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

2014/06/30 by Rostislav Devyatov, Devyatov, Rostislav
Mathematics · Computer Science · #Algebraic Geometry and Number Theory #Polynomial and algebraic computation #Commutative Algebra and Its Applications

paper · pdf · doi:10.48550/arxiv.1406.7736

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 T¹(X) is graded by the characters of T, and the zeroth graded component T¹(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 P¹ 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 T¹(X)₀ and find a formally versal equivariant deformation of X. We also establish a connection between our formula for dim T¹(X)₀ and known formulas for the dimensions of the graded components of T¹ of toric varieties.

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