Additive actions on complete toric surfaces

TL;DR

This paper classifies additive actions of the group ^n on complete toric surfaces, providing a comprehensive understanding of such symmetries in algebraic geometry.

Contribution

It offers the first complete classification of additive actions specifically on complete toric surfaces, expanding the understanding of unipotent group actions.

Findings

Classification of additive actions on all complete toric surfaces

Identification of conditions for the existence of additive actions

Explicit descriptions of additive actions in specific cases

Abstract

By an additive action on an algebraic variety $X$ we mean a regular effective action $\mathbb{G}_a^n\times X\to X$ with an open orbit of the commutative unipotent group $\mathbb{G}_a^n$. In this paper, we give a classification of additive actions on complete toric surfaces.

Equations

$$R(X)=\underset{D\in K}{⨁}{H}^0(X,D)=\underset{D\in K}{⨁}\{f\in \mathbb{K}(X{)}^{\times }\mid \mathrm{div}(f)+D⩾0\}\cup \{0\}$$

$$\begin{array}{ccc}\hat{X}& \stackrel{i}{\to }& \overline{X}=\mathrm{Spec}R(X)\\ \downarrow //H_X& & \\ X& & \end{array}$$

$$\begin{array}[]{ccc}\mathfrak{S}=\mathfrak{R}\cap-\mathfrak{R}&,&\mathfrak{U}=\mathfrak{R}\setminus\mathfrak{S}.\end{array}$$

$$t{D}_e{t}^{-1}={\overline{\chi }}^e(t)D_e.$$

$$t{D}_e{t}^{-1}=t_i^{-1}\prod _{j\ne i}{t}_j^{⟨p_j,e⟩}{D}_e=\prod _{j=1}^m{t}_j^{⟨p_j,e⟩}{D}_e={\overline{\chi }}^e(t)D_e.$$

$$0\le ⟨p_j,e_i⟩=\sum {\alpha }_{jl}{\delta }_{li}={\alpha }_{ji}.$$

$$\begin{array}[]{cccc}\varepsilon_{i}=-1&\\ \varepsilon_{l}\geq 0,&l\leq n,\,l\neq i\\ \alpha_{ji}-\sum\limits_{\begin{subarray}{c}l=1\\ l\neq i\end{subarray}}^{n}\varepsilon_{l}\alpha_{jl}\geq 0,&n<j\leq m\end{array}$$

$${\mathfrak{S}}_u^{+}=\{e\in \mathfrak{S}:⟨u,e⟩>0\},{\mathfrak{S}}_u^{-}=\{e\in \mathfrak{S}:⟨u,e⟩<0\}.$$

Full text

The author was supported by RSF grant 19-11-00172.

1. Introduction

Let $X$ be an irreducible algebraic variety of dimension $n$ over an algebraically closed field $\mathbb{K}$ of characteristic zero and $\mathbb{G}_{a}=(\mathbb{K},+)$ be the additive group of the ground field. Consider the commutative unipotent group $\mathbb{G}_{a}^{n}=\mathbb{G}_{a}\times\ldots\times\mathbb{G}_{a}$ ($n$ times). By an additive action on $X$ we mean an effective regular action $\mathbb{G}_{a}^{n}\times X\to X$ with an open orbit. In other words, an additive action on a complete variety $X$ allows to consider $X$ as a completion of the affine space $\mathbb{A}^{n}$ that is equivariant with respect to the group of parallel translations on $\mathbb{A}^{n}$.

The study of additive actions began with the work of Hassett and Tschinkel [HT]. They introduced a correspondence between additive actions on the projective space $\mathbb{P}^{n}$ and local $(n+1)$-dimensional commutative associative algebras with unit; see also [KL, Proposition 5.1] for a more general correspondence. Hassett-Tshinkel’s correspondence makes it possible to classify additive actions on $\mathbb{P}^{n}$ for $n\leq 5$; these are precisely the cases when the number of additive actions is finite.

