Mathematisches Institut, Seminars (Y. Tschinkel, ed.), p. –9 Universität Göttingen, 2004

EQUATIONS OF UNIVERSAL TORSORS AND COX RINGS

B. Hassett

Department of Mathematics, Rice University, MS 136, Houston, TX 77251-

    1892, U.S.A. • E-mail : hassett@math.rice.edu

Abstract. We discuss several constructions of universal torsors over rational surfaces.

1. Universal torsors and Cox rings

1.1. Motivating Example. All fields are supposed to be of characteristic 0.

Let X/K be a quintic Del Pezzo surface over a number field K. We have \overline{X} = X\overline{K} = Bl_{P_{1},P_{2},P_{3},P_{4}} ℙ^{2}, i.e. geometrically, X is the blow-up of ℙ^{2} in four points in general position. Without loss of generality, we may assume that

P_{1} = [1, 0, 0], P_{2} = [0, 1, 0], P_{3} = [0, 0, 1], P_{4} = [1, 1, 1].

Theorem 1 (Enriques, Swinnerton-Dyer). Even in the non-split case,

X(K) ≠ ∅.

Proof. See [Sko93].

Notes by Ulrich Derenthal, June 18th and 19th, 2004.

Since there is a unique projectivity taking arbitrary generic points P_{1}, P_{2}, P_{3}, P_{4} (i.e., distinct and no three of them collinear) to [1, 0, 0], …, [1, 1, 1] as above, the geometry behind this over \overline{K} is (where P_{5} ∈ \overline{X} is the point we want to describe):

\overline{X} = SL_{3} \\\{(P_{1}, …, P_{5}) ∈ ℙ^{2}\}

= SL_{3} \\ (\begin{matrix}x_{1} & x_{2} & x_{3} & x_{4} & x_{5} \\ y_{1} & y_{2} & y_{3} & y_{4} & y_{5} \\ z_{1} & z_{2} & z_{3} & z_{4} & z_{5} \end{matrix}) //𝔾_{m}^{5} .

Consider the Grassmannian of 3-dimensional subspaces of 5-dimensional space Gr(3, 5). Since such a subspace is described by a basis which is unique only up to an action of GL_{3}, we have GL_{3} \\M(3 × 5) ≅ Gr(3, 5), where we interpret the three rows of a 3 × 5 matrix as a basis. This implies that SL_{3} \\M(3 × 5) is the cone over this Grassmannian.

Therefore, \overline{X} ≅ Cone(Gr(3, 5))//𝔾_{m}^{5}. Here, Gr(3, 5) is embedded into ℙ^{9} by the Plücker embedding.

The “miracle” is that this generalizes to non-closed fields.

Remark 2. The permutation group S_{5} of the five points acts on the situation, and actually Aut(\overline{X}) = S_{5}.

Descent data for X is given by representations ρ : Gal(\overline{K}/K) → S_{5}. Let T_{ρ} be the nonsplit form of 𝔾_{m}^{5} corresponding to ρ. In fact, X is Cone(Gr(3, 5))//T_{ρ}.

1.2. Universal torsors. Let X be a smooth projective variety over \overline{K}. Assume Pic(X) is free of rank r. Let T_{X} be the Néron-Severi torus, i.e., its character group is χ^{∗}(T_{X}) = Pic(X).

Definition 3. A universal torsor 𝒰 is a T_{X}-principal homogeneous space

\xymatrix{T_{x} \ar[r] & 𝒰 \ar[d] \\  & X}

so that given an element λ ∈ χ^{∗}(T_{X}) (i.e., λ : T_{X} → 𝔾_{m}), then ℳ_{λ}^{×} ≅ ℒ_{λ} − {0-section} as 𝔾_{m}-bundles over X. Here, ℒ_{λ} ∈ Pic(X) is the line bundle associated to λ by χ^{∗}(T_{X}) ≅ Pic(X), and ℳ_{λ} is the associated bundle to the principal bundle 𝒰 → X induced by the representation λ.

Example 4. 1. Let X = ℙ^{n}. Then 𝒰 = 𝔸^{n+1} − {0} is the corresponding universal torsor with the torus acting diagonally. We have 𝒰 /𝔾_{m} ≅ X = ℙ^{n}.

2. Let X be the quintic Del Pezzo surface as above with the action of T_{X} on Cone(Gr(3, 5)). Then the universal torsor 𝒰 is the open subset of Cone(Gr(3, 5)) on which T_{X} acts freely.

