Gotzmann's persistence theorem for smooth projective toric varieties

Abstract

Gotzmann's persistence theorem enables us to confirm the Hilbert polynomial of a subscheme of projective space by checking the Hilbert function in just two points, regardless of the dimension of the ambient space. We generalise this result to products of projective spaces, and then extend our result to any smooth projective toric variety. The number of points to check depends solely on the Picard rank of the ambient space, with no dependence on the dimension.