volume 27, issue 2, P249-264 2002
DOI: 10.1007/s00454-001-0062-7
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Abstract: The toric Hilbert scheme is a parameter space for all ideals with the same multigraded 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 dimens…

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