L), the image of the induced rational map πΞ |ZΞ ∩WL : ZΞ ∩ WL 99K HWL is the smooth rational cubic curve SWL = SG ∩ Pk (K ∨ /L) in Pk (K ∨ /L). To state the next result, we put (X6 , P6 ) = (G, Pk (K ∨ )), (X5 , P5 ) = (Zhsi , Pk (K ∨ /hsi)) and (X4 , P4 ) = (WL , Pk (K ∨ /L)). We denote by YΞ,i ⊂ Xi × Pi , the graph of πΞ,i = πΞ : Xi 99K Pi , i = 4, 5, 6. Proposition 20. With the notation above, for i = 4, 5, 6 we have the following Sarkisov link BlSXi HXi ∼  / YΞ,i o ? _ BlΞ (ZΞ ∩ Xi ) (2.7) = PΞ (CΞ/Xi ) ✁ ❂❂ pXi ✁ ❂❂p2 ✁✁✁ ❂❂   ✁ πΞ,i ❂ / Z Ξ ∩ Xi   ✁ ❴ ❴ ❴ ❴ / Xi o❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴/  ? _ HXi o  ? _ SXi Ξ Pi o Φi where pXi : YΞ,i → Xi is the blow-up of Ξ, BlΞ (ZΞ ∩ Xi ) is the proper transform of ZΞ ∩ Xi , p2 : YΞ,i → Pi if the blow-up of SXi and BlSXi HXi is the proper transform of HXi ⊂ Pi . The birational inverse Φi of πΞ,i is given by the complete linear system of quadrics in Pi containing SXi . Proof. All the properties follow from [41] and the description above.  Example 21. With Notation 10, let V3 = he3 , e4 , e5 i, let Ξ = σ2,2 (V3 ) = {w12 = w13 = w14 = w15 = w23 = w24 = w25 = 0} be the associated plane of G and let Pk (Λ2 V ∨ ) = P9k 99K P6k = Pk (K ∨ ), [wij ]1≤i<j≤5 7→ [w12 : w13 : w14 : w15 : w23 : w24 : w25 ] be the associated linear projection. The skew-symmetric forms s = e∨ ∨ ∨ ∨ ′ ∨ ∨ ∨ ∨ 1 ∧e3 −e2 ∧e4 and s = e1 ∧e4 −e2 ∧e5 2 ∨ generate a subspace L ⊂ Λ V whose non-zero elements all have rank 4. The associated smooth linear section WL = G ∩ {w13 − w24 = 0} ∩ {w14 − w25 = 0} is the smooth fourfold in P7k ⊂ P9k with coordinates wij , (i, j) 6= (2, 4), (2, 5) defined by the equations 2    w12 w34 − w13 + w14 w23 = 0  w12 w35 − w13 w14 + w15 w23 = 0   2 w12 w45 − w14 + w13 w15 = 0 w w − w w + w15 w34 = 0  13 45 14 35    w23 w45 − w13 w35 + w14 w34 = 0   DEL PEZZO QUINTICS AS EQUIVARIANT COMPACTIFICATIONS OF VECTOR GROUPS 10 The smooth hyperplane section Zhsi = G ∩ {w13 − w24 = 0} of Example 18 and the above smooth linear section WL both contain Ξ. The images of ZΞ = G ∩ {w12 = 0}, ZΞ ∩ Zhsi and ZΞ ∩ WL by the projection πΞ and its successive restrictions are the smooth threefold SG ∼ = P1k × P2k , the smooth rational cubic surface SZhsi = PP1k (OP1k (1) ⊕ OP1k (2)) = F1 and the smooth rational normal curve SWL ∼ ∼ ∼ = P1k with equations   2  2  −w13 w24 + w14 w23 = 0  −w13 + w14 w23 = 0  −w13 + w14 w23 = 0 −w13 w25 + w15 w23 = 0 −w13 w25 + w15 w23 = 0 and −w13 w14 + w15 w23 = 0 2 −w14 w25 + w15 w24 = 0 −w14 w25 + w15 w13 = 0 −w14 + w15 w13 = 0    in P5k , P4k and P3k respectively. Remark 22. By a result attributed to Weil [42], the smooth varieties P1k × P2k ⊂ P5k , F1 ⊂ P4k and the rational normal curve P1k ⊂ P3k are the only smooth cubics which are not hypersurfaces. For a smooth codimension 2 linear section WL of G with unique σ2,2 -plane ΞL = σ2,2 (V3,L ), we infer the following description of Autk (WL ): Corollary 23. There exists a split exact sequence of k-group schemes = G4a,k ⋊ Gm,k → Autk (WL , ΞL ) = Autk (WL ) → Autk (SWL ) ∼ 0 → Autk (WL , ΞL )0 ∼ = PGLk (L) → 0, where Autk (WL , ΞL )0 is the kernel of the restriction homomorphism Autk (WL , ΞL ) → Autk (ΞL ). Proof. Since, by Lemma 16, Ξ = ΞL is the unique σ2,2 -plane contained in WL and since the intersection ZΞ ∩ WL is the union of all the σ3,1 -planes contained in WL , we have Autk (WL ) = Autk (WL , (ZΞ , Ξ)). Since the σ3,1 -planes of WL intersect Ξ ∼= Pk (V3,L ) along the smooth conic CL∨ dual to CL ∼ = Pk (L∨ ) ֒→ Pk (V3,L ∨ ) (see the proof of Lemma 16), the image of the restriction homomorphism Autk (WL , ΞL ) → Autk (ΞL ) is contained in the subgroup Aut(ΞL , CL∨ ) ∼ = Autk (CL∨ ) ∼ = PGLk (L). Since on the other hand the σ3,1 - planes of WL are the closures of the fibers of πΞ : WL \ Ξ → Pk (K ∨ /L) over the k-points of SWL , the projection πΞ : WL 99K Pk (K ∨ /L) induces an isomorphism of k-group schemes Autk (WL , (ZΞ , Ξ)) ∼ = Autk (Pk (K ∨ /L), (HWL , SWL )) which maps Autk (WL , ΞL )0 isomorphically onto the kernel G4a,k ⋊ Gm,k of the restriction homomorphism Autk (Pk (K ∨ /L), (HWL , SWL )) → Autk (SWL ). The latter homomorphism is a split surjection, which identifies Autk (SWL ) with Autk (HWL , SWL ).  3. Smooth quintic del Pezzo varieties with vector group structures Recall that a smooth quintic del Pezzo k-variety of dimension n ∈ {2, . . . 6} is a k-form X of a smooth section of the Grassmannian G(2, 5) ⊂ P9k by a linear subspace of dimension 6 − n. For n ≤ 3, the automor- phism group of Xk̄ is too small to allow the existence of a vector subgroup structure on Xk̄ . In this section, we consider the case of smooth quintic del Pezzo k-varieties of dimension 4, 5 and 6. 