UNIVERSAL TORSORS AND COX RINGS

by

Brendan Hassett and Yuri Tschinkel

Abstract. — We study the equations of universal torsors on rational surfaces.

Contents

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1. Generalities on the Cox ring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2. Generalities on toric varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 3. The E_{6} cubic surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 4. D_{4} cubic surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

Introduction

The study of surfaces over nonclosed fields k leads naturally to certain auxiliary varieties, called universal torsors. The proofs of the Hasse principle and weak approximation for certain Del Pezzo surfaces required a very detailed knowledge of the projective geometry, in fact, explicit equations, for these torsors [7], [9], [8], [22], [23], [24]. More recently, Salberger proposed using universal torsors to count rational points of bounded height, obtaining the first sharp upper bounds on split Del Pezzo surfaces of degree 5 and asymptotics on split toric varieties over ℚ [21]. This approach was further developed in the work of Peyre, de la Bretèche, and Heath-Brown [19], [20], [3], [14].

The first author was partially supported by the Sloan Foundation and by NSF Grants 0196187 and 0134259. The second author was partially supported by NSF Grant 0100277.

Colliot-Thélène and Sansuc have given a general formalism for writing down equations for these torsors. We briefly sketch their method: Let X be a smooth projective variety and {D_{j}}_{j∈J} a finite set of irreducible divisors on X such that U := X \ ∪_{j∈J}D_{j} has trivial Picard group. In practice, one usually chooses generators of the effective cone of X, e.g., the lines on the Del Pezzo surface. Consider the resulting exact sequence:

0 ⟶ k̄[U]^{∗}/k̄^{∗} ⟶ ⊕_{j∈J}ℤD_{j} ⟶ Pic(Xk̄) ⟶ 0.

Applying Hom(−, 𝔾_{m}), one obtains an exact sequence of tori

1 ⟶ T (X) ⟶ T ⟶ R ⟶ 1,

where the first term is the Néron-Severi torus of X. Suppose we have a collection of rational functions, invertible on U, which form a basis for the relations among the {D_{j}}_{j∈J}. These can be interpreted as a section U→R × U, and thus naturally induce a T (X)-torsor over U, which canonically extends to the universal torsor over X. In practice, this extension can be made explicit, yielding equations for the universal torsor.

However, when the cone generated by {D_{j}}_{j∈J} is simplicial, there are no relations and this method gives little information. In this paper, we outline an alternative approach to the construction of universal torsors and illustrate it in specific examples where the effective cone of X is simplicial.

We will work with varieties X such that the Picard and the Néron-Severi groups of X coincide and such that the ring

Cox(X) := ⨁_{L∈Pic(X)} Γ(X, L),

is finitely generated. This ring admits a natural action of the Néron-Severi torus and the corresponding affine variety is a natural embedding of the universal torsor of X. The challenge is to actually compute Cox(X) in specific examples; Cox has shown that it is a polynomial ring precisely when X is toric [10].

Here is a roadmap of the paper: In Section 1 we introduce Cox rings and discuss their general properties. Finding generators for the Cox ring entails embedding the universal torsor into affine space, which yields embeddings of our original variety into toric quotients of this affine space. We have collected several useful facts about toric varieties in Section 2. Section 3 is devoted to a detailed analysis of the unique cubic surface S with an isolated singularity of type E_{6}. We compute the (simplicial) effective cone of its minimal desingularization S̃, and produce 10 distinguished sections in Cox(S̃). These satisfy a unique equation and we show the universal torsor naturally embeds in the corresponding hypersurface in 𝔸^{10}. More precisely, we get a homomorphism from the coordinate ring of 𝔸^{10} to Cox(S̃) and the main point is to prove its surjectivity. Here we use an embedding of S̃ into a simplicial toric threefold Y, a quotient of 𝔸^{10} under the action of the Néron-Severi torus so that Cox(Y) is the polynomial ring over the above 10 generators. The induced restriction map on the level of Picard groups is an isomorphism respecting the moving cones. We conclude surjectivity for each degree by finding an appropriate birational projective model of Y and using vanishing results on it. Finally, in Section 4 we write down equations for the universal torsors (the Cox rings) of a split and a nonsplit cubic surface with an isolated singularity of type D_{4}.

For an account of the Cox rings of smooth Del Pezzo surfaces, we refer the reader to paper of Batyrev and Popov in this volume [2].

Acknowledgments: The results of this paper have been reported at the American Institute of Mathematics conference “Rational and integral points on higher dimensional varieties”. We benefited from the comments of the other participants, in particular, V. Batyrev and J.L. Colliot-Thélène. We also thank S. Keel for several helpful discussions about Cox rings and M. Thaddeus for advice about the geometric invariant theory of toric varieties.

1. Generalities on the Cox ring

For any finite subset Ξ of a real vector space, let Cone(Ξ) denote the closed cone generated by Ξ.

Let X be a normal projective variety of dimension n over an algebraically closed field k of characteristic zero. Let A_{n−1}(X) and N_{n−1}(X) denote Weil divisors on X up to linear and numerical equivalence, respectively. Let A_{1}(X) and N_{1}(X) denote the classes of curves up to equivalence. Let NE_{n−1}(X) ⊂ N_{n−1}(X)_{ℝ} denote the cone of (pseudo)effective divisors, i.e., the smallest real closed cone containing all the effective divisors of X. Let NE_{1}(X) ⊂ N_{1}(X)_{ℝ} denote the cone of effective curves and NM^{1}(X) ⊂ N_{n−1}(X)_{ℝ} the cone of nef Cartier divisors, which is dual to the cone of effective Cartier divisors. By Kleiman’s criterion, this is the smallest real closed cone containing all ample divisors of X.

Let L_{1}, …, L_{r} be invertible sheaves on X. For ν = (n_{1}, …, n_{r}) ∈ ℕ^{r} write

L^{ν} := L_{1}^{⊗n_{1}} ⊗ … ⊗ L_{r}^{⊗n_{r}}.

Consider the ring

R(X, L_{1}, …, L_{r}) := ⨁_{ν∈ℕ^{r}} Γ(X, L^{ν}),

which need not be finitely generated in general.

By definition, an invertible sheaf L on X is semiample if L^{N} is globally generated for some N > 0:

Proposition 1.1. — ([15], Lemma 2.8) If L_{1}, …, L_{r} are semiample then R(X, L_{1}, …, L_{r}) is finitely generated.

Remark 1.2. — If the L_{i} are ample then, after replacing each L_{i} by a large multiple, R(X, L_{1}, …, L_{r}) is generated by

Γ(X, L_{1}) ⊗ … ⊗ Γ(X, L_{r}).

However, this is not generally the case if the L_{i} are only semiample (despite the assertion in the second part of Lemma 2.8 of [15]). Indeed, let X→ℙ^{1} × ℙ^{1} be a double cover and L_{1} and L_{2} be the pull-backs of the polarizations on the ℙ^{1}’s to X. For suitably large n_{1} and n_{2}, L_{1}^{n_{1}} ⊗ L_{2}^{n_{2}} is very ample and its sections embed X. However,

Γ(X, L_{1}^{n_{1}}) ⊗ Γ(X, L_{2}^{n_{2}}) ≃ Γ(ℙ^{1}, O_{ℙ^{1}}(n_{1})) ⊗ Γ(ℙ^{1}, O_{ℙ^{1}}(n_{2})),

and any morphism induced by these sections factors through ℙ^{1} × ℙ^{1}.

