1999/12/02 by Diane Maclagan, Maclagan, Diane, Rekha R. Thomas +1
Computer Science · Mathematics · #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Combinatorics (math.CO) #Commutative Algebra and Its Applications #FOS: Mathematics #Polynomial and algebraic computation #math.AG #math.CO
paper · pdf · doi:10.48550/arxiv.math/9912014
26 pages, 2 figures
arxiv created 1999/12/02 · openalex publication_date 1999/12/02 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 P4 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.