The study of additive actions was originally motivated by problems of arithmetic geometry. Chambert-Loir and Tschinkel [CLT1, CLT2] gave asymptotic formulas for the number of rational points of bounded height on smooth (partial) equivariant compactifications of the vector group.

There is a number of results on additive actions on flag varieties [A1, Fe, FH, Dev], singular del Pezzo surfaces [DL], Hirzebruch surfaces [HT] and weighted projective planes [ABZ].

This work concerns the case of toric varieties. The problem of classification of additive actions on toric varieties is raised in [AS, Section 6].

It is proved in [De] that $\mathbb{G}_{a}$-actions on a toric variety $X_{\Sigma}$ normalized by the acting torus $T$ are in bijection with some elements $e\in M$, where $M$ is the character lattice of torus $T$. These vectors are called Demazure roots of the corresponding fan $\Sigma$. Cox [Cox] observed that normalized $\mathbb{G}_{a}$-actions on a toric variety can be interpreted as certain $\mathbb{G}_{a}$-subgroups of automorphisms of the Cox ring $R(X)$ of the variety $X$. In turn, such subgroups correspond to homogeneous locally nilpotent derivations of the Cox ring.

In [AR] all toric varieties admitting an additive action are described. It turns out that if a complete toric variety $X$ admits an additive action, then it admits an additive action normalized by the acting torus. Moreover, any two normalized additive actions on $X$ are isomorphic.

This work solves the problem of classification of additive actions for a complete toric surface. It was known in particular cases of the projective plane [HT, Proposition 3.2], Hirzebruch surfaces [HT, Proposition 5.5] and some weighted projective planes [ABZ, Proposition 7] that these surfaces admit exactly two non-isomorphic additive actions, the normalized and the non-normalized ones. In Theorem LABEL:main, we prove that any complete toric surface admits at most two non-isomorphic additive actions and characterize the fans of surfaces that admit precisely two additive actions.

After presenting some preliminaries on toric varieties and Cox ring (Section 2) and $\mathbb{G}_{a}$-actions and Demazure roots (Section 3), we describe the results of [AR] (Section 4). In Section 5, we prove some facts about Demazure roots of a toric variety admitting an additive action. In Section LABEL:mainsection, we formulate and prove the main theorem of this work. In Section LABEL:example, we give several explicit examples of additive actions on toric surfaces in Cox ring coordinates and discuss the further research.

The author is grateful to his supervisor Ivan Arzhantsev for posing the problem and permanent support and to Yulia Zaitseva for useful discussions and comments.

2. Toric varieties and Cox rings

In this section we introduce basic notation of toric geometry, see [Fu, CLS] for details.

Definition 1**.**

A toric variety is a normal variety $X$ containing a torus $T\simeq(\mathbb{K}^{*})^{n}$ as a Zariski open subset such that the action of $T$ on itself extends to an action of $T$ on $X$.

Let $M$ be the character lattice of $T$ and $N$ be the lattice of one-parameter subgroups of $T$. Let $\langle\cdot\,,\cdot\rangle:N\times M\to\mathbb{Z}$ be the natural pairing between the lattice N and the lattice M. It extends to the pairing $\langle\cdot\,,\cdot\rangle_{\mathbb{Q}}:N_{\mathbb{Q}}\times M_{\mathbb{Q}}\to\mathbb{Q}$ between the vector spaces $N_{\mathbb{Q}}=N\otimes\mathbb{Q}$ and $M_{\mathbb{Q}}=M\otimes\mathbb{Q}$.

Definition 2**.**

A fan $\Sigma$ in the vector space $N_{\mathbb{Q}}$ is a finite collection of strongly convex polyhedral cones $\sigma$ such that:

1. (1)

For all cones $\sigma\in\Sigma$, each face of $\sigma$ is also in $\Sigma$. 2. (2)

For all cones $\sigma_{1},\sigma_{2}\in\Sigma$, the intersection $\sigma_{1}\cap\sigma_{2}$ is a face of the cones $\sigma_{1}$ and $\sigma_{2}$.