3.1. Toric vector groups structures on linear sections of G(2, 5). Let V be a k-vector space of dimen- sion 5 and, with the notation introduced in § 2.3, let X6 = Gk (V ∨ , 2), let V3 ⊂ V be a 3-dimensional k-vector subspace with associated plane Ξ = σ2,2 (V3 ) of X6 and let K6 = Λ2 V /Λ2 V3 . Let hsi ⊂ K6∨ and L ⊂ K6∨ be respectively a 1-dimensional and a 2-dimensional linear subspace of skew-symmetric bilinear forms on V whose non-zero elements all have maximal rank. Put K5∨ = K6∨ /hsi, K4∨ = K6∨ /L and let X5 = Zhsi and X4 = WL be the smooth linear sections of X6 containing Ξ defined by hsi and L respectively. Let Fi = Homk (Ki∨ /Λ2 (V /V3 )∨ , Λ2 (V /V3 )∨ ) ∼ = k ⊕i , i = 4, 5, 6, and let Vk (Fi∨ ) be the associated vector group. We derive from the exact sequence a b 0 → Λ2 (V /V3 )∨ → Ki∨ → Ki∨ /Λ2 (V /V3 )∨ → 0, i = 4, 5, 6 a faithful homomorphism of k-group schemes Vk (Fi∨ ) → GLk (Ki∨ ), f 7→ idKi∨ + a ◦ f ◦ b correspond- ing to a Vk (Fi∨ )-action on Pk (Ki∨ ) restricting to the trivial action on the invariant hyperplane HXi = Pk (Ki∨ /Λ2 (V /V3 )∨ ) and having Pk (Ki∨ ) \ HXi as an open orbit. By Proposition 20, this action lifts through the birational projection πΞ : Xi 99K Pk (Ki∨ ) from the plane Ξ to a Vk (Fi∨ )-action on Xi with open orbit Xi \ (ZΞ ∩ Xi ) and whose restriction to Ξ is trivial. As a consequence, we obtain the following: Proposition 24. Every smooth section of G(2, 5) ⊂ P9k by a linear subspace of codimension ≤ 3 admits a vector group structure. Example 25. With the notation of Example 21, let V3 = he3 , e4 , e5 i with associated linear projection P9k 99K P6k , [wij ]1≤i<j≤5 7→ [w12 : w13 : w14 : w15 : w23 : w24 : w25 ], let s = e∨ ∨ ∨ ∨ ∨ ∨ ′ ∨ ∨ ∨ ∨ 1 ∧ e3 − e2 ∧ e4 ∈ K6 and L ⊂ K6 be the subspace generated by s and s = e1 ∧ e4 − e2 ∧ e5 .  DEL PEZZO QUINTICS AS EQUIVARIANT COMPACTIFICATIONS OF VECTOR GROUPS 11 • For the basis tij = (e∨ ∨ ∨ i ∧ ej ) ⊗ (ē1 ∧ ē2 ), i = 1, 2, j = 3, 4, 5 of F6 , the corresponding action of ∨ ∼ Vk (F6 ) = Spec(k[tij ]) on Pk (K6 ) = Pk with open orbit Pk \ {w12 = 0} is the “toric” G6a -structure on P6k ∨ 6 6 given by w12 7→ w12 and wij 7→ wij + tij w12 for (i, j) 6= (1, 2).2 Its lift to X6 = G is the restriction of the Vk (F6∨ )-action on Pk (Λ2 V ∨ ) induced by the second exterior power of the representation   1 0 −t23 −t24 −t25  0 1 t13 t14 t15  .   ∨ ∨  . . .  (3.1) ρ6 : Vk (F6 ) → GLk (V ), (t13 , t23 , t14 , t24 , t15 , t25 ) 7→  .  . 1 0 0 .  . .  .. ..  1 0  0 ··· ··· 0 1 Comparing with the classification in [39, Chapter 3, § 3, IV], the image of ρ6 is one of the two maximal abelian unipotent subgroups of GL5 corresponding to cases N2 and N3 . The other one, given by the representation dual to ρ6 , corresponds to a vector group structure on Gk (V, 2) ∼ = Gk (V ∨ , 3). ∨ ∼ • The corresponding action of Vk (F5 ) = Spec(k[t13 , t23 , t14 , t15 , t25 ]) on Pk (K5∨ ) = P5k with open orbit P5k \ {w12 = 0} is the toric G5a -structure on P5k given by w12 7→ w12 and wij 7→ wij + tij w12 for (i, j) 6= (1, 2), (2, 4). Its lift to X5 = Zhsi is the restriction of the Vk (F5∨ )-action on Pk (Λ2 V ∨ ) preserving Zhsi = G∩{w13 −w24 = 0} induced by the second exterior power of the representation   1 0 −t23 −t13 −t25  0 1 t13 t14 t15  .   .. . .. ρ5 : Vk (F5∨ ) → GLk (V ∨ ), (t13 , t23 , t14 , t15 , t25 ) 7→    (3.2)  1 0 0 .   . .  .. ..  1 0  0 ··· ··· 0 1 The stabilizer Stab(he∨ ∨ ∨ 5 i) of the subspace he5 i is the subgroup Spec(k[t15 , t25 ]) of Vk (F5 ) and the induced ∨ ∨ ∼ ∨ ∨ action of the vector group Vk (F5 )/Stab(he5 i) = Spec(k[t̄13 , t̄23 , t̄14 ]) on V̄ = (V /he5 i) endowed with the basis dual to that determined by the images ēi of the ei , i = 1, . . . , 4, is given by the representation   1 0 −t̄23 −t̄13  0 1 t̄13 t̄14  (3.3) ρ̄5 : Vk (F5∨ )/Stab(he∨ ∨ 5 i) → Sp4 (V̄ , s̄), (t̄13 , t̄23 , t̄14 ) 7→   0 0 , 1 0  0 0 0 1 where s̄ = ē∨ ∨ ∨ ∨ 1 ∧ ē3 − ē2 ∧ ē4 is the symplectic form on V̄ induced by s. • The corresponding action of Vk (F4∨ ) ∼ = Spec(k[t13 , t23 , t14 , t15 , ]) on Pk (K4∨ ) = P4k with open orbit Pk \ {w12 = 0} is the toric Ga -structure on P4k given by w12 7→ w12 and wij 7→ wij + tij w12 for (i, j) 6= 4 4 (1, 2), (2, 4), (2, 5). Its lift to X4 = WL is the restriction of the Vk (F4∨ )-action on Pk (Λ2 V ∨ ) preserving WL = G ∩ {w13 − w24 = w14 − w25 = 0} induced by the second exterior power of the representation   1 0 −t23 −t13 −t14  0 1 t13 t14 t15  .   (3.4) ρ4 : Vk (F4∨ ) → GLk (V ∨ ), (t13 , t23 , t14 , t15 ) 7→   .. . . . 1 0 0  .  .    .. . ..  1 0  0 ··· ··· 0 1 3.2. Proof of Theorem 1. We now proceed to the proof of Theorem 1, each case n ∈ {4, 5, 6} is treated separately in the next subsections. 3.2.1. Proof of Theorem 1 for sixfolds. A smooth quintic del Pezzo sixfold X is a k-form of the Grassman- nian G(2, 5). Recall [4] that isomorphism classes of k-forms of a projective k-variety X are in one-to-one correspondence with the elements of the Galois cohomology set H 1 (Γ, Autk̄ (Xk̄ )) of continuous Galois 1- cocycles γ : Γ = Gal(k̄/k) → Autk̄ (Xk̄ ), where Autk̄ (Xk̄ ) is endowed with the discrete topology and the natural action of Γ by conjugation. Since Autk̄ (G(2, 5)k̄ ) ∼ = PGL5 (k̄) = Autk̄ (P4k̄ ) (see § 1.2), isomorphism classes of k-forms of G(2, 5) are in one-to-one correspondence with isomorphism classes of k-forms P of P4k . In view of Lemma 6, the existence of a k-rational point is a necessary condition for the existence of a vector group structure on X. Conversely, for a k-form X of G(2, 5) containing a k-rational point, the corresponding k-form P of P4k contains a closed sub-variety C defined over k whose base extension Ck̄ is a line in Pk̄ ∼ = P4k̄ . 