The abstract approach to universal torsors is as follows: Choose a minimal set ℒ_{1}, …, ℒ_{r} generating Pic(X) over ℤ. Denote ℒ_{j} −{0-section} by ℒ_{j}^{×}. Let 𝒰 = ℒ_{1}^{×} × ⋯ × ℒ_{r}^{×}. Then T_{X} → 𝒰 → X is a T_{X}-principal bundle defining the universal torsor.

However, this abstract definition is not very useful, e.g., for number theoretic applications.

Remark 5. Over non-closed fields, we may not be able to descend the universal torsor 𝒰 .

For example, consider a non-split conic X. It is geometrically isomorphic to ℙ^{1}, but it has no line bundle isomorphic to 𝒪_{ℙ^{1}}(1) over the ground field. It only has line bundles of even degree, so there cannot exist a universal torsor over the ground field.

1.3. Total coordinate rings / Cox rings.

Definition 6. Let X be a projective variety with properties as above. Let ℒ_{1}, …, ℒ_{r} be a basis of Pic(X). Then the Cox ring of X is defined as

Cox(X) = ⨁_{(v_{1},...,v_{r})∈ℤ^{r}} Γ(X, ℒ_{1}^{v_{1}} ⊗ ⋯ ⊗ ℒ_{r}^{v_{r}}).

Properties of Cox(X) are:

1. It is graded by Pic(X): for λ ∈ χ^{∗}(T_{X}) ≅ Pic(X), the part of degree λ is given by Cox(X)_{λ} = Γ(X, ℒ_{λ}).

2. The torus T_{X} acts naturally on Cox(X): For t ∈ T_{X}, s ∈ Cox(X)_{λ}, this action is given by t(s) := λ(t) · s.

3. Cox(X) is independent of the choice of generators ℒ_{i} of the Picard group. Given two sets of generators ℒ_{i} and ℳ_{j}, the induced isomorphism of rings is canonical only up to the action of the torus T_{X}. The reason is that the isomorphism depens on a choice of isomorphisms

L_{j} ≅ ℳ_{1}^{n_{1}} ⊗ ⋯ ⊗ ℳ_{r}^{n_{r}}, j ∈ {1, …, r}.

However, such an isomorphism is not canonical: ℒ_{j} has automorphisms given by scalar multiplication. For details, see [HT04].

The existence of non-trivial automorphisms makes the descent of universal torsors an interesting question.

4. The graded pieces of Cox(X) which are non-zero correspond to effective divisors on X.

The Cox ring does not need to be finitely generated:

Example 7 (Mukai). Let X = Bl_{P_{1},...,P_{n}} ℙ^{r−1} be the blowup of projective space in n points in general position. If \frac{1}{2} + \frac{1}{r} + \frac{1}{n−r} ≥ 1, then Cox(X) is not finitely generated (i.e., for ℙ^{2}: n ≥ 9; for ℙ^{3}: n ≥ 8). Details can be found in [Muk01].

However, it is finitely generated if one of the following conditions is true:

1. The cone of effective divisors NE(X) is generated by a finite collection of semi-ample line bundles (e.g., X = G/P where P is parabolic subgroup of an algebraic group G).

2. X is (log) Fano of dimension ≤ 3.

3. X is toric. In this case, for X − 𝔾_{m}^{dim X} = ⋃_{j=1}^{N} D_{j} where the D_{j} are subvarieties of codimension 1, and s_{j} ∈ Γ(𝒪_{X}(D_{j})) is non-zero, then Cox(X) ≅ K[s_{1}, …, s_{N}].

1.4. Relations between universal torsors and Cox rings. From now on, assume that Cox(X) is finitely generated. Let 𝒱 = Spec(Cox(X)). It is affine with T_{X}-action T_{X} × 𝒱 → 𝒱 . Fix an open subset 𝒰 on which T_{X} acts freely. The basic fact is that 𝒰 is a T_{X}-principal bundle over X:

\xymatrix{T_{X} \ar[r] & 𝒰 \ar[d] \\  & X}

and 𝒰 is a universal torsor.

The punchline is that this way, the universal torsor 𝒰 is naturally a quasiaffine variety. Therefore, giving equations for 𝒰 is equivalent to giving generators and relations for Cox(X). This can be done by algebro-geometric methods, which may be seen as an improvement to the existing number theoretic method to calculate universal torsors.

