Bounds for Betti numbers and the degrees of the generators of syzygies in graded resolutions

Abstract

The main goal of this paper is to size up the minimal graded free resolution of a homogeneous ideal in terms of its generating degrees, by itself an ambitious objective. As understood, sizing up means looking closely at the two available parameters: the shifts and the Betti numbers. Since, in general, bounds for the shifts can behave quite steeply, we filter the difficulty by the subadditivity of the syzygies. The method we applied is hopefully new and sheds additional light on the structure of the minimal free resolution. For the Betti numbers, we apply the Boij–Söderberg techniques in order to get polynomial upper bounds for them. It is expected that the landscape of hypersurface singularities already contains most of the difficult corners of the arbitrary case. In this regard, we treat some facets of this case, including an improvement of one of the basic results of a recent work by Busé–Dimca–Schenck–Sticlaru.