Toric Q-Gorenstein Singularities

TL;DR

This paper explores the deformation theory of affine toric Q-Gorenstein varieties, linking infinitesimal deformations to Minkowski summands of faces of the defining polytope, and discusses rigidity conditions and examples.

Contribution

It establishes a relationship between deformation spaces and Minkowski decompositions of polytopes, providing new insights into the rigidity and deformation of toric singularities.

Findings

Rigid unless Gorenstein and 3-dimensional

Deformations correspond to Minkowski decompositions of the polytope

Examples include cones over toric Del Pezzo surfaces

Abstract

For an affine, toric Q-Gorenstein variety Y (given by a lattice polytope Q) the vector space T^1 of infinitesimal deformations is related to the complexified vector spaces of rational Minkowski summands of faces of Q. Moreover, assuming Y to be an isolated, at least 3-dimensional singularity, Y will be rigid unless it is even Gorenstein and dim Y=3 (dim Q=2). For this particular case, so-called toric deformations of Y correspond to Minkowski decompositions of Q into a sum of lattice polygons. Their Kodaira-Spencer-map can be interpreted in a very natural way. We regard the projective variety P(Y) defined by the lattice polygon Q. Data concerning the deformation theory of Y can be interpreted as data concerning the Picard group of P(Y). Finally, we provide some examples (the cones over the toric Del Pezzo surrfaces). There is one such variety yielding Spec C[e]/e^2 as the base space of the semi-universal deformation.