Non-coherent Components of the Toric Hilbert Scheme

TL;DR

This paper investigates the geometry of non-coherent components of the toric Hilbert scheme, providing explicit constructions of their associated polytopes and advancing understanding of their structure.

Contribution

It introduces an explicit method to construct polytopes for the normalisation of non-coherent components, expanding knowledge beyond the coherent component.

Findings

Explicit construction of polytopes for non-coherent components

Enhanced understanding of the geometry of the toric Hilbert scheme

Connection between local equations and component structure

Abstract

We want to understand the geometry of all irreducible components of the toric Hilbert scheme. Until now it is known that the coherent component is (up to normalisation) the toric variety associated to the state polytope of the toric ideal. For the non-coherent components it was only known that there exists such a polytope describing the normalisation. Using the local equations and various facts about toric Hilbert schemes, we will derive an explicit construction of the polytope corresponding to the normalisation of the underlying reduced structure of a given non-coherent component of the toric Hilbert scheme.