There is one-to-one correspondence between normal toric varieties $X$ and fans $\Sigma$ in the vector space $N_{\mathbb{Q}}$, see [CLS, Section 3.1] for details.

Here we recall basic notions of the Cox construction, see [ADHL, Chapter 1] for more details. Let $X$ be a normal variety. Suppose that the variety $X$ has free finitely generated divisor class group $\operatorname{Cl}(X)$ and there are only constant invertible regular functions on $X$. Denote the group of Weil divisors on $X$ by $\operatorname{WDiv}(X)$ and consider a subgroup $K\subseteq\operatorname{WDiv}(X)$ which maps onto $\operatorname{Cl}(X)$ isomorphically. The  Cox ring of the variety $X$ is defined as

[TABLE]

and multiplication on homogeneous components coincides with multiplication in the field of rational functions $\mathbb{K}(X)$ and extends to the Cox ring $R(X)$ by linearity. It is easy to see that up to isomorphism the graded ring $R(X)$ does not depend on the choice of the subgroup $K$.

Suppose that the Cox ring $R(X)$ is finitely generated. Then ${\overline{X}:=\operatorname{Spec}R(X)}$ is a normal affine variety with an action of the torus $H_{X}:=\operatorname{Spec}\mathbb{K}[\operatorname{Cl}(X)]$. There is an open $H_{X}$-invariant subset $\widehat{X}\subseteq\overline{X}$ such that the complement $\overline{X}\backslash\widehat{X}$ is of codimension at least two in $\overline{X}$, there exists a good quotient $p_{X}\colon\widehat{X}\rightarrow\widehat{X}/\!/H_{X}$, and the quotient space $\widehat{X}/\!/H_{X}$ is isomorphic to $X$, see [ADHL, Construction 1.6.3.1]. Thus, we have the following diagram

[TABLE]

It is proved in [Cox] that if $X$ is toric, then $R(X)$ is a polynomial algebra $\mathbb{K}[x_{1},\ldots,x_{m}]$, where the variables $x_{i}$ correspond to $T$-invariant prime divisors $D_{i}$ on $X$ or, equivalently, to the rays $\rho_{i}$ of the corresponding fan $\Sigma$. The $\operatorname{Cl}(X)$-grading on $R(X)$ is given by $\deg(x_{i})=[D_{i}]$. In this case, $\overline{X}$ is isomorphic to $\mathbb{K}^{m}$, and $\overline{X}\setminus\widehat{X}$ is a union of some coordinate subspaces in $\mathbb{K}^{m}$ of codimension at least two. Denote by $\mathbb{T}$ the torus $(\mathbb{K}^{*})^{m}$ acting on the variety $\overline{X}$. Each $w\in M$ gives a character $\chi^{w}:T\to\mathbb{K}^{*}$, and hence $\chi^{w}$ is a rational function on $X$. By [CLS, Theorem 4.1.3], the function $\chi^{w}$ defines the principal divisor ${\operatorname{div}}(\chi^{w})=-\sum_{\rho}\langle w,u_{\rho}\rangle D_{\rho}$. Let us consider the map $M\longrightarrow\mathbb{Z}^{m}$ defined by $w\mapsto(\langle w,u_{\rho_{1}}\rangle,\ldots,\langle w,u_{\rho_{m}}\rangle)$, where $\rho_{1},\ldots,\rho_{m}$ are one-dimensional cones of $\Sigma$. We identify the group $\mathbb{Z}^{m}$ with the character lattice of the torus $(\mathbb{K}^{*})^{m}$. Thus, every element $w\in M$ corresponds to the character $\overline{\chi}^{w}$ of the torus $\mathbb{T}$. Moreover, for any $w,w^{\prime}\in M$ the equality $w=w^{\prime}$ holds if and only if $\overline{\chi}^{w}=\overline{\chi}^{w^{\prime}}$.

