Gregory G. Smith: Identifying nonempty Hilbert schemes

Abstract: A Hilbert scheme parametrizes all closed subschemes of a given projective variety with a fixed Hilbert polynomial. How do the combinatorical properties of the polynomial distinguish the nonempty Hilbert schemes? In this talk, we will first review the answer when the ambient variety is projective space. We will then speculate about possible generalizations when projective space is replaced by another smooth projective toric variety.