Bounds for the degree and Betti sequences along a graded resolution

TL;DR

This paper establishes polynomial upper bounds for the Betti numbers of homogeneous ideals using Boij-S"oderberg techniques, and explores the structure of minimal free resolutions with implications for hypersurface singularities.

Contribution

It introduces a new approach to bounding Betti numbers and shifts in graded resolutions, improving existing results and applying Boij-S"oderberg methods to this problem.

Findings

Polynomial upper bounds for Betti numbers derived.

Enhanced understanding of the structure of minimal free resolutions.

Improved bounds related to hypersurface singularities.

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 and large, this is too ambitious an 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. We use the Boij-S\"oderberg techniques for the Betti numbers 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 Bus\'e--Dimca--Schenck--Sticlaru.