To sketch a proof of these results, observe that X is naturally a Geometric Invariant Theory quotient (𝒱 //T_{X})_{λ} (by Keel–Hu, [HK00]) after specifying a linearization λ ∈ χ^{∗}(T_{X}) so that ℒ_{λ} is an ample line bundle on X.

Note that we need to mix affine invariant theory and the usual projective Geometric Invariant Theory to interpret (𝒱 //T_{X})_{λ}: First take the affine quotient under the action of ker(λ), which gives an affine variety. Then take Proj using the grading coming from the character λ.

Then Proj(⨁_{n≥0} Cox(X)_{nλ}) = (𝒱 //T_{X})_{λ} by Geometric Invariant Theory. The left hand side is Proj(⨁_{n≥0} Γ(X, ℒ_{λ}^{⊗n})), which is just X since ℒ_{λ} is ample.

A second observation is that given λ ∈ χ^{∗}(T_{X}), i.e., λ : T_{X} → 𝔾_{m}, the associated bundle induces ℒ_{λ}^{−1}. Therefore, it suffices to check the claim for ample λ.

We have an inclusion ⨁_{n≥0} Cox(X)_{nλ} → Cox(X) which induces a dominant map 𝒱 → Cone(X ⊂ ℙ^{N}, ℒ_{λ}). Therefore, we have

𝒱 ⟶  Cone(X ⊂ ℙ^{N})
↑  ↑
𝒰 ⟶ (Cone(X ⊂ ℙ^{N}) − {0}) ≅ (ℒ_{λ}^{−1})^{×}

The point is: One gets hold of the universal torsor by embedding it into the affine variety Spec(Cox(X)).

2. Equations of universal torsors

From now on, let X be a smooth projective variety over on algebraically closed field K of characteristic 0 with Pic(X) ≅ ℤ^{r} whose Cox ring is finitely generated. Therefore, the cone of effective divisors NE(X) is finitely generated.

2.1. The method of Colliot-Thélène and Sansuc. This approach to the calculation of Cox rings can be found in [CTS87].

On X, choose effective divisors D_{1}, …, D_{N} generating Pic(X). Let W = X\(D_{1} ∪ ⋯ ∪ D_{N}. Since removing these generators kills the Picard group, Pic(W) = 0.

We have an exact sequence

0 → K[W]^{∗}/K^{∗} → ⨁_{j=1}^{N} ℤD_{j} → Pic(X) → 0

where K[W]^{∗}/K^{∗} describes the linear equivalences among {D_{1}, …, D_{N}}. Dualizing this sequence by applying Hom(·, 𝔾_{m}), we obtain

1 → T_{X} → 𝔾_{m}^{N} ⟶^{q} R_{W} → 1.

Remark 8. A morphism φ : Z → R_{W} gives a T_{X}-torsor:

\xymatrix{T_{X} \ar[r] & 𝔾_{m}^{N} ×_{R_{W}} Z \ar[d] & ⊃ & q^{−1}(φ(z)) \ar[d] \\  & Z & ∋ & z}

The strategy is to construct a T_{X}-torsor 𝒰_{W} over W which extend to a universal torsor over X. This strategy works well in many cases, but not in general.

The morphism φ : W → R_{W} is constructed by constructing a splitting σ to the quotient

\xymatrix{K[W]^{∗} \ar[r] & K[W]^{∗}/K^{∗} : \ar[l]_{σ}}

Note that σ induces a K-algebra homomorphism

K[R_{W}] = K[t_{1}, t_{1}^{−1}, …, t_{N−r}, t_{N−r}^{−1}] → K[W], t_{j} ↦ σ(t_{j}),

where the t_{j} form a basis for χ^{∗}(R_{W}) and r = Rank(Pic(X)). Since R_{W} is affine, such a homomorphism corresponds to a K-morphism W → R_{W}, which defines φ.

The key fact is that the morphism φ extracted from σ gives a torsor T_{X} → 𝒰_{W} → W on W admitting an extension to a universal torsor T_{X} → 𝒰 → X over X.

\xymatrix{T_{X} \ar[r] & 𝒰_{W} \ar[d] \\  & W}  ⇝  \xymatrix{T_{X} \ar[r] & 𝒰 \ar[d] \\  & X}

An explicit method for constructing such an extension is not known. Only the existence is proven in [CTS87].