Proposition 1.3. — Let L_{1}, …, L_{r} be a set of invertible sheaves on X such that L_{j} is generated by sections s_{j,0}, ..., s_{j,d_{j}}. Assume that the induced morphism X→ ∏_{j} ℙ^{d_{j}} is birational into its image. Then the ring generated by the s_{j,k}’s has the same fraction field as R(X, L_{1}, …, L_{r}).

Proof. — Both rings have fraction field k(X)(t_{1}, ..., t_{r}), where t_{j} is a nonzero section of L_{j}.

Definition 1.4. — [15] Let X be a nonsingular projective variety so that Pic(X) is a free abelian group of rank r. The Cox ring for X is defined

Cox(X) := R(X, L_{1}, …, L_{r}),

where L_{1}, …, L_{r} are lines bundles so that 1. the L_{i} form a ℤ-basis of Pic(X);

2. the cone Cone({L_{1}, …, L_{r}}) contains NE_{n−1}(X). This ring is naturally graded by Pic(X): for ν ∈ Pic(X) the ν-graded piece is denoted Cox(X)_{ν}.

Proposition 1.5. — [15] The ring Cox(X) does not depend on the choice of generators for Pic(X).

Proof. — Consider two sets of generators L_{1}, …, L_{r} and M_{1}, …, M_{r}. Since Cone({L_{i}}) and Cone({M_{i}}) contain all the effective divisors, the nonzero graded pieces of both R(X, L_{1}, …, L_{r}) and R(X, M_{1}, …, M_{r}) are indexed by the effective divisor classes in Pic(X). Choose isomorphisms

M_{j} ≃ L^{(a_{1j},...,a_{rj})}, i = 1, …, r, A = (a_{ij})

which naturally induce isomorphisms

Γ(M^{ν}) ≃ Γ(L^{Aν}), Aν = (a_{11}ν_{1} + … + a_{1r}ν_{r}, …, a_{r1}ν_{1} + … + a_{rr}ν_{r}).

Thus we find R(X, L_{1}, …, L_{r}) ≃ R(X, M_{1}, …, M_{r}).

As Cox(X) is graded by Pic(X), a free abelian group of rank r, the torus

T (X) := Hom(Pic(X), 𝔾_{m})

acts on Cox(X). Indeed, each ν ∈ Pic(X) naturally yields a character χ_{ν} of T (X), and the action is given by

t · ξ = χ_{ν}(t)ξ, ξ ∈ Cox(X)_{ν}, t ∈ T (X).

Thus the isomorphism constructed in Proposition 1.5 is not canonical: Two such isomorphisms differ by the action of an element of T (X). It is precisely this ambiguity that makes descending the universal torsor to nonclosed fields an interesting question.

The following conjecture is a special case of 2.14 of [15]:

Conjecture 1.6 (Finiteness of Cox ring). — Let X be a log Fano variety. Then Cox(X) is finitely generated.

Remark 1.7. — Note that if Cox(X) is finitely generated it follows trivially that NE_{n−1}(X) is finitely generated. Moreover, the nef cone NM^{1}(X) is also finitely generated.

Indeed, the nef cone corresponds to one of the chambers in the group of characters of T (X) governed by the stability conditions for points v ∈ Spec(Cox(X)). These chambers are bounded by finitely many hyperplanes (see Theorem 0.2.3 in [11] for more details).

It has been conjectured by Batyrev [1] that the pseudo-effective cone of a Fano variety is finitely generated. However, the finiteness of the Cox ring is not a formal consequence of the finiteness of the pseudo-effective cone.

Example 1.8. — Let p_{1}, …, p_{9} ∈ H ⊂ ℙ^{3} be nine distinct coplanar points given as a complete intersection of two generic cubic curves in the hyperplane H, and let X be the blow-up of ℙ^{3} at these points. Then NE^{1}(X) is finitely generated but Cox(X) is not. Indeed, X is an equivariant compactification of the additive group 𝔾_{a}^{3}, acting by translation on the affine space ℙ^{3} − H. The group action can be used to show that NE^{1}(X) is generated by the boundary components (see [13]). Similarly, one can show that the cone NE_{1}(X) is generated by classes of curves in the boundary components, e.g., the proper transform H̃ ⊂ X of H. It is well-known that NE_{1}(H̃) is infinite [17] §1.23(4): The pencil of cubic plane curves with base locus p_{1}, …, p_{9} induces an elliptic fibration,

H̃→ℙ^{1},

for which the nine exceptional curves of H̃→H are sections. Addition in the group law gives an infinite number of sections, which are also (−1)- curves and generators of NE_{1}(H̃). These are also generators of NE_{1}(X), since the sections (other than the nine exceptional curves) intersect H̃ negatively. It follows that NE_{1}(X) and NM^{1}(X) are not finitely generated and hence Cox(X) is not finitely generated (see Remark 1.7).

Proposition 1.9. — Let X be a nonsingular projective variety whose anticanonical divisor −K_{X} is nef and big. Suppose that D is a nef divisor on X. Then H^{i}(X, O_{X}(D)) = 0 for each i > 0 and D is semiample.

Proof. — The first assertion is a consequence of Kawamata-Viehweg vanishing [17] §2.5. The second is a special case of the Kawamata Basepointfreeness Theorem [17] §3.2.

Proposition 1.9 largely determines the Hilbert function of the Cox ring:

Corollary 1.10. — Retain the assumptions of Proposition 1.9. Then for nef classes ν we have

dim Cox(X)_{ν} = χ(O_{X}(ν)).

Remark 1.11. — In practice, this will help us to find generators of Cox(X).

2. Generalities on toric varieties

We recall quotient constructions of toric varieties, following Brion-Procesi [4], Cox [10], and Thaddeus [25].

Let T ≃ 𝔾_{m}^{r} be a torus with character group 𝔛^{∗}(T). Suppose that T acts faithfully on the polynomial ring k[x_{1}, …, x_{n+r}] by the formula

t(x_{j}) = χ_{j}(t)x_{j}, t ∈ T,

where {χ_{1}, …, χ_{n+r}} ⊂ 𝔛^{∗}(T). Define M as the kernel of the surjective morphism

χ := (χ_{1}, …, χ_{n+r}) : ℤ^{n+r}→𝔛^{∗}(T).

We interpret M as the character group of the quotient torus 𝔾_{m}^{n+r}/T. Set N = Hom(M, ℤ) so that dualizing gives

(ℤ^{n+r})^{∗}→N→0.

Let e_{1}, …, e_{n+r} and e_{1}^{∗}, …, e_{n+r}^{∗} denote the coordinate vectors in ℤ^{n+r} and (ℤ^{n+r})^{∗}; let ē_{1}^{∗}, …, ē_{n+r}^{∗} ∈ N denote the images of the e_{i}^{∗} in N. Concretely, the ē_{i}^{∗} are the columns of the n×(n+r) matrix of dependence relations among the χ_{j}.

Consider a toric n-fold X associated with a fan having one-skeleton {ē_{1}^{∗}, …, ē_{n+r}^{∗}}. In particular, we assume that none of ē_{i}^{∗} is zero or a positive multiple of any of the others. The variety X is a categorical quotient of an invariant open subset U ⊂ 𝔸^{n+r} under the action of T described above (see [10] 2.1). Elements ν ∈ 𝔛^{∗}(T) classify T -linearied invertible sheaves L_{ν} on 𝔸^{n+r} and

Γ(𝔸^{n+r}, L_{ν}) ≃ k[x_{1}, …, x_{n+r}]_{ν}.

