Bottom complexes

TL;DR

This paper introduces three explicit infinite families of lattice polytopal complexes called bottom complexes, which define flat deformations of affine toric varieties, advancing the understanding of their algebraic and geometric structures.

Contribution

It provides explicit constructions of infinite families of bottom complexes that induce scheme-theoretic flat deformations of affine toric varieties.

Findings

Three infinite families of bottom complexes are explicitly described.

These complexes define scheme-theoretic flat deformations.

The work advances understanding of deformations of affine toric varieties.

Abstract

The bottom complex of a finite polyhedal pointed rational cone is the lattice polytopal complex of the compact faces of the convex hull of nonzero lattice points in the cone. The algebra, associated to the bottom complex of a cone, defines a flat deformation of the affine toric variety, associated to the polyhedral cone, set-theoretically. We describe three explicit infinite families of abstract polytopal complexes, defining such flat deformations scheme-theoretically.