Bigraded Castelnuovo-Mumford regularity and Gr\"obner bases

TL;DR

This paper explores the connection between bigraded Castelnuovo-Mumford regularity and the bidegrees of Gr"obner bases for bihomogeneous ideals, extending classical results to a multigraded setting and providing bounds and certification methods.

Contribution

It introduces a bounding region for the bidegrees of minimal generators of bihomogeneous Gr"obner bases and relates this region to the bigraded Castelnuovo-Mumford regularity.

Findings

Established a bounding region for minimal generators' bidegrees.

Connected the bounding region to the bigraded Castelnuovo-Mumford regularity.

Provided methods to certify generators near the boundary.

Abstract

We study the relation between the bigraded Castelnuovo-Mumford regularity of a bihomogeneous ideal $I$ in the coordinate ring of the product of two projective spaces and the bidegrees of a Gr\"obner basis of $I$ with respect to the degree reverse lexicographical monomial order in generic coordinates. For the single-graded case, Bayer and Stillman unraveled all aspects of this relationship forty years ago and these results led to complexity estimates for computations with Gr\"obner bases. We build on this work to introduce a bounding region of the bidegrees of minimal generators of bihomogeneous Gr\"obner bases for $I$. We also use this region to certify the presence of some minimal generators close to its boundary. Finally, we show that, up to a certain shift, this region is related to the bigraded Castelnuovo-Mumford regularity of $I$.