We have A_{n−1}(X) ≃ 𝔛^{∗}(T) and we can identify

Γ(O_{X}(D)) ≃ k[x_{1}, …, x_{n}]_{ν(D)},

where ν(D) ∈ 𝔛^{∗}(T) is associated with the divisor class of D. The variables x_{i} are associated with the irreducible torus-invariant divisors D_{i} on X (see [12] §3.4), and the cone of effective divisors NE_{n−1}(X) is generated by {D_{1}, …, D_{n+r}}. Geometrically, the effective cone in 𝔛^{∗}(T) is the image of the standard simplicial cone generated by e_{1}, …, e_{n+r} under the projection homomorphism χ : ℤ^{n+r}→𝔛(T).

Recall that the moving cone

Mov(X) ⊂ NE_{n−1}(X)

is defined as the smallest closed subcone containing the effective divisors on X without fixed components.

Proposition 2.1. — Retaining the notation and assumptions above,

Mov(X) = ⋂_{i=1,...,n+r} Cone(χ_{1}, …, χ_{i−1}, χ_{i+1}, …, χ_{n+r})

and has nonempty interior.

Proof. — The fixed components of Γ(X, O_{X}(D)) are necessarily invariant under the torus action, hence are taken from {D_{1}, …, D_{n+r}}. Moreover, D_{i} is fixed in each Γ(X, O_{X}(dD)), d > 0 if and only if x_{i} divides each element of k[x_{1}, …, x_{n+r}]_{dν(D)}. This is the case exactly when

ν(D) ∈ Cone(χ_{1}, …, χ_{n+r}) − Cone(χ_{1}, …, χ_{i−1}, χ_{i+1}, …, χ_{n+r}).

Suppose that the interior of the moving cone is empty. After permuting indices there are two possibilities: Either Cone(χ_{2}, …, χ_{n+r}) has no interior, or the cones Cone(χ_{2}, …, χ_{n+r}) and Cone(χ_{1}, χ_{3}, …, χ_{n+r}) have nonempty interiors but meet in a cone with positive codimension. As the T -action is faithful, the χ_{i} span 𝔛^{∗}(T). In the first case, χ_{2}, …, χ_{n+r} span a codimension-one subspace of 𝔛^{∗}(T) that does not contain χ_{1}, so that each dependence relation

c_{1}χ_{1} + … + c_{n+r}χ_{n+r} = 0

has c_{1} = 0. This translates into ē_{1}^{∗} = 0, a contradiction. In the second case, χ_{3}, …, χ_{n+r} span a hyperplane, and χ_{1} and χ_{2} are on opposite sides of this hyperplane. Putting the dependence relations among the χ_{i} in row echelon form, we obtain a unique relation with nonzero first and second entries, and these two entries are both positive. This translates into the proportionality of ē_{1}^{∗} and ē_{2}^{∗}.

We now seek to characterize the projective toric n-folds X with oneskeleton {ē_{1}^{∗}, …, ē_{n+r}^{∗}}. These are realized as Geometric Invariant Theory quotients 𝔸^{n+r}//T associated with the various linearizations of our T - action. We consider the graded ring

R := ∑_{d≥0} Γ(𝔸^{n+r}, L_{dν}) = ∑_{d≥0} k[x_{1}, …, x_{n+r}]_{dν}.

Proposition 2.2 (see [25] §2,3). — Retain the notation above and set X := Proj(R).

1. X is projective over k if and only if 0 is not contained in the convex hull of {χ_{1}, …, χ_{n+r}}.

2. X is toric of dimension n if ν is in the interior of

Cone(χ_{1}, …, χ_{n+r}).

3. In this case, the one-skeleton of X is contained in {ē_{1}^{∗}, …, ē_{n+r}^{∗}}. Equality holds if ν is in the interior of the moving cone

⋂_{i=1,...,n+r} Cone(χ_{1}, …, χ_{i−1}, χ_{i+1}, …, χ_{n+r}).

Remark 2.3. — Our proof will show that X may still be of dimension n even when ν is contained in a facet of

Cone(χ_{1}, …, χ_{n+r}).

Similary, the one-skeleton of X may still be {ē_{1}^{∗}, …, ē_{n+r}^{∗}} even when ν is contained in a facet of

⋂_{i=1,...,n+r} Cone(χ_{1}, …, χ_{i−1}, χ_{i+1}, …, χ_{n+r}).

Proof. — The monomials which appear in R are in one-to-one correspondence to solutions of

a_{1}χ_{1} + … + a_{n+r}χ_{n+r} = dν, a_{i} ∈ ℤ_{≥0}.

In geometric terms, the monomials appearing in R coincide with the elements of ℤ^{n+r} in the cone

χ^{−1}(Cone(ν)) ∩ Cone(e_{1}, …, e_{n+r}).

By Gordan’s Lemma in convex geometry, R is generated as a k-algebra by a finite set of monomials x^{m_{1}}, …, x^{m_{s}}. The monomials appearing in the dth graded piece R_{d} coincide with elements of ℤ^{n+r} in the polytope

P_{dν} := χ_{ℝ}^{−1}(dν) ∩ Cone(e_{1}, …, e_{n+r}).

Note that χ^{−1}(dν) is a translate of M.

For the first part, recall that Proj(R) is projective over Spec(R_{0}), where R_{0} is the degree-zero part. Now 0 is in the convex hull of {χ_{1}, …, χ_{n+r}} if and only if there are nonconstant elements of R of degree zero. Our hypothesis just says that R_{0} = k and thus is equivalent to the projectivity of X over k.

As for the second part, T acts on R by homotheties and thus acts trivially on Proj(R), so we have an induced action of 𝔾_{m}^{n+r}/T on Proj(R). We claim this action is faithful, so the quotient is toric of dimension n. Let μ_{1}, …, μ_{n} be generators for M = 𝔛^{∗}(𝔾_{m}^{n+r}/T). Choose v ∈ ℤ^{n+r} in the interior of Cone(e_{1}, …, e_{n+r}) so that χ_{ℝ}(Cone(v)) = Cone(ν).

Replacing v by a suitably large integral multiple, we may assume each v+μ_{i}, i = 1, …, n, is in Cone(e_{1}, …, e_{n+r}). If χ(v) = dν then R_{d} contains a set of generators for M, so the induced representation of 𝔾_{m}^{n+r}/T on R_{d} is faithful.

For the third part, we extract the fan classifying X from P_{dν}, following [12] §1.5 and 3.4: For each face Q of P_{dν}, consider the cone

σ_{Q} = {v ∈ N_{ℝ} : ⟨u, v⟩ ≤ ⟨u′, v⟩ for all u ∈ Q, u′ ∈ P_{dν}}.

This assignment is inclusion reversing, so the one-dimensional cones of the fan correspond to facets of P_{dν}. Moreover, each facet of P_{dν} is induced by one of the facets of Cone(e_{1}, …, e_{n+r}). The corresponding one-dimensional cone in N_{ℝ} is spanned by ē_{i}^{∗}. It remains to verify that each facet of Cone(e_{1}, …, e_{n+r}) actually induces a facet of P_{dν}. The hypothesis that ν is in the moving cone means that P_{dν} intersects each of the Cone(e_{1}, …, e_{i−1}, e_{i+1}, …, e_{n+r}). If ν is in the interior of the moving cone then the intersection of P_{dν} with Cone(e_{1}, …, e_{i−1}, e_{i+1}, …, e_{n+r}) meets the relative interior of this cone, hence this cone induces a facet of P_{dν}.

