Multigraded Castelnuovo-Mumford regularity and Gröbner bases

We study the relation between the multigraded Castelnuovo-Mumford regularity of a multihomogeneous ideal $I$ and the multidegrees of a Gr\"obner basis of $I$ with respect to the degree reverse lexicographical monomial order in generic coordinates. For the single graded case, forty years ago, Bayer and Stillman unravelled all aspects of this relation, which in turn the use to complexity estimates for the computation with Gr\"obner bases. We build on their work to introduce a bounding region of the multidegrees of minimal generators of multigraded 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 multigraded Castelnuovo-Mumford regularity of $I$.