Bigraded Castelnuovo-Mumford regularity and Groebner bases

Abstract

International audienceWe study the relation between the bigraded Castelnuovo-Mumford regularity of abihomogeneous ideal $I$ in the coordinate ring of the product of two projective spaces and the bidegrees of a Groebner 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 Groebner bases. We build on this work to introduce a bounding region of the bidegrees of minimal generators of bihomogeneous Groebner bases for $I$. We also use this region to certify the presence of some minimalgenerators close to its boundary. Finally, we show that, up to a certain shift, this region is related to the bigraded Castelnuovo-Mumford regularity of $I$