Equivariant deformations of algebraic varieties with an action of an algebraic torus of complexity 1

TL;DR

This paper studies the space of equivariant infinitesimal deformations of 3-dimensional affine varieties with a 2-dimensional torus action, providing explicit dimension formulas and constructing a versal deformation.

Contribution

It offers a new explicit computation of the dimension of the space of equivariant first order deformations for certain algebraic varieties with torus actions and links it to known results for toric varieties.

Findings

Computed the dimension of the space of equivariant first order deformations.

Constructed a formally versal equivariant deformation.

Connected deformation dimensions to known formulas for toric varieties.

Abstract

Let $X$ be a 3-dimensional affine variety with a faithful action of a 2-dimensional torus $T$. Then the space of first order infinitesimal deformations $T^1(X)$ is graded by the characters of $T$, and the zeroth graded component $T^1(X)_0$ consists of all equivariant first order (infinitesimal) deformations. Suppose that using the construction of such varieties from [1], one can obtain $X$ from a proper polyhedral divisor $\mathscr D$ on $\mathbb P^1$ such that the tail cone of (any of) the used polyhedra is pointed and full-dimensional, and all vertices of all polyhedra are lattice points. Then we compute $\dim T^1(X)_0$ and find a formally versal equivariant deformation of $X$. We also establish a connection between our formula for $\dim T^1(X)_0$ and known formulas for the dimensions of the graded components of $T^1$ of toric varieties.