2By [2], every complete toric variety admitting a vector group structure has such a structure which is normalized by the torus. We used the term “toric” here to indicate the fact that the given G6a -action is normalized by the toric structure on P6k .  DEL PEZZO QUINTICS AS EQUIVARIANT COMPACTIFICATIONS OF VECTOR GROUPS 12 Since the restriction of a canonical divisor KP of P to C has odd degree, C is the trivial k-form of P1k and hence, P is the trivial k-form of P4k . This implies in turn that X is isomorphic to G(2, 5). By Proposition 24, G(2, 5) admits at least one vector group structure. The following proposition completes the proof of Theorem 1 in the case n = 6. Proposition 26. The Grassmannian G(2, 5) has a unique class of vector group structure. Proof. Write G(2, 5) = Gk (V ∨ , 2) = G for some 5-dimensional k-vector space V . Through the isomorphism PGLk (V ∨ ) ∼ = Autk (G) of (1.1), a vector group structure on G is given by the projective representation associated to a faithful representation ρ : U → GLk (V ∨ ) of a vector group U. Let V1 ⊂ V be a 1-dimensional k-vector subspace of V that is invariant for the representation dual to ρ, its existence being guaranteed by Lemma 3(3). Since the action of GLk (V ) on such 1-dimensional subspaces is transitive, up to the choice of a basis of V as in Notation 10 and up to changing ρ by its conjugate by a suitable automorphism of G we henceforth assume without loss of generality that V1 = he5 i and put V̄ = V /V1 ∼ = he1 , . . . , e4 i. The image of ρ is then contained in the stabilizer ∆V1 of the subspace V̄ ∨ ⊂ V ∨ , see Notation 13. Let U′ be the kernel of the induced representation U → GLk (V̄ ∨ ) and let ρ̄ : Ū = U/U′ → GLk (V̄ ∨ ) be the induced faithful representation. With the notation of § 2.2 and Example 18, the projection πΠ : G 99K Ḡ = {w12 w 34 − w 13 w 24 + w 14 w 23 = 0} ⊂ P5k from the solid Π = σ3,0 (V1 ) is then U-equivariant for the action of U on Ḡ factoring through the action of Ū determined under the isomorphism Autk (Ḡ) ∼ = PGLk (V̄ ∨ ) by the projective representation induced by ρ̄. The U-action on G lifts to a vector group structure on the blow-up YΠ → G of Π and, by Proposition 8 applied to the induced morphism pḠ : YΠ → Ḡ, the Ū-action on Ḡ defines a vector group structure on it. In particular, Ū is 4-dimensional, say Ū = Spec(k[t13 , t23 , t14 , t24 ]). By [37], Ḡ admits a unique vector group structure given up to isomorphism by the projective representation associated to the representation   1 0 −t23 −t24  0 1 t13 t14  Ū → GLk (V̄ ∨ ), t = (tij )i=1,2,j=3,4 7→ A(t) =  0 0 . 1 0  0 0 0 1 ′ Write U = Spec(k[s1 , s2 ]), U =∼ Spec(k[t̄][s1 , s2 ]) =∼ Ū × U and put u = (t, s1 , s2 ). Then, with Notation 13, ′ the representation ρ : U → ∆V1 lifting ρ̄ : Ū → GLk (V̄ ∨ ) has the form u 7→ M (u) = M (A(t̄), 1, t L(u)) for some row matrix L(u) = (fi (u))i=1,...4 of elements of k[u] such that M (u+u′ ) = M (u)M (u′ ) = M (u)M (u′ ) = M (u′ + u). By direct computation, this identity implies that f3 = f4 = 0 and that f1 and f2 are linear polynomials. Moreover, since ρ is injective, we must have k[t̄, f1 , f2 ] = k[t̄, s1 , s2 ]. Denoting t̄ij , f1 and f2 anew by tij , −t25 and t15 then identifies ρ with the representation (3.1) in Example 25.  3.2.2. Proof of Theorem 1 for fivefolds. The following proposition establishes the first part of the assertion of Theorem 1 for n = 5. Proposition 27. A smooth quintic del Pezzo fivefold is isomorphic to a smooth hyperplane section of G(2, 5) and all these sections are isomorphic. Proof. Let X be smooth quintic del Pezzo fivefold. By [19], Xk̄ is isomorphic to a smooth hyperplane section Zhsi of G = Gk̄ (V ∨ , 2) for some 5-dimensional k̄-vector space V . By Lemma 14, X contains a unique 3- dimensional sub-scheme Π whose base extension Πk̄ to k̄ is a σ3,0 -solid of Xk̄ . Let IΠk̄ ⊂ OXk̄ be the ideal sheaf of Πk̄ , let OXk̄ (1) = Λ2 Q|Xk̄ and consider the projection πΠ : X = Zhsi 99K Qhs̄i ⊂ P (H 0 (X , IΠ (1)) ∼ k̄ k̄ k̄ k̄ = P4 k̄ k̄ from Πk̄ . Since the action of the Galois group Γ = Gal(k̄/k) on Xk̄ maps smooth hyperplane sections of Xk̄ to smooth hyperplane sections, the projective space Pk̄ (H 0 (Xk̄ , IΠk̄ (1)) inherits a natural continuous linear Galois action of Γ. By Galois descent for quasi-projective varieties and rational maps between these, the map πΠk̄ thus descends to a rational map πΠ : X 99K Q ⊂ P4 whose image is a k-form Q of Qhs̄i in a k-form P4 of P4k̄ . The divisor −KP4 − 2Q being defined over k and of degree 1, P4 is the trivial form of P4k . Let pX : Y → X be the blow-up of Π. Then, by Proposition 17, the induced morphism pQ : Y → Q is an étale locally trivial P2 -bundle whose base extension to k̄ is isomorphic to PQhsi (S ⊕ OQhs̄i ) where S denotes the spinor sheaf on Qhs̄i . Furthermore, the restriction