Combinatorics of the toric Hilbert scheme

Thumbnail Image

Date

Authors

Journal Title

Journal ISSN

Volume Title

Publisher

Abstract

Description

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.
26 pages, 2 figures

Keywords

Algebraic Geometry, Combinatorics

Citation

URI

http://salesiana.dossiersoluciones.com/handle/123456789/79530

Consulte el texto completo en el siguiente enlace:

https://arxiv.org/abs/math/9912014
 http://arxiv.org/abs/math/9912014

Collections

arXiv