Remark 9 (Batyrev). Given a point P ∈ W, we get a natural splitting σ_{P} : K[W]^{∗}/K^{∗} → K[W]^{∗}: for every element of K[W]^{∗}/K^{∗}, choose a representing f satisfying f(P) = 1.

2.2. The example of the quintic Del Pezzo surface. Let X = Bl_{P_{1},...,P_{4}} ℙ^{2} be again the blow-up of ℙ^{2} in

P_{1} = [1, 0, 0], P_{2} = [0, 1, 0], P_{3} = [0, 0, 1], P_{4} = [1, 1, 1].

We will see how to obtain the Plücker equations defining the universal torsor by this method.

Consider the exceptional divisors E_{i} and the transforms l_{ij} of the lines through P_{i} and P_{j} (i ≠ j ∈ {1, …, 4}). Choose coordinates [x, y, z] and let u = \frac{x}{z}, v = \frac{y}{z}.

Consider

div(u = x/z) = l_{23} + E_{3} − l_{12} − E_{1}
div(v = y/z) = l_{13} + E_{3} − l_{12} − E_{2}
div(u − 1) = l_{24} + E_{4} − l_{12} − E_{1}
div(v − 1) = l_{14} + E_{4} − l_{12} − E_{2}
div(u − v) = l_{34} + E_{3} + E_{4} − l_{12} − E_{1} − E_{2}

Next, we normalize these functions by constructing a section σ_{P} from a chosen point, say P = [3, 2, 1]. This gives a morphism φ : W → R_{W} as above.

Consider the sections λ_{ij} corresponding to l_{ij} and η_{i} to E_{i}. Using the normalization, we obtain:

\frac{u}{3} = \frac{λ_{23}η_{3}}{λ_{12}η_{1}} , \frac{v}{2} = \frac{λ_{13}η_{3}}{λ_{12}η_{2}} , \frac{u − 1}{2} = \frac{λ_{24}η_{4}}{λ_{12}η_{1}} , v−1 = \frac{λ_{14}η_{4}}{λ_{12}η_{2}} , u−v = \frac{λ_{34}η_{3}η_{4}}{λ_{12}η_{1}η_{2}} .

Then the relations between the sections u, v, u − 1, v − 1, u − v give relations between the sections λ_{ij}, η_{i}:

3\frac{u}{3} − 2\frac{v}{2} = u − v ⇝ −(3λ_{23})η_{2} + (2λ_{13})η_{1} + λ_{34}η_{4} = 0
2\frac{v}{2} = (v − 1) + 1 ⇝ λ_{14}η_{4} − (2λ_{13})η_{3} + λ_{12}η_{2} = 0
2\frac{u − 1}{2} − (v − 1) = u − v ⇝ λ_{34}η_{3} − (2λ_{24})η_{4} + λ_{14}η_{1} = 0
3\frac{u}{3} = 2\frac{u − 1}{2} + 1 ⇝ (2λ_{24})η_{4} − (3λ_{23})η_{3} + λ_{12}η_{1} = 0
−(u − v) + v(u − 1) − (v − 1)u = 0 ⇝ λ_{12}λ_{34} − (2λ_{13})(2λ_{24}) + (3λ_{23})λ_{14} = 0

Replacing 3λ_{23}, 2λ_{13}, 2λ_{24} by new variables exactly gives the Plücker relations.

2.3. The Cox ring approach. Consider a different example:

X = Bl_{P_{1},P_{2},P_{3}} ℙ^{2} where P_{1} = [1, 0, 0], P_{2} = [1, 1, 0], P_{3} = [0, 1, 0],

i.e., X is the blow-up of ℙ^{2} in three points lying on a line. Let l_{123} be the transform of this line.

Basic facts on X are:

1. NE(X) = ⟨l_{123}, E_{1}, E_{2}, E_{3}⟩ is a simplicial cone, i.e., there are no relations between its generators. Therefore, the previous method does not work.

We have W = X − {E_{1}, E_{2}, E_{3}, l_{123}} ≅ 𝔸^{2}, and X is an equivariant compactification of 𝔾_{a}^{2}, acting on 𝔸^{2} by translation.

2. The ample cone, which is the dual of the effective cone, is generated by

{l_{123} + E_{1} + E_{2} + E_{3}, l_{123} + E_{1} + E_{2}, l_{123} + E_{1} + E_{3}, l_{123} + E_{2} + E_{3}}.

