Combinatorics of the toric Hilbert scheme

TL;DR

This paper investigates the connectivity of the toric Hilbert scheme by constructing a flip graph of monomial ideals, establishing a link to the Baues graph, and providing evidence that the scheme may be disconnected.

Contribution

It introduces the flip graph as a tool to analyze the connectedness of the toric Hilbert scheme and relates it to the Baues graph of triangulations.

Findings

The toric Hilbert scheme's connectedness is equivalent to the flip graph's connectedness.

The flip graph maps into the Baues graph of triangulations.

Evidence suggests the toric Hilbert scheme may be disconnected.

Abstract

The toric Hilbert scheme is a parameter space for all ideals with the same multi-graded Hilbert function as a given toric ideal. Unlike the classical Hilbert scheme, it is unknown whether toric Hilbert schemes are connected. We construct a graph on all the monomial ideals on the scheme, called the flip graph, and prove that the toric Hilbert scheme is connected if and only if the flip graph is connected. These graphs are used to exhibit curves in P^4 whose associated toric Hilbert schemes have arbitrary dimension. We show that the flip graph maps into the Baues graph of all triangulations of the point configuration defining the toric ideal. Inspired by the recent discovery of a disconnected Baues graph, we close with results that suggest the existence of a disconnected flip graph and hence a disconnected toric Hilbert scheme.