pQ : E → Q of pQ to the exceptional divisor of pX is an étale locally trivial P1 -sub-bundle of pQ : Y → Q, whose base extension to k̄ is isomorphic to the sub-bundle PQhs̄i (S ) of PQhs̄i (S ⊕ OQhs̄i ). Thus, considering the direct image by pQ of the exact sequence 0 → OY → OY (E) → OE (E) → 0 on Y , we conclude that pQ : Y → Q is isomorphic to the P2 -bundle PQ (E) where E = (pQ )∗ OY (E) is a locally free sheaf of rank 3 on Q. Furthermore, SQ = (pQ )∗ OE (E) is a locally  DEL PEZZO QUINTICS AS EQUIVARIANT COMPACTIFICATIONS OF VECTOR GROUPS 13 free sheaf of rank 2 whose base extension to k̄ is isomorphic to the spinor sheaf S on Qk̄ ∼ = Qhs̄i . Since H 0 (Qhs̄i , S ∨ ) ∼ = H 0 (Q, SQ∨ ) ⊗k k̄ by flat base change and since S ∨ is globally generated by sections whose zero schemes are lines of Qhs̄i ⊂ P4k̄ (see e.g. [32]), we infer that Q contains a line of P4 ∼ = P4k . A smooth 4 quadric of Pk containing a line being unique up to isomorphism and equal to a hyperplane section of the smooth quadric G(2, 4) ⊂ P5k , we conclude by reading the Sarkisov link of Proposition 20 backwards that X is isomorphic to a smooth hyperplane section of G(2, 5) over k. The transitivity of the action of Autk (G)(k) on the set of such smooth sections, see § 2.1.2, implies that they are all isomorphic.  Since by Proposition 24 every smooth hyperplane section of G(2, 5) ⊂ P9k admits a vector group structure, the following proposition completes the proof of Theorem 1 in the case n = 5. Proposition 28. A smooth hyperplane section of G(2, 5) has a unique class of vector group structure. Proof. Write G(2, 5) = Gk (V ∨ , 2) = G for some 5-dimensional k-vector space V . The action of Autk (G) on the set of smooth hyperplane sections being transitive, we are reduced without loss of generality to prove that the smooth hyperplane section Zhsi associated to the skew-symmetric form s = e∨ ∨ ∨ 1 ∧ e3 − e2 ∧ e4 ∨ ⊥ with V = he5 i considered in Example 25 admits a unique class of vector group structure. Under the isomorphism of Corollary 19, a vector group structure on Zhsi is given by a certain faithful representation ρ : U → GLk (V ∨ ) of a vector group U with image contained in the subgroup of the stabilizer ∆V ⊥ of the subspace V̄ ∨ ⊂ V ∨ consisting of matrices of the form   A4 U M (A4 , λ, U ) = with A4 ∈ Sp4 (V̄ ∨ , s̄), U ∈ Homk ((V ⊥ )∨ , V̄ ∨ ), λ ∈ GLk ((V ⊥ )∨ ) = Gm,k . 0 λ Let U′ be the kernel of the induced representation U → Sp4 (V̄ ∨ , s̄) and let ρ̄ : Ū = U/U′ → Sp4 (V̄ ∨ , s̄) be the induced injective homomorphism. With the notation of § 2.2 and Example 18, the projection πΠ : Zhsi 99K Qhs̄i = {w12 w34 − w 213 + w14 w23 = 0} from the solid Π = σ3,0 (V ⊥ ) is then U-equivariant for the action of U on Qhs̄i factoring through the action of Ū determined under the isomorphism Autk (Qhs̄i ) ∼ = PSpk (V̄ ∨ , s̄) by the projective representation induced by ρ̄. The U-action on Zhsi lifts to a vector group structure on the blow-up YΠ → Zhsi of Π and, by Proposition 8 applied to the induced morphism pQhs̄i : YΠ → Qhs̄i , the Ū-action on Qhs̄i defines a vector group structure on it. In particular, Ū is 3-dimensional, say Ū = Spec(k[t13 , t23 , t24 ]). By [37], Qhs̄i admits a unique vector group structure up to isomorphism, which, with our choice of coordinates, is induced by the representation ρ̄5 : Vk (F5∨ )/Stab(he∨ ∨ ′ 5 i) → Sp4 (V̄ , s̄) of (3.3) in Example 25. Writing U = Spec(k[s1 , s2 ]), U∼ = Spec(k[t̄][s1 , s2 ]) ∼ ′ = Ū×U and u = (t, s1 , s2 ), the same argument as in the proof of Proposition 26 implies that the lift ρ : U → ∆V ⊥ ⊂ GLk (V ∨ ) of ρ̄ : Ū → Sp4 (V̄ ∨ , s̄) has the form u 7→ M (u) = M (A(t̄), 1, t L(s)) where L(s) = (f1 (s), f2 (s), 0, 0) is a row matrix of linear elements of k[s] with the property that k[t̄, f1 , f2 ] = k[t̄, s1 , s2 ]. Denoting the t̄ij , f1 and f2 anew by tij , −t25 and t15 respectively, we get that ρ : U → GLk (V ∨ ) is the representation (3.2) of Example 25.  3.2.3. Proof of Theorem 1 for fourfolds. The following proposition completes the proof of Theorem 1. Proposition 29. A smooth quintic del Pezzo fourfold is isomorphic to a smooth section of G(2, 5) ⊂ P9k by a linear subspace of codimension 2. Furthermore, all such sections are isomorphic and admit exactly one class of vector group structure. Proof. Let X be smooth quintic del Pezzo fourfold. By [19], we can assume that Xk̄ is a smooth section WL of G(2, 5)k̄ by a linear subspace of codimension 2. By Lemma 16, X contains a unique 2-dimensional sub-scheme Ξ whose base extension Ξk̄ to k̄ is a σ2,2 -plane of WL . Let IΞk̄ ⊂ OXk̄ be the ideal sheaf of Ξk̄ , let OXk̄ (1) = Λ2 Q|Xk̄ and consider the projection πΞ : X = WL 99K P (H 0 (X , IΞ (1)) = ∼ P4 k̄ k̄ k̄ k̄ k̄ k̄ 0 from Ξk̄ as in § 2.3. The projective space Pk̄ (H (Xk̄ , IΞk̄ (1)) inherits a natural continuous linear action of the Galois group Γ = Gal(k̄/k), which stabilizes the hyperplane corresponding to the unique section of IΞk̄ (1) whose zero scheme is the special hyperplane section ZΞk̄ ∩ Xk̄ of Xk̄ spanned by the σ3,1 -planes of Xk̄ . In particular, the divisor ZΞk̄ ∩ Xk̄ is defined over k, say ZΞk̄ ∩ Xk̄ = (ZΞ )k̄ for some geometrically irreducible divisor ZΞ ⊂ X containing Ξ. In view of Proposition 20, the projection πΞk̄ thus descends to a birational map πΞ : X 99K P4 , with image equal to a k-form P4 of P4k , which contract ZΞ onto a smooth curve C contained in a hypersurface P3 ⊂ P4 such that the triple (P4 , P3 , C) is a k-form of the triple (P4k , P3k , C3 ), where P3k is a hyperplane of P4k and C3 ⊂ P3k is a smooth rational cubic curve. Thus, P4 and P3 are trivial forms of P4k and P3k respectively, and, since the intersection of C with a hyperplane of P3k is a divisor of degree  DEL PEZZO QUINTICS AS EQUIVARIANT COMPACTIFICATIONS OF VECTOR GROUPS 14 3 on C, it follows that C ∼ = P1k . Reading the Sarkisov link of Proposition 20 backwards, we conclude that X is isomorphic to a smooth section of G(2, 5) by a linear subspace of codimension 2, and the uniqueness up to isomorphism follows from the transitivity of the action PGL5 (k) on triples (P4k , P3k , C3 ) where C3 ⊂ P3k is a smooth rational cubic. Proposition 24 implies that X admits a vector group structure. To show the uniqueness, we observe that being unique, the σ2,2 -plane Ξ is stable under any faithful action of a unipotent abelian group U on X defining a vector group structure on X. This vector group structure lifts to the blow-up pΞ : Y → X of Ξ and then, by Proposition 8, descends via the contraction p2 : Y → P4k of the proper transform of ZΞ to a faithful U-action on P4k defining a vector group structure, for which the pair (P3k , C) is globally U-stable. By the classification of vector group structures on P4k , see [21] or [22, Corollary 3.6], the unique class of vector group structure with this property is the toric G4a -structure on P4k described in Example 25.  Remark 30. A by-product of Proposition 29 is that among the four non-isomorphic compactifications of A4C into the smooth quintic del Pezzo fourfold classified by Prokhorov [34, Theorem 3.1], only the case (i) can be endowed with a vector group structure making it an equivariant compactification of G4a,C . Remark 31. By [14, § 2.2], smooth sections of G(2, 5) ⊂ P9k by linear subspaces of codimension 3 have in general non-trivial k-forms, whose isomorphism classes are parametrized by equivalence classes of non- degenerate ternary quadratic forms over k. In contrast, Proposition 27 and Proposition 29 imply that smooth del Pezzo quintics of dimension n = 4, 5 do not have non-trivial k-forms. Since these are compactifications of Ank , one deduces from the techniques in loc. cit. that a proper morphism f : X → Y between normal varieties over an algebraically closed field of characteristic zero whose general closed fibers are smooth quintic del Pezzo varieties of dimension n = 4, 5 contains a vertical An -cylinder in the sense of [14]. Similarly, a proper morphism f : X → Y whose general closed fibers are isomorphic to G(2, 5) and which has a rational section contains a vertical A6 -cylinder. 4. Vector group structures on terminal quintic del pezzo threefolds and canonical quintic del Pezzo surfaces Classifying vector group structures on all del Pezzo quintics, including singular ones, is a challenging problem. For instance, many families of singular del Pezzo quintics of dimension n = 3, 4, 5 endowed with a vector group structure can be constructed as hyperplanes sections of G = G(2, 5) with respect to the Plücker embedding containing a σ2,2 -plane Ξ of G (see § 2.3 for the notation) which is fixed by the G6a,k -structure on G corresponding under the birational projection πΞ : G = G(2, 5) 99K P6k = P(H 0 (G, IΞ (1))) to the toric G6a,k -structure on P6k described in Example 25. Namely, every linear subspace L of codimension m = 1, 2, 3 of P6k which intersects the complement H∞ = {w12 = 0} of the open orbit transversely has stabilizer isomorphic 6−m to Gm a,k and the induced effective action of the quotient group Ga,k on L defines a vector group structure on it. The proper transform of L by πΞ is then a linear section XL of G of dimension 6 − m, singular in general, endowed with a vector structure for which the inclusion XL ֒→ G is equivariant for the induced action of a subgroup G6−m 6 a,k of the group Ga,k which acts with an open orbit on G. Here, we consider mildly singular del Pezzo quintics of dimension 3 and 2, a case of special interest due to the fact that no smooth del Pezzo quintics in these dimensions admit a vector group structure. 