Deformations of Smooth Toric Surfaces

TL;DR

This paper studies the deformation theory of smooth toric surfaces, showing their unobstructedness, characterizing rigidity as Fano property, and constructing explicit deformations via Minkowski decompositions.

Contribution

It provides a detailed description of the deformation space for smooth toric surfaces and introduces a method to construct and analyze deformations using polyhedral subdivisions.

Findings

Smooth toric surfaces are unobstructed.

A smooth toric surface is rigid if and only if it is Fano.

Constructed deformations span the entire deformation space T_Y^1.

Abstract

For a complete, smooth toric variety Y, we describe the graded vector space T_Y^1. Furthermore, we show that smooth toric surfaces are unobstructed and that a smooth toric surface is rigid if and only if it is Fano. For a given toric surface we then construct homogeneous deformations by means of Minkowski decompositions of polyhedral subdivisions, compute their images under the Kodaira-Spencer map, and show that they span T_Y^1.