Bigraded Castelnuovo-Mumford regularity and Gr\"obner bases
Abstract
We study the relation between the bigraded Castelnuovo-Mumford regularity of a bihomogeneous ideal in the coordinate ring of the product of two projective spaces and the bidegrees of a Gr\"obner basis of 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 . 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 .
Cite
@article{arxiv.2407.13536,
title = {Bigraded Castelnuovo-Mumford regularity and Gr\"obner bases},
author = {Matías Bender and Laurent Busé and Carles Checa and Elias Tsigaridas},
journal= {arXiv preprint arXiv:2407.13536},
year = {2025}
}
Comments
Revised version: we restrict to the bigraded case. More details in Section 6 and several additional remarks