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

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.