Proposition 2.2 yields the following nice consequence:

Proposition 2.4. — Let X be a complete toric variety and ν a divisor class in the interior of Mov(X). Then there exists a projective toric variety Y_{ν}, with the same one-skeleton as X, and polarized by ν.

For generic T -linearized invertible sheaves on 𝔸^{n+r}, all semistable points are actually stable; hence Y_{ν} is a simplicial toric variety for generic ν (see [4] 1.2 and [10] 2.1). For the special values ν_{0}, contained in the walls of the chamber decomposition of [25], this fails to be the case. However, for each special ν_{0}, there exists a generic ν so that Cone(ν) is very close to Cone(ν_{0}) and there is a projective, torus-equivariant morphism Y_{ν}→Y_{ν_{0}} [25] 3.11. The polarization associated to ν_{0} pulls back to Y_{ν}, so we obtain the following:

Proposition 2.5. — Let X be a complete toric variety and ν_{0} a divisor class in the moving cone of X. Then there exists a simplicial projective toric variety Y, with the same one-skeleton as X, so that ν_{0} is semiample on Y.

Of course, ν_{0} is big when it is in the interior of the effective cone.

3. The E_{6} cubic surface

By definition, the E_{6} cubic surface is given by the homogeneous equation

(3.1)  S = {(w, x, y, z) : xy^{2} + yw^{2} + z^{3} = 0} ⊂ ℙ^{3}.

We recall some elementary properties (see [5] for more details on singular cubic surfaces):

Proposition 3.1. —

1. The surface S has a single singularity at the point p := (0, 1, 0, 0), of type E_{6}.

2. S is the unique cubic surface with this property, up to projectivity.

3. S contains a unique line, satisfying the equations y = z = 0.

Any smooth cubic surface may be represented as the blow-up of ℙ^{2} at six points in ‘general position’. There is an analogous property of the E_{6} cubic surface:

Proposition 3.2. — The E_{6} cubic surface S is the closure of the image of ℙ^{2} under the linear series

w = a^{2}c x = −(ac^{2} + b^{3}) y = a^{3} z = a^{2}b,

where

Γ(ℙ^{2}, O_{ℙ^{2}}(1)) = ⟨a, b, c⟩.

This map is the inverse of the projection of S from the double point p. The affine open subset

𝔸^{2} := {a ≠ 0} ⊂ ℙ^{2}

is mapped isomorphically onto S − ℓ. In particular, S \ ℓ ≃ 𝔸^{2}, so the E_{6} cubic surface is a compactification of 𝔸^{2}.

Remark 3.3. — Note that S is not an equivariant compactification of 𝔾_{a}^{2}, so the general theory of [6] does not apply.

Indeed, if S were an equivariant compactification of 𝔾_{a}^{2} then the projection from p would be 𝔾_{a}^{2}-equivariant (see [13]). Therefore, the map ℙ^{2} −−→ S given above has to be 𝔾_{a}^{2}-equivariant. The only 𝔾_{a}^{2}-action

Figure 1. Dynkin diagram of E_{6}

on ℙ^{2} under which a line is invariant is the standard translation action [13]. However, the linear series above is not invariant under the standard translation action

b ↦ b + βa c ↦ c + γa.

We proceed to compute the effective cone of the minimal resolution ϕ_{ℓ} : S̃→S. Let ℓ ⊂ S̃ be the proper transform of the line mentioned in Proposition 3.1.

Proposition 3.4. — The Picard group Pic(S̃) is a free abelian group of rank seven, generated by ℓ and the exceptional curves of ϕ_{ℓ}. For a suitable ordering {F_{1}, F_{2}, F_{3}, F_{4}, F_{5}, F_{6}} of the exceptional curves, the intersection pairing takes the form

(3.2)  \begin{aligned}& F_{1} & F_{2} & F_{3} & ℓ & F_{4} & F_{5} & F_{6} \\ F_{1} & −2 & 0 & 1 & 0 & 0 & 0 & 0 \\ F_{2} & 0 & −2 & 0 & 0 & 0 & 0 & 1 \\ F_{3} & 1 & 0 & −2 & 0 & 0 & 0 & 1 \\ ℓ & 0 & 0 & 0 & −1 & 1 & 0 & 0 \\ F_{4} & 0 & 0 & 0 & 1 & −2 & 1 & 0 \\ F_{5} & 0 & 0 & 0 & 0 & 1 & −2 & 1 \\ F_{6} & 0 & 1 & 1 & 0 & 0 & 1 & −2 \end{aligned} .

Proposition 3.5. — The effective cone NE(S̃) is simplicial and generated by Φ := {F_{1}, F_{2}, F_{3}, ℓ, F_{4}, F_{5}, F_{6}}. Each nef divisor is contained in the monoid generated by the divisors:

A_{1} =  F_{2} + F_{3} + 2ℓ + 2F_{4} + 2F_{5} + 2F_{6}
A_{2} = F_{1} + F_{2} + 2F_{3} + 3ℓ + 3F_{4} + 3F_{5} + 3F_{6}
A_{3} = F_{1} + 2F_{2} + 2F_{3} + 4ℓ + 4F_{4} + 4F_{5} + 4F_{6}
A_{ℓ} = 2F_{1} + 3F_{2} + 4F_{3} + 3ℓ + 4F_{4} + 5F_{5} + 6F_{6}
A_{4} = 2F_{1} + 3F_{2} + 4F_{3} + 4ℓ + 4F_{4} + 5F_{5} + 6F_{6}
A_{5} = 2F_{1} + 3F_{2} + 4F_{3} + 5ℓ + 5F_{4} + 5F_{5} + 6F_{6}
A_{6} = 2F_{1} + 3F_{2} + 4F_{3} + 6ℓ + 6F_{4} + 6F_{5} + 6F_{6}

Moreover A_{ℓ} is the anticanonical class −KS̃ and its sections induce the resolution morphism ϕ_{ℓ} : S̃→S.

Proof. — The intersection form in terms of A := {A_{1}, …, A_{6}} is:

(3.3)  \begin{aligned}& A_{1} & A_{2} & A_{3} & A_{ℓ} & A_{4} & A_{5} & A_{6} \\ A_{1} & 0 & 1 & 1 & 2 & 2 & 2 & 2 \\ A_{2} & 1 & 1 & 2 & 3 & 3 & 3 & 3 \\ A_{3} & 1 & 2 & 2 & 4 & 4 & 4 & 4 \\ A_{ℓ} & 2 & 3 & 4 & 3 & 4 & 5 & 6 \\ A_{4} & 2 & 3 & 4 & 4 & 4 & 5 & 6 \\ A_{5} & 2 & 3 & 4 & 5 & 5 & 5 & 6 \\ A_{6} & 2 & 3 & 4 & 6 & 6 & 6 & 6 \end{aligned} .

This is the inverse of the intersection matrix (3.2) written in terms of the basis Φ, so the A_{i} generate the dual to Cone(Φ). Observe that all the entries of matrix (3.3) are nonnegative and

Cone(A) ⊂ Cone(Φ).

Suppose that D is an effective divisor on S̃. We write D as a sum of the fixed components contained in {F_{1}, …, F_{6}, ℓ} and the parts moving relative to Φ:

D = M_{Φ} + a_{1}F_{1} + … + a_{6}F_{6} + a_{ℓ}ℓ, a_{1}, …, a_{6}, a_{ℓ} ≥ 0.