4.1. Vector group structures on terminal quintic del Pezzo threefolds. Over algebraically closed fields of characteristic zero, non-smooth terminal quintic del Pezzo threefolds are classified in [35]. By Corollary 5.3 in loc. cit., the main invariant of such a threefold X is the rank r(X) = rkZ Cl(X) of its divisor class group, which is either 2, 3 or 4. Furthermore, all the singularities of X are ordinary double points and, by [35, Corollary 8.3.1], the number of such nodes equals r(X) − 1. Lemma 32. A terminal quintic del Pezzo threefold X over k whose base extension Xk̄ to k̄ has one or two nodes does not admit a vector group structure. Proof. Since it is enough to show that Xk̄ does not admit any vector group structure, we henceforth assume that k = k̄. Assume that X has a vector group structure. Then the latter lifts to a vector group structure on a small Q-factorialization ξ : X̂ → X of X. If X has a unique node then, by [25, Theorem 3.6] or [35, Theorem 5.2], X̂ is isomorphic to the total space of a P1 -bundle π : X̂ ∼ = PP2k (E) → P2k for some stable, 2 hence simple, locally free sheaf E of rank 2 on Pk . We would thus obtain a vector group structure on PP2k (E) which is impossible by Corollary 9. If X has two nodes then, by [35, Theorem 8.1 and Corollary 8.1.1], X̂ is isomorphic to the blow-up σ : X̂ → X̃ of the total space of the projective bundle π : X̃ = PP2k (TP2k (−1)) → P2k at a point. By Proposition 8, the vector group structure on X̂ descends to a vector group structure on X̃. But again, this is impossible since TP2k (−1) is a simple sheaf.   DEL PEZZO QUINTICS AS EQUIVARIANT COMPACTIFICATIONS OF VECTOR GROUPS 15 We now consider the case of terminal quintic del Pezzo threefolds whose base extensions to k̄ possess exactly three nodes, which we henceforth call for short trinodal quintic del Pezzo threefolds. Proposition 33. A trinodal quintic del Pezzo threefold X3 is isomorphic to a section of G(2, 5) ⊂ P9k by a linear subspace of codimension 3. It contains a unique σ2,2 -plane Ξ of G(2, 5) and the projection πΞ : G(2, 5) 99K P6k from Ξ induces the following Sarkisov link (4.1) BlSX3 HX3  / BlΞ X3 ∼ = BlSX3 P3k ❑❑❑ pX3 rrr ❑❑p❑2 rr rrr ❑❑   yr ❴r ❴ ❴ ❴ ❴ ❴πΞ❴ ❴ ❴ ❴ ❴❑❑❴❑/% r Ξ / X3 o❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ P3 o ? _ HX3 o ? _ SX3 k Φ where pX3 : BlΞ X3 → X3 is the blow-up of Ξ, p2 : BlΞ X3 → P3k is the blow-up of a smooth 0-dimensional sub-scheme SX3 of length 3 of P3k not contained in a line, HX3 is the unique hyperplane of P3k containing SX3 and BlSX3 HX3 is its proper transform. Furthermore, the birational inverse Φ of πΞ is given by the complete linear system of quadrics of P3k containing SX3 . Proof. By [35, Theorem 7.1], X3,k̄ is isomorphic to a threefold obtained from P3k̄ as the blow-up σ : Y → P3k̄ of three non-colinear closed points, say p1 , p2 , p3 , followed by the contraction ξ : Y → X3,k̄ of the proper transforms by σ of the lines in P3k̄ passing through pi and pj , 1 ≤ i < j ≤ 3 to the nodes of X3,k̄ . The class group of X3,k̄ is freely generated by the classes of the proper transform H̃ of the unique hyperplane H ⊂ P3k̄ containing the points pi and of the images Fi , 1 ≤ i ≤ 3, of the exceptional divisors of σ. By [35, Theorem 7.2], P and the Fi are the only planes contained in X3,k̄ . Their union is thus defined over k and since P is the only plane among these which fully contains the singular locus of X3,k̄ it is defined over k as well, say P = Ξk̄ for some closed sub-scheme Ξ of X3 . This implies in turn that the union of the Fi is defined over k, S P3 say Fi = Fk̄ for some closed sub-scheme F of X3 . Since the divisor (Ξ + F )k̄ = P + i=1 Fi is linearly equivalent to the proper transform in X3,k̄ of any hyperplane H ′ ⊂ P3k̄ not passing through the points blown- P3 up, it is Cartier. The invertible sheaf OX3,k̄ (P + i=1 Fi ) is then the base extension to k̄ of the invertible sheaf OX3 (1) := OX3 (Ξ + F ) and, letting I ⊂ OX3 be the ideal sheaf of the singular locus of X3 , the rational map σ ◦ ξ −1 : X3,k̄ 99K P3k̄ is the base extension of the birational map π : X3 99K P(H 0 (X3 , I(1)) ∼ = P3k . The 3 latter maps Ξ to a hyperplane HX3 ⊂ Pk and contracts F to a smooth closed sub-scheme SX3 ⊂ HX3 of length 3 whose base extension to k̄ equals the union of the points pi . Since SX3 is not contained in a line, there exists a smooth rational cubic curve SX4 ⊂ P3k whose scheme-theoretic intersection with HX3 equals SX3 . Considering P3k as a hyperplane HX4 of P4k , it follows from Proposition 20 that the image of the rational map P4k 99K P7k given by the complete linear system of quadrics containing SX4 ⊂ HX4 is a smooth quintic del Pezzo fourfold X4 containing X3 as a hyperplane section and which has the proper transform Ξ of HX4 as its unique σ2,2 -plane. By construction, the birational map π : X3 99K P3k then coincides with the restriction to X3 of the projection πΞ : X4 99K P4k from Ξ, which completes the proof.  