Uniform bounds on multigraded regularity

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.