Hassett 1: Equations of Universal Torsors

Cox Rings

This follows an exposition due to Hu and Keel (Yi Hu and Seán Keel, Mori dream spaces and GIT, Michigan Math. J., 48 (2000), 331-348). Let $K=\mathbb{C}$, and $X$ a projective smooth variety over $K$. Let $L_1,\dots,L_r$ be line bundles on $X$. If $\nu=(n_1,\dots,n_r) \in \mathbb{N}^r$, then $L^\nu=L_1^{\otimes n_1} \otimes \dots \otimes L_r^{\otimes n_r}$.

We have a multiplication map $\Gamma(X,L^{\nu_1}) \otimes \Gamma(X,L^{\nu_2}) \to \Gamma(X,L^{\nu_1+\nu_2})$. We have a ring $R(X,L_1,\dots,L_r)=\bigoplus_{\nu \in \mathbb{N}^r} \Gamma(X,L^\nu)$, which is often not finitely generated.

If $L_1,\dots,L_r$ are (semi)ample (a semiample bundle is a pullback of an ample bundle), then $R(X,L_1,\dots,L_r)$ is finitely generated.

Definition. Let $X$ be smooth and projective, and assume ${\mathrm{Pic}}(X) \cong \mathbb{Z}^r$ (e.g. a Fano variety). The Cox ring ${\mathrm{Cox}}(X)$ is ${\mathrm{Cox}}(X)=R(X,L_1,\dots,L_r)$ where:

1. $L_1,\dots,L_r$ is a basis for ${\mathrm{Pic}}(X)$;
2. Every effective divisor $D$ can be written $D=n_1[L_1]+\dots+n_r[L_r]$ for $n_j \geq 0$.

Remark.

1. ${\mathrm{Cox}}(X)$ is (multi)graded by ${\mathrm{Pic}}(X)$; for $\nu \in {\mathrm{Pic}}(X)$, there is a $\nu$-graded piece ${\mathrm{Cox}}(X)_\nu$. This gives a natural action by $T_{NS}$, the Neron-Severi torus, $t(\xi)=\chi_v(t)\xi$, $t \in T$, $\chi_v \in X^*(T_{NS})$ corresponding to $v \in {\mathrm{Pic}}(X)$.

2. Graded pieces which are nonzero are in one-to-one correspondence with effective divisor classes of $X$.

3. Definition does not depend on choice of $\{L_j\}$; $R(X,L_1,\dots,L_\nu) \simeq R(X,M_1,\dots,M_v)$ is natural up to the action of the Neron-Severi torus.

The Hilbert function $h(\nu)=\dim {\mathrm{Cox}}(X)_\nu=\chi(\mathscr{O}_X(v))$ if $\nu$ has no higher cohomology, so is a `polynomial in $\nu$'. For example, if $\nu \in K_X+($ample cone$)$, Kodaira vanishing implies that $h(\nu)=\chi(\mathscr{O}_X(\nu))$.

If $-K_X$ is nef ($D$ is nef if $D\cdot C \geq 0$ for every curve $C$) and big ($D$ is big if $D$ is in the interior of the effective cone), for example if $X$ is Fano, then $h(\nu)=\chi(\mathscr{O}(\nu))$ for all $\nu$ nef and big. Our basic strategy: use knowledge of the Hilbert function to read off the structure of ${\mathrm{Cox}}(X)$. There is hope that the ring will be finitely generated from this polynomial expression.

Finite Generation

What are necessary conditions for ${\mathrm{Cox}}(X)$ to be finitely generated? (Part of a theorem of Hu, Keel which give necessary and sufficient conditions.) We must have:

1. The effective cone of $X$ should be finitely generated. (It is an open problem if the effective cone of a Fano variety is finitely generated, either in the sense that the closed cone is rational polyhedral or the associated monoid is finitely generated.)
2. The nef cone is finitely generated.

There are also sufficient conditions:

1. ${\mathrm{Cox}}(X) \cong k[x_\sigma]$ for $\sigma \in \Sigma(1)$ if $X$ is a toric variety.
2. If $X$ is (log) Fano of dimension $\leq 3$. Then ${\mathrm{Cox}}(X)$ is finitely generated (Shokurov).

Remark. The universal torsor $T \subset {\mathrm{Spec}}{\mathrm{Cox}}(X)$ as an explicitly defined open subset, if ${\mathrm{Cox}}(X)$ is finitely generated.

$E_6$ Cubic Surface

This is joint work with Tschinkel. The $E_6$ cubic surface $S$ is defined by the equation $xy^2+yw^2+z^3=0$ embedded in $\mathbb{P}^3$; it contains a unique line $\ell:y=z=0$, and a unique singularity $P:x=y=z=0$.