3. Demazure roots and locally nilpotent derivations

Let $X_{\Sigma}$ be a toric variety of dimension $n$ and $\Sigma$ be the fan of the variety $X_{\Sigma}$. Let $\Sigma(1)=\{\rho_{1},\ldots,\rho_{m}\}$ in $N$ be the set of rays of the fan $\Sigma$ and $p_{i}$ be the primitive lattice vector on the ray $\rho_{i}$.

For any ray $\rho_{i}\in\Sigma(1)$ we consider the set $\mathfrak{R}_{i}$ of all vectors $e\in M$ such that

1. (1)

$\langle p_{i},e\rangle=-1$ and $\langle p_{j},e\rangle\geq 0$ for $j\neq i$, $1\leq j\leq n$; 2. (2)

if $\sigma$ is a cone of $\Sigma$ and $\langle v,e\rangle=0$ for all $v\in\sigma$, then the cone generated by $\sigma$ and $\rho_{i}$ is in $\Sigma$ as well.

Elements of the set $\mathfrak{R}=\bigcup\limits_{i=1}^{m}\mathfrak{R}_{i}$ are called Demazure roots of the fan $\Sigma$ (see [De, Section 3.1] or [Oda, Section 3.4]). Let us divide the roots $\mathfrak{R}$ into two classes:

[TABLE]

Roots in $\mathfrak{S}$ and $\mathfrak{U}$ are called semisimple and unipotent respectively.

A derivation $\partial$ of an algebra $A$ is said to be locally nilpotent (LND) if for every $f\in A$ there exists $k\in\mathbb{N}$ such that $\partial^{k}(f)=0$. For any LND $\partial$ on $A$ the map ${\varphi_{\partial}:\mathbb{G}_{a}\times A\rightarrow A}$, ${\varphi_{\partial}(s,f)=\exp(s\partial)(f)}$, defines a structure of a rational $\mathbb{G}_{a}$-algebra on $A$. A derivation $\partial$ on a graded ring $A=\bigoplus\limits_{\omega\in K}A_{\omega}$ is said to be homogeneous if it respects the $K$-grading. If ${f,h\in A\backslash\operatorname{ker}\partial}$ are homogeneous, then ${\partial(fh)=f\partial(h)+\partial(f)h}$ is homogeneous too and ${\deg\partial(f)-\deg f=\deg\partial(h)-\deg h}$. So any homogeneous derivation $\partial$ has a well-defined degree given as $\deg\partial=\deg\partial(f)-\deg f$ for any homogeneous $f\in A\backslash\operatorname{ker}\partial$.

Every locally nilpotent derivation of degree zero on the Cox ring $R(X_{\Sigma})$ induces a regular action $\mathbb{G}_{a}\times X_{\Sigma}\to X_{\Sigma}$. In fact, any regular $\mathbb{G}_{a}$-action on $X_{\Sigma}$ arises this way, see [Cox, Section 4] and [ADHL, Theorem 4.2.3.2]. If a $\mathbb{G}_{a}$-action on a variety $X_{\Sigma}$ is normalized by the acting torus $T$, then the lifted $\mathbb{G}_{a}$-action on $\overline{X}_{\Sigma}=\mathbb{K}^{m}$ is normalized by the diagonal torus $\overline{T}=(\mathbb{K}^{*})^{m}$. Conversely, any $\mathbb{G}_{a}$-action on $\mathbb{K}^{m}$ normalized by the torus $(\mathbb{K}^{\times})^{m}$ and commuting with the subtorus $H_{X_{\Sigma}}$ induces a $\mathbb{G}_{a}$-action on $X_{\Sigma}$. This shows that $\mathbb{G}_{a}$-actions on $X_{\Sigma}$ normalized by $T$ are in bijection with locally nilpotent derivations of the Cox ring $\mathbb{K}[x_{1},\ldots,x_{m}]$ that are homogeneous with respect to the standard grading by the lattice $\mathbb{Z}^{m}$ and have degree zero with respect to the $\operatorname{Cl}(X_{\Sigma})$-grading.