A priori, M_{Φ} may have fixed components, but they are not contained in Φ (however, see Lemma 3.6). It follows that M_{Φ} intersects each element of Φ nonnegatively, i.e., it is contained in Cone(A) and thus in Cone(Φ). We conclude that D ∈ Cone(Φ). Since A_{1}, ..., A_{6}, A_{ℓ} generate N_{1}(S̃) over ℤ each nef divisor can be written as a nonnegative linear combination of these divisors.

To see that A_{ℓ} is the anticanonical divisor, we apply adjunction

KS̃F_{i} = 0, i = 1, …, 6 KS̃ℓ = −1.

Nondegeneracy of the intersection form implies A_{ℓ} = −KS̃. Since S has rational double points, the resolution map ϕ_{ℓ} is crepant, i.e., ϕ_{ℓ}^{∗}K_{S} = KS̃. Thus

Γ(A_{ℓ}) = Γ(−KS̃) = Γ(−ϕ_{ℓ}^{∗}K_{S}) = Γ(ϕ_{ℓ}^{∗}O_{S}(+1))

so the sections of A_{ℓ} induce ϕ_{ℓ}.

Choose nonzero sections ξ_{1}, …, ξ_{ℓ} generating Γ(F_{1}), …, Γ(ℓ):

Γ(F_{1}) = ⟨ξ_{1}⟩ , …, Γ(F_{6}) = ⟨ξ_{6}⟩ , Γ(ℓ) = ⟨ξ_{ℓ}⟩ .

These are canonical up to scalar multiplication. Each effective divisor

D = b_{1}F_{1} + b_{2}F_{2} + b_{3}F_{3} + b_{ℓ}ℓ + b_{4}F_{4} + b_{5}F_{5} + b_{6}F_{6}

has a distinguished nonzero section

ξ^{(b_{1},b_{2},b_{3},b_{ℓ},b_{4},b_{5},b_{6})} := ξ_{1}^{b_{1}} … ξ_{6}^{b_{6}}ξ_{ℓ}^{b_{ℓ}}.

The distinguished section of A_{j} is denoted ξ^{α(j)}. Note that we have an injective ring homomorphism

(3.4)  k[ξ_{1}, …, ξ_{6}, ξ_{ℓ}]→Cox(S̃).

There is a partial order on the monoid of effective divisors of S̃: D_{1} ≺ D_{2} if D_{2} − D_{1} is effective. The restriction of this order to the generators of the nef cone is illustrated in the diagram below:

A_{6}

|
A_{5}
|
A_{4}
⧸  ⧹
A_{ℓ}  A_{3}
⧹  ⧸
A_{2}
|
A_{1}

Whenever D_{1} ≺ D_{2} we have an inclusion

Γ(D_{1})↪Γ(D_{2})

which is natural up to scalar multiplication. Indeed, express

D_{1} − D_{2} = b_{1}F_{1} + b_{2}F_{2} + … + b_{6}F_{6} + b_{ℓ}ℓ, b_{j} ≥ 0

so we have

s_{1} ↦ ξ^{(b_{1},b_{2},b_{3},b_{ℓ},b_{4},b_{5},b_{6})}s_{1}
Γ(D_{1}) ↪ Γ(D_{2}).

The homomorphism (3.4) is not surjective, and we now look for generators of Cox(S̃) beyond the ξ_{j}. Consider the subring

Cox_{a}(S̃) = ⨁_{ν∈NM(S̃)} Cox(S̃)_{ν}

obtained by restricting to degrees corresponding to nef classes on S̃. The following lemma implies that any homogeneous element s_{D} ∈ Cox(S̃) can be written in the form

s_{D} = m_{D}ξ_{1}^{b_{1}} ⋯ ξ_{6}^{b_{6}}ξ_{ℓ}^{b_{ℓ}}

with nonnegative exponents and m_{D} ∈ Cox_{a}(S̃).

Lemma 3.6. — Let D be an effective divisor on S̃ with fixed part F_{D} and moving part M_{D}. Then F_{D} is supported in {F_{1}, …, F_{6}, ℓ}, and M_{D} is a linear combination of A_{1}, …, A_{6}, A_{ℓ} with nonnegative coefficients.

Proof. — Clearly M_{D} is nef, so the description of the nef divisors in Proposition 3.5 gives the expression in terms of the A_{i}. Proposition 1.9 shows M_{D} is semiample with vanishing higher cohomology; the last part of Proposition 3.5 gives the requisite positivity of the anticanonical class.

Let F be a fixed component of D not supported in {F_{1}, …, F_{6}, ℓ}. To arrive at a contradiction, we need to show that h^{0}(OS̃(M_{D} + F)) > h^{0}(OS̃(M_{D})). Since M_{D} has vanishing higher cohomology and

h^{2}(OS̃(F + M_{D})) = h^{0}(OS̃(KS̃ − F − M_{D})) = 0

it suffices to show that

χ(OS̃(F + M_{D})) > χ(OS̃(M_{D})).

By Riemann-Roch, it suffices to show that

F^{2} + 2M_{D}F − KS̃F > 0

or, equivalently,

F^{2} + KS̃F + 2M_{D}F − 2KS̃F = 2g(F) − 2 + 2M_{D}F − 2KS̃F > 0.

Since F is irreducible, g(F) ≥ 0 and M_{D}F ≥ 0 and −KS̃F > 1, as M_{D} is nef and −KS̃ is nonpositive only along the exceptional curves and has degree 1 only on the line ℓ (see Proposition 3.1).

Corollary 1.10 gives the dimensions of the graded pieces of Cox_{a}(S̃). We focus first on the generators of the nef cone, introducing sections τ_{j} ∈ Γ(A_{j}) as needed to achieve the prescribed dimensions:

Γ(A_{1}) = ⟨ξ^{α(1)}, τ_{1}⟩
Γ(A_{2}) = ⟨ξ^{α(2)}, ξ^{α(2)−α(1)}τ_{1}, τ_{2}⟩
Γ(A_{ℓ}) = ⟨ξ^{α(ℓ)}, ξ^{α(ℓ)−α(1)}τ_{1}, ξ^{α(ℓ)−α(2)}τ_{2}, τ_{ℓ}⟩

The sections of A_{ℓ} induce ϕ_{ℓ} : S̃→S ⊂ ℙ^{3} by Proposition 3.5, and can be identified with the coordinates w, x, y, z of Equation (3.1). Since A_{1} ≺ A_{2} ≺ A_{ℓ}, we have

Γ(A_{1}) ↪ Γ(A_{2}) ↪ Γ(A_{ℓ}).

We can identify Γ(A_{1}) = ⟨y, z⟩; these correspond to projecting S from the line ℓ = {y = z = 0} and induce a conic bundle structure

ϕ_{1} : S̃→ℙ^{1}.

We have Γ(A_{2}) = ⟨x, y, z⟩; these correspond to projecting S from the singularity p = {w = y = z = 0} and induce the blow-up realization

ϕ_{2} : S̃→ℙ^{2}.

Therefore, we may choose τ_{1}, τ_{2}, and τ_{ℓ} so that

y = ξ^{α(ℓ)} w = ξ^{α(ℓ)−α(2)}τ_{2} z = ξ^{α(ℓ)−α(1)}τ_{1} x = τ_{ℓ}.