Analysis of the singularity: In affine coordinates, we have $Y^2+YW^2+Z^3=0$. We rewrite this as $(Y+W^2/2)^2-(1/4)W^4+Z^3=0$, and up to analytic equivalence, this is $Y'^2+W'^4+Z^3=0$, which is the normal form of an $E_6$ singularity.

The resolution $\beta:\widetilde{S} \to S$ has six $-2$ exceptional curves in an $E_6$-diagram, and ${\mathrm{Pic}}(\widetilde{S})$ has intersection form

$$\begin{array}{ccccccc} F_1 & F_2 & F_3 & \ell & F_4 & F_5 & ... ...0 & 0 & 1 & -2 & 1 \\ 0 & 1 & 1 & 0 & 0 & 1 & -2 \end{array}$$

with numbering as

$$\xymatrix{ & & 2 \ar@{-}[d] \\ 1 \ar@{-}[r] & 3 \ar@{-}[r] & 6 \ar@{-}[r] & 5 \ar@{-}[r] & 4 \ar@{-}[r] & \ell. }$$

The inverse of this matrix is given by

$$\begin{array}{ccccccc} A_1 & A_2 & A_3 & L & A_4 & A_5 & A_6... ... & 4 & 5 & 5 & 5 & 6 \\ 2 & 3 & 4 & 6 & 6 & 6 & 6 \end{array}$$

The inverse proves that: Proposition. The effective cone of $X$ is generated by $\{F_j,\ell\}$; the nef cone is generated by $\{A_j,L\}$.

We have

$${\mathrm{Cox}}(\widetilde{S})=\bigoplus_{n_1,\dots,n_6}\Gamma(\mathscr{O}_{\widetilde{S}}(n_1F_1+\dots+n_6F_6+n_\ell \ell));$$

we choose $\xi_j$ generating $\Gamma(\mathscr{O}_{\widetilde{S}}(F_j))$. (Descent to $k$ involves rescaling these $\xi_j$ such that the relations are defined over $k$.)

We have $k[\xi_1,\dots,\xi_6,\xi_\ell] \hookrightarrow {\mathrm{Cox}}(X)$, but this is not surjective. We need additional generators. We let $\xi^{\alpha(j)} \in \Gamma(\mathscr{O}(A_j))$; since $A_1$ is semiample, we have $\dim \Gamma(\mathscr{O}(A_j))=2$ by Riemann-Roch, and $\widetilde{S} \to \mathbb{P}^1$ is a conic bundle. Then

$$\Gamma(\mathscr{O}(A_1))=\langle \xi^{\alpha(1)},\tau_1 \rangle .$$

Similarly, $\dim \Gamma(\mathscr{O}(A_2))=3$, so we have an additional generator

$$\Gamma(\mathscr{O}(A_2))=\langle \xi^{\alpha(2)},\xi^{\alpha(2)-\alpha(1)}\tau_1,\tau_2 \rangle .$$

Since $\dim \Gamma(\mathscr{O}(A_\ell))=4$, we have a fourth generator

$$\Gamma(\mathscr{O}(A_\ell))=\langle \xi^{\alpha(\ell)},\xi^{\alpha(\ell)-\alpha(1)}\tau_1,\xi^{\alpha(\ell)-\alpha(2)}\tau_2,\tau_\ell \rangle .$$

Fact. We have a surjection $\mathbb{C}[\xi_1,\dots,\xi_\ell,\tau_1,\tau_2,\tau_\ell] \to {\mathrm{Cox}}(X)$.

Now we look for relations among these generators. By the Hilbert function, we know that $\dim \Gamma(\mathscr{O}(A_6))=7$, but we have $8$ elements of degree $\alpha(\mathscr{O})$ in the polynomial ring:

$$\{\xi^{\alpha(6)},\xi^{\alpha(6)-\alpha(1)}\tau_1,\dots,\xi^{\alpha(6)-\alpha(\ell)}\tau_\ell\}.$$

Since $\mathscr{O}(A_\ell)=\mathscr{O}_{S}(+1)$, $Y=\xi^{\alpha(\ell)}$, $Z=\xi^{\alpha(\ell)-\alpha(1)}\tau_1$, $W=\xi^{\alpha(\gamma)-\alpha(2)}\tau_2$ after renormalization, so the original equation $Y^2+YW^2+Z^3=0$ gives the relation

$$F:\tau_\ell\xi_\ell^3\xi_4^2\xi_5+\tau_2^2\xi_2+\tau_1^3\xi_1^2\xi_3=0.$$

With this,

$${\mathrm{Cox}}(\widetilde{S})=\mathbb{C}[\xi_1,\dots,\xi_\ell,\tau_1,\tau_2,\tau_\ell]/\langle F \rangle ;$$

by a computation of the Hilbert function of the quotient, there are no more relations.

This method should also work for other very singular cubic surfaces.

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