Corollary 34. Isomorphism classes of trinodal quintic del Pezzo threefolds are in one-to-one correspondence with PGL2 (k)-orbits of smooth 0-dimensional sub-schemes of P1k of length three. Furthermore, every such threefold admits a unique class of vector group structure. Proof. By Proposition 33, two trinodal quintic del Pezzo threefolds X3 and X3′ are isomorphic if and only if there exists an automorphism of P3k which maps the pair (HX3 , SX3 ) onto the pair (HX3′ , SX3′ ). Being of length 3 and not contained in a line, the schemes SX3 and SX3′ are contained in smooth k-rational conics CX3 and CX3′ of HX3 and HX3′ , respectively. Since Autk (P3k ) acts transitively on pairs (H, C) consisting of a hyperplane H of P3k and a smooth k-rational conic C ∼ = P1k in it and since for such pairs the restriction homomorphism Autk (H, C) → Autk (C) is an isomorphism, we conclude that X3 and X3′ are isomorphic if and only if there exists an isomorphism ϕ : CX3 → CX3′ which maps SX3 onto SX3′ . This holds if and only if SX3 and ϕ−1 (SX3′ ) belong to the same orbit of the action of Autk (CX3 )(k) ∼ = PGL2 (k). For the second assertion, since the singular locus of X3 and the unique σ2,2 -plane Ξ of X3 are stable under any vector group action on X3 , it follows from Proposition 8 applied to the birational morphism BlΞ X → P3k that the Sarkisov link of Proposition 33 is equivariant for any vector group structure on X3 and that the corresponding vector group structure on P3k stabilizes the non-linear closed sub-scheme SX3 of length 3. By the classification [21] of vector group structures on P3k = Projk (k[x0 , x1 , x2 , x3 ]), the unique class of vector group structure with this property is that of the toric G3a -structure defined by x0 7→ x0 and xi 7→ xi + ti x0 , 1 ≤ i ≤ 3. Conversely, this structure lifts to a vector group structure on the blow-up of SX3 , which, by Proposition 8 again, descends in turn to a vector group structure on X3 .   DEL PEZZO QUINTICS AS EQUIVARIANT COMPACTIFICATIONS OF VECTOR GROUPS 16 Remark 35. By Proposition 33 and Corollary 34, every trinodal quintic del Pezzo threefold contains the affine 3-space A3k as a Zariski open subset. In contrast, there exist in general k-forms of smooth quintic del Pezzo threefolds contains which do not contain A3k , see [14, Theorem 12]. Example 36. With Notation 10, let V3 = he3 , e4 , e5 i and let Ξ = σ2,2 (V3 ) be the associated plane of G = G(2, 5) ⊂ P9k as in Example 21. For every β ∈ k ∗ , the linear section X3 (β) = G ∩ {w13 − w24 = 0} ∩ {βw14 − w25 = 0} ∩ {βw15 + w23 = 0} is a trinodal quintic del Pezzo threefold containing Ξ, isomorphic to the sub-variety in P6k with coordinates wij , (i, j) 6= (2, 3), (2, 4), (2, 5) defined by the equations 2    w12 w34 − w13 − βw14 w15 = 0 2 w12 w35 − βw13 w14 − βw15 =0    2 w12 w45 − βw14 + w13 w15 = 0 w13 w45 − w14 w35 + w15 w34 = 0     −βw15 w45 − w13 w35 + βw14 w34 = 0  Its singular locus Sing(X3 (β)) is the closed sub-scheme of Ξ ∼ = Proj (k[w34 , w35 , w45 ]) with equations k 2 2 2 βw34 w45 − w35 = w34 w35 − βw45 = w35 w45 − w34 = 0. Letting λ, ǫ ∈ k̄ be respectively a third root of β and a primitive third root of unity, the singular locus of X3 (β)k̄ is the union of the three closed points [λǫm : (λǫm )2 : 1], 0 ≤ m ≤ 2, of Ξ. Thus, according to whether β is cube in k ∗ or not and k ∗ contains a primitive primitive third root of unity or not, Sing(X3 (β)) consists either of a single closed point, or the union of a k-point and a single other closed point, or the union of three k-points. The image of Ξ by the restriction X3 (β) 99K P3k = Projk (k[w12 , w13 , w14 , w15 ]) of the projection from Ξ is the hyperplane HX3 (β) = {w12 = 0}. The associated smooth 0-dimensional sub-scheme SX3 (β) of length 3 is the closed sub-scheme of HX3 (β) defined by the equations 2 2 2 βw14 w15 + w13 = w13 w14 + w15 = w13 w15 − βw14 = 0. ∼ The restriction to X3 (β) of the action of the sub-group U = Ga,k of the vector group Vk (F6∨ ) of (3.1) in 3 Example 25 defined by t24 = t13 , t25 = βt14 and t23 = −βt15 induces a vector group structure    w12 7−→ w12 w13 7−→ w13 + t13 w12      w14 7−→ w14 + t14 w12   w15 7−→ w15 + t15 w12 w34 7−→ w34 + 2t13 w13 + β(t15 w14 + t14 w15 ) + (t213 + βt14 t15 )w12     w35 7−→ w35 + β(t14 w13 + t13 w14 + 2t15 w15 ) + β(t13 t14 + t215 )w12    w45 7−→ w45 + (2βt14 w14 − t15 w13 − t13 w15 ) + (βt214 − t15 t13 )w12   on X3 (β) with open orbit X3 (β) \ {w12 = 0}. 4.2. Vector group structures on canonical quintic del Pezzo surfaces. Del Pezzo surfaces with canonical singularities admitting a vector group structure are classified in [12], see also [30]. For the sake of completeness, we record the following consequence of the classification in the quintic case: Proposition 37. Up to isomorphism, there exist two quintic del Pezzo surfaces with canonical singularities which admit a vector group structure: a) A surface S with an A3 -singularity, whose neutral component Aut0 (S) of the automorphism group is isomorphic to G2a,k ⋊ Gm,k and which admits a unique class of vector group structure. b) A surface S ′ with an A4 -singularity, whose neutral component Aut0 (S ′ ) of the automorphism group is isomorphic to U3 ⋊ Gm,k , where U3 is a maximal unipotent subgroup of PGL3 (k), and which admits exactly two classes of vector group structures. Proof. All the properties but those concerning the actual number of equivalence classes of vector group structures are established in [12] and for the description of the automorphism groups in [30, Table 1], cases 5E and 5F for S and S ′ respectively. We briefly recall the principle of the argument in loc. cit. and explain how derive from it the equivalence