For any element $e\in\mathfrak{R}_{i}$ we consider a locally nilpotent derivation $\prod\limits_{j\neq i}x_{j}^{\langle p_{j},e\rangle}\frac{\partial}{\partial x_{i}}$ on the algebra $R(X_{\Sigma})$. This derivation has degree zero with respect to the grading by the group $\operatorname{Cl}(X_{\Sigma})$. This way one obtains a bijection between the Demazure roots in $\mathfrak{R}$ and locally nilpotent derivations of degree zero on the ring $R(X_{\Sigma})$, which in turn are in bijection with $\mathbb{G}_{a}$-actions on $X_{\Sigma}$ normalized by the acting torus.

Lemma 1**.**

Let $D_{e}$ be a homogeneous LND that corresponds to the Demazure root $e\in M$ and $t$ be an element of maximal torus $\mathbb{T}$. Then,

[TABLE]

Proof.

By definition, the derivation $D_{e}$ is equal to $\prod\limits_{j\neq i}x_{j}^{\langle p_{j},e\rangle}\frac{\partial}{\partial x_{i}}$. Let us consider the image $tD_{e}t^{-1}(x_{i})$ of an element $x_{i}$. It is equal to $t_{i}^{-1}\prod\limits_{j\neq i}t_{j}^{\langle p_{j},e\rangle}\prod\limits_{j\neq i}x_{j}^{\langle p_{j},e\rangle}$. Thus, we get

[TABLE]

∎

4. Complete toric varieties admitting an additive action

In this section, we shortly present the results of [AR]. Let $X_{\Sigma}$ be a toric variety of dimension $n$ admitting an additive action and $\Sigma$ be the fan of the variety $X_{\Sigma}$. Denote primitive vectors on the rays of the fan $\Sigma$ by $p_{i}$, where $1\leq i\leq m$.

Definition 3**.**

A set $e_{1},\ldots,e_{n}$ of Demazure roots of a fan $\Sigma$ of dimension $n$ is called a complete collection if $\langle p_{i},e_{j}\rangle=-\delta_{ij}$ for all $1\leq i,j\leq n$ for some ordering of $p_{1},\ldots,p_{m}$.

An additive action on a toric variety $X_{\Sigma}$ is said to be normalized if the image of the group $\mathbb{G}_{a}^{n}$ in $\operatorname{Aut}(X_{\Sigma})$ is normalized by the acting torus.

Theorem 1**.**

[AR, Theorem 1]* Let $X_{\Sigma}$ be a toric variety. Then normalized additive actions on $X_{\Sigma}$ normalized are in bijection with complete collections of Demazure roots of the fan $\Sigma$.*

Corollary 1**.**

A toric variety $X_{\Sigma}$ admits a normalized additive action if and only if there is a complete collection of Demazure roots of the fan $\Sigma$.

Theorem 2**.**

[AR, Theorem 3]* Let $X_{\Sigma}$ be a complete toric variety with an acting torus $T$. The following conditions are equivalent.*

• (1)

There exists an additive action on $X_{\Sigma}$.

• (2)

There exists a normalized additive action on $X_{\Sigma}$.

Here we prove a proposition that will be used below.

Definition 4**.**

The negative octant of the rational vector space $V$ with respect to a basis $f_{1},\ldots,f_{n}$ is the cone $\left\{\sum\limits_{i=1}^{n}\lambda_{i}f_{i}\mid\lambda_{i}\leq 0\right\}\subset V$.

Proposition 1**.**

Let $X_{\Sigma}$ be a complete toric variety. The following statements are equivalent.

