Multivariate Hilbert Polynomials For Smooth Projective Toric Varieties

Abstract

Generalizing Hilbert polynomials for varieties embedded in projective space, we develop the theory of multivariate Hilbert polynomials for varieties embedded in a smooth projective toric variety. These polynomials can be understood either as an Euler characteristic, or as a vector partition function. We give a characterization of toric projective bundles via local cohomology. We use this characterization to show that, for smooth projective toric surfaces, the multivariate Hilbert polynomial of the Cox ring is reducible if and only if the Picard rank of the surface is less than or equal to 2. For smooth projective toric varieties of dimension greater than 2, we investigate the connection between reducibility of the Hilbert polynomial of the Cox ring, and the splitting of the fan of the variety.