We obtain the following induced sections for A_{3}, A_{4}, A_{5}, and A_{6}:

Γ(A_{3}) = ⟨ξ^{α(3)}, ξ^{α(3)−α(1)}τ_{1}, ξ^{α(3)−α(2)}τ_{2}, ξ^{α(3)−2α(1)}τ_{1}^{2}⟩
Γ(A_{4}) = ⟨ξ^{α(4)}, ξ^{α(4)−α(1)}τ_{1}, ξ^{α(4)−α(2)}τ_{2}, ξ^{α(4)−α(ℓ)}τ_{ℓ}, ξ^{α(4)−2α(1)}τ_{1}^{2}⟩
Γ(A_{5}) = ⟨ξ^{α(5)}, ξ^{α(5)−α(1)}τ_{1}, ξ^{α(5)−α(2)}τ_{2}, ξ^{α(5)−α(ℓ)}τ_{ℓ}, ξ^{α(5)−2α(1)}τ_{1}^{2},
ξ^{α(5)−α(1)−α(2)}τ_{1}τ_{2}⟩
Γ(A_{6}) = ⟨ξ^{α(6)}, ξ^{α(6)−α(1)}τ_{1}, ξ^{α(6)−α(2)}τ_{2}, ξ^{α(6)−α(ℓ)}τ_{ℓ}, ξ^{α(6)−2α(1)}τ_{1}^{2},
ξ^{α(6)−α(1)−α(2)}τ_{1}τ_{2}, ξ^{α(6)−2α(2)}τ_{2}^{2}, ξ^{α(6)−3α(1)}τ_{1}^{3}⟩

Equation (3.1) gives the relation

τ_{ℓ}ξ^{2α(ℓ)} + τ_{2}^{2}ξ^{3α(ℓ)−2α(2)} + τ_{1}^{3}ξ^{3α(ℓ)−3α(1)} = 0.

Dividing by a suitable monomial ξ^{β}, we obtain

τ_{ℓ}ξ_{ℓ}^{3}ξ_{4}^{2}ξ_{5} + τ_{2}^{2}ξ_{2} + τ_{1}^{3}ξ_{1}^{2}ξ_{3} = 0,

a dependence relation in Γ(A_{6}). This is the only such relation: Any other relation, after multiplying through by ξ^{β}, yields a cubic form vanishing on S ⊂ ℙ^{3}, but equation (3.1) is the only such form. It follows that the sections given above for A_{1}, …, A_{5} form bases for Γ(A_{1}), …, Γ(A_{5}).

Since

A_{3} ≺ A_{4} ≺ A_{5} ≺ A_{6} ≺ 2A_{ℓ}

we have

Γ(A_{3}) ↪ Γ(A_{4}) ↪ Γ(A_{5}) ↪ Γ(A_{6})
↪ Γ(2A_{ℓ}) = ⟨w^{2}, wx, wy, x^{2}, xy, xz, y^{2}, yz, z^{2}⟩

and identifications

Γ(A_{3}) = ⟨y^{2}, yz, wy, z^{2}⟩
Γ(A_{4}) = ⟨y^{2}, yz, wy, xy, z^{2}⟩
Γ(A_{5}) = ⟨y^{2}, yz, wy, xy, z^{2}, wz⟩
Γ(A_{6}) = ⟨y^{2}, yz, wy, xy, z^{2}, wz, w^{2}⟩ .

The sections of A_{3} induce a morphism

ϕ_{3} : S̃→ℙ^{3}

onto a quadric surface with a single ordinary double point. The sections of A_{4} induce a morphism

ϕ_{4} : S̃→ℙ^{4}

with image a quartic Del Pezzo surface with a rational double point of type D_{5}. The sections of A_{5} induce a morphism

ϕ_{5} : S̃→ℙ^{5}

with image a quintic Del Pezzo surface with a rational double point of type A_{4}. The sections of A_{6} induce a morphism

ϕ_{6} : S̃→ℙ^{6}

with image a sextic Del Pezzo surface with two rational double points, of types A_{1} and A_{2}.

We summarize this analysis in the following proposition

Proposition 3.7. — Every section of A_{j}, j = 1, 2, 3, ℓ, 4, 5, 6, can be expressed as a polynomial in ξ_{1}, …, ξ_{6}, ξ_{ℓ}, τ_{1}, τ_{2}, τ_{6}. The only dependence relation among these is

τ_{ℓ}ξ_{ℓ}^{3}ξ_{4}^{2}ξ_{5} + τ_{2}^{2}ξ_{2} + τ_{1}^{3}ξ_{1}^{2}ξ_{3} = 0

in Γ(A_{6}). Each A_{j} is globally generated and induces a morphism

ϕ_{j} : S̃→ℙ^{χ−1}, χ = χ(OS̃(A_{j})).

The remainder of this section is devoted to proving the following:

Theorem 3.8. — The homomorphism

ϱ : k[ξ_{1}, ..., ξ_{6}, ξ_{ℓ}, τ_{1}, τ_{2}, τ_{ℓ}]/⟨τ_{ℓ}ξ_{ℓ}^{3}ξ_{4}^{2}ξ_{5} + τ_{2}^{2}ξ_{2} + τ_{1}^{3}ξ_{1}^{2}ξ_{3}⟩→Cox(S̃)

is an isomorphism.

If ϱ were not injective, its kernel would have nontrivial elements in degree ν = dA_{ℓ}, for some d sufficiently large. These translate into homogeneous polynomials of degree d vanishing on S ⊂ ℙ^{3}. All such polynomials are multiples of the cubic form defining S, which itself is a multiple of the relation we already have.

It remains to show that ϱ is surjective. By Proposition 3.5, Lemma 3.6 and the analysis of the sections of the A_{i}, it suffices to prove:

Proposition 3.9. — ϱ is surjective in degrees corresponding to nef divisor classes of S̃.

Lemma 3.10. — For any positive integers c_{1}, c_{2}, c_{3}, c_{ℓ}, c_{4}, c_{5}, c_{6}, the image of

Γ(A_{1})^{⊗c_{1}} ⊗ … ⊗ Γ(A_{ℓ})^{⊗c_{ℓ}} ⊗ … ⊗ Γ(A_{6})^{⊗c_{6}} ⟶ Γ(c_{1}A_{1} + … + c_{6}A_{6})

is a linear series embedding S̃.

Proof. — Proposition 3.7 says that each A_{j} is globally generated, so if the image of

Γ(A_{1}) ⊗ … ⊗ Γ(A_{6}) ⟶ Γ(A_{1} + … + A_{6})

embed S̃ then the general result follows. We use the standard criterion: a linear series gives an embedding iff any length-two subscheme Σ ⊂ S̃ imposes two independent conditions on the linear series.

First, suppose the support of Σ is not contained in the exceptional locus of ϕ_{ℓ} : S̃→S, i.e., the curves F_{1}, F_{2}, F_{3}, F_{4}, F_{5}, F_{6}. Then ϕ_{ℓ} maps Σ to a subscheme of length two, which imposes independent conditions on Γ(A_{ℓ}), and thus independent conditions on the linear series in question. Second, suppose that Σ ⊂ F_{j} for some j (resp. Σ ⊂ ℓ). Since A_{j} · F_{j} = 1 (resp. A_{ℓ} · ℓ = 1), ϕ_{j} maps F_{j} (resp. ℓ) isomorphically onto a line. It follows that Σ imposes independent conditions on Γ(A_{j}). Third, suppose that Σ is reduced with support in F_{i} and F_{j}, but is not contained in either F_{i} or F_{j}. Consider the chain of rational curves containing F_{i} and F_{j} (see Figure 3.) There exists a curve F_{k} in this chain so that ϕ_{k}(F_{i}) ≠ ϕ_{k}(F_{j}), so Σ imposes independent conditions on Γ(A_{k}). Fourth, suppose that Σ is nonreduced and supported in F_{j} but not contained in any F_{i} or ℓ. The morphism ϕ_{j} ramifies at points where F_{j} meets one of the other exceptional curves, and the kernel of the tangent morphism dϕ_{j} consists of the tangent vectors to the curves contracted by ϕ_{j}. It follows that ϕ_{j}(Σ) has length two and imposes independent conditions on Γ(A_{j}).