1. (1)

There is a complete collection of Demazure roots of the fan $\Sigma$. 2. (2)

We can order rays of the fan $\Sigma$ in such a way that the primitive vectors on the first $n$ rays form a basis of the lattice $N$ and the remaining rays lie in the negative octant with respect to this basis. 3. (3)

There exists an additive action on $X_{\Sigma}$.

Proof.

Let us prove implication $(1)\Rightarrow(2)$. Assume that the vectors $p_{1},\ldots,p_{n}$ are linearly dependent, i.e. there exists a non-trivial linear relation ${\alpha_{1}p_{1}+\ldots+\alpha_{n}p_{n}=0}$. Then we get $-\alpha_{i}={\langle\alpha_{1}p_{1}+\ldots+\alpha_{n}p_{n},e_{i}\rangle}=0$ for all $1\leq i\leq n$, a contradiction. Consider an arbitrary vector $v=\sum_{i=1}^{n}\nu_{i}p_{i}$ of the lattice $N$. By definition of a complete collection, we get $\langle v,e_{i}\rangle=-\nu_{i}\in\mathbb{Z}$. Therefore, the vectors $p_{1},\ldots,p_{n}$ form the basis of the lattice $N$.

All other vectors $p_{j},\,j>m,$ are equal to $-\sum_{l=1}^{n}\alpha_{jl}p_{l}$ for some integer $\alpha_{jl}$. By definition of a Demazure root, we obtain

[TABLE]

The converse implication is straightforward.

Equivalence $(1)\Leftrightarrow(3)$ follows from Theorems 1 and 2. ∎

5. Demazure roots of a variety admitting an additive action

Let $X_{\Sigma}$ be a complete toric variety of dimension $n$ admitting an additive action and $\Sigma$ be the fan of the variety $X_{\Sigma}$. Denote primitive vectors on the rays of the fan $\Sigma$ by $p_{i}$, where $1\leq i\leq m$.

From Proposition 1 it follows that we can order $p_{i}$ in such a way that the first $n$ vectors form a basis of the lattice $N$ and the remaining vectors $p_{j}$ $(n<j\leq m)$ are equal to $\sum_{i=1}^{n}-\alpha_{ji}p_{i}$ for some non-negative integers $\alpha_{ji}$.

Let us denote the dual basis of the basis $p_{1},\ldots,p_{n}$ by $p_{1}^{*},\,\ldots,p_{n}^{*}$.

Lemma 2**.**

Consider $1\leq i\leq n$. The set ${\mathfrak{R}}_{i}$ is a subset of the set ${-p_{i}^{*}+\sum\limits_{l=1,l\neq i}^{n}\mathbb{Z}_{\geq 0}p_{j}^{*}}$ and the vector $-p_{i}^{*}$ is a Demazure root from the set ${\mathfrak{R}}_{i}$.

Proof.

Let $e=\sum\limits_{i=1}^{n}\varepsilon_{i}p_{i}^{*}$ be a Demazure root from ${\mathfrak{R}}_{i}$. By the definition, the Demazure roots from $\mathfrak{R}_{i}$ are defined by the following equations:

[TABLE]

It is clear that all possible solutions lie in the set ${-p_{i}^{*}+\sum\limits_{\begin{subarray}{c}l=1\\ l\neq i\end{subarray}}^{n}\mathbb{Z}_{\geq 0}p_{l}^{*}}$, and the vector $-p_{i}^{*}$ satisfies them. ∎

Consider the set $\operatorname{Reg}(\mathfrak{S})=\{u\in N:\langle u,e\rangle\neq 0\text{ for all }e\in\mathfrak{S}\}$. Any element $u$ from $\operatorname{Reg}(\mathfrak{S})$ divides the set of semisimple roots $\mathfrak{S}$ into two classes as follows:

[TABLE]

At this point, any element of ${\mathfrak{S}_{u}}^{+}$ is called positive and any element of $\mathfrak{S}_{u}^{-}$ is called negative.