Uniform bounds on multigraded regularity

TL;DR

This paper establishes an effective uniform bound on multigraded regularity for subschemes in smooth projective toric varieties, introducing a new combinatorial tool called Stanley filtration, with applications to Hilbert schemes.

Contribution

It introduces Stanley filtration as a novel combinatorial tool and provides a new proof of Gotzmann's regularity theorem, extending bounds to multigraded settings.

Findings

Derived a uniform bound on multigraded regularity

Introduced Stanley filtration for monomial ideals

Applied bounds to multigraded Hilbert schemes

Abstract

We give an effective uniform bound on the multigraded regularity of a subscheme of a smooth projective toric variety X with a given multigraded Hilbert polynomial. To establish this bound, we introduce a new combinatorial tool, called a Stanley filtration, for studying monomial ideals in the homogeneous coordinate ring of X. As a special case, we obtain a new proof of Gotzmann's regularity theorem. We also discuss applications of this bound to the construction of multigraded Hilbert schemes.