The polynomial ring

k[ξ_{1}, …, ξ_{6}, ξ_{ℓ}, τ_{1}, τ_{2}, τ_{ℓ}]

is graded by the Néron-Severi group of S̃

deg(ξ_{j}) = F_{j}, j = 1, …, 6 deg(ξ_{ℓ}) = ℓ deg(τ_{j}) = A_{j}, j = 1, 2, ℓ.

This gives an action of the Néron-Severi torus T (S̃) on the corresponding affine space 𝔸^{10}.

We consider the projective toric varieties that arise as quotients of 𝔸^{10} by T (S̃). As sketched in §2, these varieties have the one-skeleton

x_{1} = (0, 1, 2), x_{2} = (1, 1, 3), x_{3} = (1, 2, 4), x_{ℓ} = (2, 3, 3), x_{4} = (2, 3, 4)

x_{5} = (2, 3, 5), x_{6} = (2, 3, 6), t_{1} = (−1, 0, 0), t_{2} = (0, −1, 0), t_{ℓ} = (0, 0, −1) where the x_{j} correspond to the ξ_{j} and the t_{j} correspond to the τ_{j}.

Lemma 3.11. — Let X be a toric threefold with one-skeleton {x_{1}, …, t_{ℓ}} and divisor class-group 𝔛^{∗}(T (S̃)) = N_{1}(S̃). Then

Mov(X) = Cone(A_{1}, …, A_{6}, A_{ℓ}).

Proof. — Proposition 2.1 reduces this to computing the intersection of the cones generated by subsets of

{F_{1}, …, F_{6}, ℓ, A_{1}, A_{2}, A_{ℓ}}

with nine elements. Since A_{1}, A_{2}, A_{ℓ} are effective combinations of the classes F_{1}, …, F_{6}, and ℓ, it suffices to compute

Cone(F_{1}, …, ℓ̂, …, A_{ℓ}) ⋂ ( ⋂_{i=1,...,6} Cone(F_{1}, …, F̂_{i}, …, A_{ℓ})) .

This intersection obviously contains A_{1}, A_{2}, and A_{ℓ}, and it is a straightforward computation to show that it also contains A_{3}, A_{4}, A_{5}, A_{6}. For the reverse inclusion, suppose that D is contained in the intersection. Considering D as a divisor on S̃, we see that

D · F_{1}, …, D · F_{6}, D · ℓ

are all nonnegative. Thus D is an effective sum of A_{j} by Proposition 3.5.

Combining Lemmas 3.11 and 3.10 with Propositions 2.2 and 3.7, we obtain the following

Proposition 3.12. — Let ν be an ample divisor on S̃. Then there exists a projective toric variety Y_{ν} with one-skeleton {x_{1}, …, t_{ℓ}} and polarization ν, and an embedding S̃ ↪ Y_{ν} with the following properties:

1. the divisor class group of Y_{ν} is isomorphic to the divisor class group of S̃ so that the moving cone of Y_{ν} is identified with the nef cone of S̃;

2. the equation for S̃ in the Cox ring of Y_{ν} is

τ_{ℓ}ξ_{ℓ}^{3}ξ_{4}^{2}ξ_{5} + τ_{2}^{2}ξ_{2} + τ_{1}^{3}ξ_{1}^{2}ξ_{3} = 0

and [S̃] = A_{6} in the divisor class group of Y_{ν};

3. Cox(Y_{ν}) = k[ξ_{1}, …, τ_{ℓ}] and is mapped isomorphically to the image of the homomorphism ϱ.

For each toric variety Y_{ν}, we can consider the exact sequence of sheaves

0→IS̃→O_{Y_{ν}}→OS̃→0,

where IS̃ ≃ O_{Y_{ν}}(−A_{6}) is the ideal sheaf of S̃. Given an element θ in the divisor class group of Y_{ν}, we can twist to obtain

0→IS̃(θ)→O_{Y_{ν}}(θ)→OS̃(θ)→0.

We should make precise what we mean by the twist F(θ) of a coherent sheaf F on Y_{ν}: Realize F as the sheafification of a graded module F over Cox(Y_{ν}) (which exists by [18] Theorem 1.1, [10] Proposition 3.1), shift F by θ, and then resheafify the shifted module to obtain F(θ). Twisting respects exact sequences [10] 3.1.

The anticanonical divisor of a toric variety is the sum of the invariant divisors [12] p. 89, so

−K_{Y_{ν}} = F_{1} + … + F_{6} + ℓ + A_{1} + A_{2} + A_{ℓ} = A_{ℓ} + A_{6}

and we can rewrite our exact sequence as

0→O_{Y_{ν}}(K_{Y_{ν}} + A_{ℓ} + θ)→O_{Y_{ν}}(θ)→OS̃(θ)→0.

Suppose that θ corresponds to a nef class on S̃; we shall prove that ϱ is surjective in degree θ, thus proving Proposition 3.9 and Theorem 3.8. Since

Γ(O_{Y_{ν}}(θ)) ≃ k[ξ_{1}, …, ξ_{6}, ξ_{ℓ}, τ_{1}, τ_{2}, τ_{ℓ}]_{θ}

it suffices to show that

H^{1}(O_{Y_{ν}}(K_{Y_{ν}} + A_{ℓ} + θ)) = 0.

We apply Proposition 2.5, with ν_{0} = A_{ℓ} + θ, to get a simplicial toric variety Y_{ν} on which A_{ℓ} + θ is nef. As A_{ℓ} is in the interior of the effective cone of Y_{ν}, A_{ℓ}+θ is also big. Note that Y_{ν} has finite-quotient singularities, which are log terminal [17] §5.2. The desired vanishing follows from Theorem 2.17 of [16]. Alternately, we could apply Theorem 0.1 of [18], which applies in arbitrary characteristic and obviates the need to pass to a simplicial model.

4. D_{4} cubic surface

The strategy of the previous section can applied to other surfaces as well. Here we illustrate it in the case of a cubic surface given by the homogeneous equation

S = {(x_{1}, x_{2}, x_{3}, w) : w(x_{1} + x_{2} + x_{3})^{2} = x_{1}x_{2}x_{3}} ⊂ ℙ^{3}.

We summarize its properties:

1. S has a single singularity at the point p = (0, 0, 0, 1) of type D_{4}.

2. S contains 6 lines with the equations

\begin{matrix}ℓ′_{1} := \{w = x_{1} = 0\} & m′_{1} := \{x_{1} = x_{2} + x_{3} = 0\} \\ ℓ′_{2} := \{w = x_{2} = 0\} & m′_{2} := \{x_{2} = x_{1} + x_{3} = 0\} \\ ℓ′_{3} := \{w = x_{3} = 0\} & m′_{3} := \{x_{3} = x_{1} + x_{2} = 0\} \end{matrix}

3. S is the closure of the image of ℙ^{2} under the linear series

x_{1} = u_{1}(u_{1} + u_{2} + u_{3})^{2}, x_{2} = u_{2}(u_{1} + u_{2} + u_{3})^{2}, x_{3} = u_{3}(u_{1} + u_{2} + u_{3})^{2},

w = u_{1}u_{2}u_{3},
where ⟨u_{1}, u_{2}, u_{3}⟩ = Γ(ℙ^{2}, O_{ℙ^{2}}(1)).

