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Abstract: The toric Hilbert scheme parameterizes all ideals in a
polynomial ring
with the same multigraded Hilbert series as a given toric ideal. Such
ideals were first introduced in a special case by Arnold, and in
generality by Sturmfels. The ideals and scheme have also been studied by
Korkina, Peeva and Gasharov, and Peeva and Stillman. One central open
problem on toric Hilbert schemes is whether they are always connected. I
will describe joint work with Rekha Thomas (Texas A&M) which connects this
question to the Baues problem of geometric combinatorics. We construct a
graph on the monomial ideals in scheme which is connected if and only if
the scheme is connected.
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