3. The anticanonical divisor −K_{X} is nef and big. Therefore, X is (log) Del Pezzo.

Next, we are looking for generators and relations of Cox(X). Generators are λ_{123} ∈ Γ(𝒪_{X}(l_{123})) ⊂ Cox(X)_{l_{123}} which is vanishing exactly along l_{123}, and η_{j} ∈ Γ(𝒪_{X}(E_{j})) ⊂ Cox(X)_{E_{j}} for j ∈ {1, 2, 3}.

These sections do not generate the Cox ring – in cases where they generate it, the method of Colliot-Thélène works well, but not here. We must choose additional generators: Γ(𝒪_{X}(l_{123} + E_{1} + E_{2})) corresponds to linear forms in x, y, z vanishing at P_{3}, i.e., it is K^{2} ≅ ⟨x, z⟩. Besides λ_{123}η_{1}η_{2}, which can be identified as z, we can choose another section ξ_{3} such that ξ_{3}η_{3} = −x.

Similarly, we have ξ_{1} ∈ Γ(𝒪_{X}(l_{123} + E_{2} + E_{3})) such that y = ξ_{1}η_{1} and ξ_{2} ∈ Γ(𝒪_{X}(l_{123} + E_{1} + E_{3})) such that x − y = ξ_{2}η_{2}.

This gives a homomorphism

ψ : K[λ_{123}, η_{1}, η_{2}, η_{3}, ξ_{1}, ξ_{2}, ξ_{3}]/⟨η_{1}ξ_{1} + η_{2}ξ_{2} + η_{3}ξ_{3}⟩ → Cox(X),

and since the dimension of both of these is 6 (dim(X) = 2 and Rank(Pic(X)) = 4), it is reasonable to hope that this is an isomorphism.

Remark 10. Then η_{1}ξ_{1} + η_{2}ξ_{2} + η_{3}ξ_{3} is the equation of the universal torsor T_{X} → 𝒰 → X in the sense that

𝒰 ⊂ 𝒱 := Spec K[λ_{123}, η_{1}, η_{2}, η_{3}, ξ_{1}, ξ_{2}, ξ_{3}]/⟨η_{1}ξ_{1} + η_{2}ξ_{2} + η_{3}ξ_{3}⟩.

Strategy of the proof. First, consider ψ in degrees ν corresponding to a nef line bundles on X. Such line bundles are semi-ample and in this case even globally generated. By induction on the effective monoid or by application of a vanishing theorem, we can prove that ψ is surjective in these nef degrees.

In degrees ν corresponding to not necessarily nef divisors ν, we reduce to the nef case the following way: Given s ∈ Cox(X)_{ν} = Γ(X, ℒ_{ν}), there exists a nef line bundle m, a section μ ∈ Cox(X)_{m} and a, b_{1}, b_{2}, b_{3} ∈ ℤ_{≥0} so that s = μλ_{123}^{a}η_{1}^{b_{1}}η_{2}^{b_{2}}η_{3}^{b_{3}}. This follows from the geometric fact that, given effective D on X, we can write D = M + F for a base point free divisor M and a fixed divisor F supported in {l_{123}, E_{1}, E_{2}, E_{3}}.

References

[CTS87] J.-L. Colliot-Thélène & J.-J. Sansuc – La descente sur les variétés rationnelles. II, Duke Math. J. 54 (1987), no. 2, p. 375–492.

[HK00] Y. Hu & S. Keel – Mori dream spaces and GIT, Michigan Math. J. 48 (2000), p. 331–348, Dedicated to William Fulton on the occasion of his 60th birthday.

[HT04] B. Hassett & Y. Tschinkel – Universal torsors and Cox rings, Arithmetic of higher-dimensional algebraic varieties (Palo Alto, CA, 2002), Progr. Math., vol. 226, Birkhäuser Boston, Boston, MA, 2004, p. 149–173.

[Muk01] S. Mukai – Counterexample to Hilbert’s Fourteenth Problem for the 3dimensional additive group, RIMS preprint 1343, 2001.

[Sko93] A. N. Skorobogatov – On a theorem of Enriques-Swinnerton-Dyer, Ann. Fac. Sci. Toulouse Math. (6) 2 (1993), no. 3, p. 429–440.