Kieran Hilmer

Purdue University

Bounds for Regularity of Toric Varieties in Terms of Multiplicity

The Eisenbud-Goto conjecture states that for a homogeneous prime ideal over an algebraically closed field, the Castelnuovo-Mumford regularity is bounded above by the multiplicity minus height plus one. In 2017, McCullough and Peeva showed that the regularity of an arbitrary homogeneous prime cannot be bounded by any polynomial function of the multiplicity. We present two exponential bounds for the regularity of a toric ideal in terms of only the multiplicity. One uses discrete optimization techniques to find a small generating set of the associated lattice, which in turn gives a small generating set of the ideal. The other uses homological techniques to construct a regular sequence with small support of maximal length in the ideal. We then generalize this to a bound for the regularity of any prime generated by polynomials with bounded support.