Deformations of equivelar Stanley-Reisner abelian surfaces

TL;DR

This paper investigates the deformation spaces of Stanley-Reisner schemes linked to equivelar torus triangulations, revealing binomial-defined deformation spaces, toric smoothings, and connections to abelian surface moduli, including a Calabi-Yau 3-fold example.

Contribution

It provides a detailed description of the deformation space structure for these schemes, especially highlighting the M"obius torus case and its geometric implications.

Findings

Deformation space defined by binomials

Existence of a toric smoothing component

Construction of a Calabi-Yau 3-fold with Euler number 6

Abstract

The versal deformation of Stanley-Reisner schemes associated to equivelar triangulations of the torus is studied. The deformation space is defined by binomials and there is a toric smoothing component which I describe in terms of cones and lattices. Connections to moduli of abelian surfaces are considered. The case of the M\"obius torus is especially nice and leads to a projective Calabi-Yau 3-fold with Euler number 6.