classes of vector group structures. Equivalence classes of vector group structures on a del Pezzo surface S with canonical singularities are in one-to-one correspondence with those on its minimal desingularization S̃ → S, which is obtained from S by performing a finite sequence of successive blow-ups of singular loci of intermediate surfaces and normalizations. Indeed, a vector group structure stabilizes singular loci, hence canonically lifts to their blow-ups and, by universal property, canonically lifts as well to normalizations. Conversely, Proposition 8 ensures that every vector group structure on S̃  DEL PEZZO QUINTICS AS EQUIVARIANT COMPACTIFICATIONS OF VECTOR GROUPS 17 descend to S. Here, the surfaces S̃ are weak del Pezzo surfaces of degree 5 whose base extensions to k̄ are obtained from P2k̄ by performing certain finite sequences of blow-ups of closed points. In the two cases under consideration, the respective dual graphs of the unions of the (−1)-curves and (−2)-curves in S̃k̄ have the following structure e1 e2 e3 ℓ˜2 e1 e4 e3 e2 ◦ ◦ ◦ • ◦ ◦ ◦ ◦ • • ℓ˜1 ℓ̃ in which the vertices • and ◦ correspond respectively to (−1)-curves which are the proper transforms of the lines in Sk̄ and to (−2)-curves which are the exceptional divisors of the desingularization S̃k̄ → Sk̄ . Since these diagrams have no symmetries, all the curves displayed are defined over k, corresponding to irreducible smooth k-rational curves in S̃ with the same self-intersection numbers. In the case of an A3 -singularity, the successive contractions of ℓ̃2 , and then of the exceptional divisors e3 , e2 and e1 yield a birational morphism σ : S̃ → P2k = Projk (k[u0 , u1 , u2 ]) which maps ℓ̃1 onto a line ℓ ⊂ P2k and contracts ℓ̃2 ∪ e1 ∪ e2 ∪ e3 onto a k-point p ∈ ℓ , say, up to composition by a suitable automorphism of P2k , ℓ = {u2 = 0} and p = [1 : 0 : 0]. A vector group structure on S and its canonical lift to S̃ being given, Proposition 8 implies the existence of a unique vector group structure on P2k for which σ : S̃ → P2k is equivariant. The latter stabilizes ℓ as well as the proper and infinitely near base points of σ −1 . By [21, Proposition 3.2] there are two classes of vector group structures on P2k fixing ℓ and p: the “toric” structure given by the G2a,k -action [u0 : u1 : u2 ] 7→ [u0 + t0 u2 : u1 + t1 u2 : u2 ] and the “non-toric” one given by the G2a,k -action [u0 : u1 : u2 ] 7→ [u0 + t1 u1 + ( 21 t21 + t0 )u2 : u1 + t1 u2 : u2 ]. A direct verification shows that the lift of the toric structure to the surface S̃1 obtained from S̃ by contracting ℓ̃2 acts transitively on e3 \ e2 , hence that this structure cannot be induced by a vector group structure on S̃. On the other hand, the lift to S̃1 of the other structure fixes e3 point wise, hence is descended via the contraction of ℓ̃2 from a vector group structure on S̃. Thus, S̃, whence S, has a unique class of vector group structure. In the case of an A4 -singularity, the successive contractions of ℓ̃, and then of the exceptional divisors e4 , e3 and e2 yield birational morphism σ : S̃ → P2k which maps e1 onto a line ℓ ⊂ P2k and ℓ̃ ∪ e4 ∪ e3 ∪ e2 onto a k-point p ∈ ℓ. Up to composing by a suitable automorphism of P2k as above, we again infer that a vector group structure on S̃ is equivalent to the lift via σ of one of the two equivalence classes of such structures on P2k described above. Noting that for both structures the first three points blown-up by σ are fixed and that the lifts of these two structures to the resulting surface both fix e4 point wise, we conclude that both structures lift to S̃. These two structures descend in turn on S, showing that S has at most two equivalence classes of vector group structures. The conclusion follows from the observation that two so-constructed induced structures have non-isomorphic fixed point schemes, hence are not equivalent.  Remark 38. With the notation of the proof of Proposition 37, in the case of the del Pezzo surface S with an A3 -singularity, the contractions of ℓ̃1 , e1 , e2 and ℓ̃2 yield another birational morphism σ ′ : S̃ → P2k which maps e3 onto a line ℓ′ and contracts ℓ̃1 ∪ e2 ∪ e1 and ℓ̃2 onto a pair of disctinct k-points of ℓ. In contrast with the morphism σ : S̃ → P2k constructed in the proof of Proposition 37 which is equivariant with respect to the non-toric G2a,k -structure on P2k , the birational morphism σ ′ is equivariant with respect to the toric G2a,k -structure on P2k . The toric and non-toric structure on P2k thus become equivalent on S and hence, are birationally conjugated on P2k by the birational automorphism σ ′ ◦ σ −1 . References 1. I. V. Arzhantsev, Flag varieties as equivariant compactifications of Gn a , Proc. Amer. 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Email address: [email protected] Department of Mathematics, Faculty of Science, Saitama University, Saitama 338-8570, Japan Email address: [email protected] Departamento de Matemática Universidad Técnica Federico Santa María Avenida España 1680, Valparaíso, Chile Email address: [email protected]