Remark 4.1. — There are two isomorphism classes of cubic surfaces with a D_{4} singularity [5] Lemma 4. The other class is

S_{0} = {(x_{1}, x_{2}, x_{3}, w) : w(x_{1} + x_{2} + x_{3})^{2} = x_{1}x_{2}(−x_{1} − x_{2})};

it is obtained from S by substituting

(w, x_{1}, x_{2}, x_{3}) ↦ (t^{−2}w, x_{1}, x_{2}, tx_{3} + (t − 1)x_{1} + (t − 1)x_{2})

and letting t→0 in the resulting equation.

We can distinguish S and S_{0} geometrically: In S, the three lines not containing p do not share a common point. In S_{0}, the analogous lines

{w = x_{1} = 0}, {w = x_{2} = 0}, {w = x_{1} + x_{2} = 0} ⊂ S_{0}

are coincident at w = x_{1} = x_{2} = 0.

Let β : S̃→S denote the minimal desingularization of S and

ℓ_{1}, ℓ_{2}, ℓ_{3}, m_{1}, m_{2}, m_{3}

the strict transforms of the lines. The rational map S −−→ ℙ^{2} induces a morphism S̃→ℙ^{2} and let L denote the pullback of the hyperplane class. Let E_{0}, E_{1}, E_{2}, E_{3} be the exceptional divisors of β, ordered so that we have the following intersection matrix:

(4.1)  \begin{aligned}& L & E_{1} & E_{2} & E_{3} & m_{1} & m_{2} & m_{3} \\ L & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ E_{1} & 0 & −2 & 0 & 0 & 1 & 0 & 0 \\ E_{2} & 0 & 0 & −2 & 0 & 0 & 1 & 0 \\ E_{3} & 0 & 0 & 0 & −2 & 0 & 0 & 1 \\ m_{1} & 0 & 1 & 0 & 0 & −1 & 0 & 0 \\ m_{2} & 0 & 0 & 1 & 0 & 0 & −1 & 0 \\ m_{3} & 0 & 0 & 0 & 1 & 0 & 0 & −1 \end{aligned} .

This is a rank seven unimodular matrix; since the Picard group of S̃ has rank seven, it is generated by L, E_{1}, E_{2}, E_{3}, m_{1}, m_{2}, m_{3}. In particular, we have

E_{0} = L − (E_{1} + E_{2} + E_{3} + m_{1} + m_{2} + m_{3}) and ℓ_{j} = L − E_{j} − 2m_{j}.

The anticanonical class is given by

−KS̃ = 3L − (E_{1} + E_{2} + E_{3}) − 2(m_{1} + m_{2} + m_{3}) = ℓ_{1} + ℓ_{2} + ℓ_{3}.

Proposition 4.2. — The effective cone NE(S̃) is generated by

Ξ := {E_{0}, E_{1}, E_{2}, E_{3}, m_{j}, ℓ_{j}}.

Proof. — Each effective divisor D can be expressed as a sum

D = M_{Ξ} + b_{E_{0}}E_{0} + b_{E_{1}}E_{1} + … + b_{ℓ_{3}}ℓ_{3},

with nonnegative coefficients, where M_{Ξ} intersects each of the elements in Ξ nonnegatively and thus is in the dual cone to Cone(Ξ). Direct computation shows that the dual to Cone(Ξ) has generators

L, L − E_{i} − m_{i}, 2L − E_{i} − 2m_{i}, 2L − E_{i} − E_{j} − 2m_{i} − 2m_{j},
2L − E_{i} − E_{j} − m_{i} − 2m_{j}.

Each of these is contained in Cone(Ξ):

\begin{aligned}L & = & ℓ_{i} + E_{i} + 2m_{i}, \\ 2L − E_{i} − 2m_{i} & = & 2ℓ_{i} + E_{i} + 2m_{i}, \\ 2L − E_{i} − E_{j} − m_{i} − 2m_{j} & = & ℓ_{i} + ℓ_{j} + m_{i}, \\ L − E_{i} − m_{i} & = & ℓ_{i} + m_{i}, \\ 2L − E_{i} − E_{j} − 2m_{i} − 2m_{j} & = & ℓ_{i} + ℓ_{j}. \end{aligned}

It follows that M_{Ξ} and D are sums of elements in Ξ with nonnegative coefficients.

Each of the divisors m_{i}, ℓ_{i} and E_{i} has a distinguished nonzero section (up to a constant), denoted μ_{i}, λ_{i} and η_{i}, respectively. We have

{λ_{i}η_{i}μ_{i}^{2}, η_{0}η_{1}η_{2}η_{3}μ_{1}μ_{2}μ_{3}} ⊂ Γ(L),

and we may identify

u_{i} = λ_{i}η_{i}μ_{i}^{2} and u_{1} + u_{2} + u_{3} = η_{0}η_{1}η_{2}η_{3}μ_{1}μ_{2}μ_{3}

after suitably normalizing the μ_{i}, λ_{i}, and η_{i}. The dependence relation among the sections in Γ(L) translates into

(4.2)  λ_{1}η_{1}μ_{1}^{2} + λ_{2}η_{2}μ_{2}^{2} + λ_{3}η_{3}μ_{3}^{2} = η_{0}η_{1}η_{2}η_{3}μ_{1}μ_{2}μ_{3}.

An argument similar to the one given at the end of Section 3 proves that the natural homomorphism

k[η_{0}, ..., η_{3}, μ_{i}, λ_{i}]/⟨λ_{1}η_{1}μ_{1}^{2} + λ_{2}η_{2}μ_{2}^{2} + λ_{3}η_{3}μ_{3}^{2} − η_{0}η_{1}η_{2}η_{3}μ_{1}μ_{2}μ_{3}⟩→Cox(S̃)

is an isomorphism.

The cubic surface S admits an 𝔖_{3}-action on the coordinates x_{1}, x_{2}, x_{3}. In particular, it admits nonsplit forms over nonclosed ground fields. They can be expressed as follows: Let K/k be a cubic extension with Galois closure E/k. Fix a basis {γ, γ′, γ′′} for K over k so that elements Y ∈ K can be represented as

Y = yγ + y′γ′ + y′′γ′′

with y, y′, y′′ ∈ k. Choose σ ∈ Gal(E/k) so that σ and σ^{2} are coset representatives Gal(E/k) modulo Gal(E/K). Then

w · Tr_{K/k}(Y)^{2} = N_{K/k}(Y)

is isomorphic, over E, to S:

\begin{aligned}x_{1} & = & Y = yγ + y′γ′ + y′′γ′′ \\ x_{2} & = & σ(Y) = yσ(γ) + y′σ(γ′) + y′′σ(γ′′) \\ x_{3} & = & σ^{2}(Y) = yσ^{2}(γ) + y′σ^{2}(γ′) + y′′σ^{2}(γ′′) \end{aligned} .

Assigning elements U, V, W ∈ K to η_{1}, μ_{1} and λ_{1}, respectively, the torsor equation (4.2) takes the form

Tr_{K/k}(UV^{2}W) = η_{0}N_{K/k}(UV).

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