Restricting linear syzygies: algebra and geometry

TL;DR

This paper explores how long linear syzygies in the minimal free resolution of schemes in projective space influence geometric properties and invariants, providing bounds and characterizations.

Contribution

It establishes new geometric consequences of linear syzygies, linking algebraic properties to geometric intersections and homological invariants.

Findings

Bounds on homological invariants of projective varieties

Characterization of quadratic monomial ideals with long linear syzygies

Geometric implications of linear syzygies in projective schemes

Abstract

In this paper we derive geometric consequences from the presence of a long strand of linear syzygies in the minimal free resolution of a closed scheme in projective space whose homogeneous ideal is generated by quadrics. These consequences are given in terms of intersections with arbitrary linear subspaces. We use our results to bound homological invariants of some well-known projective varieties, to give a combinatorial characterization of quadratic monomial ideals with a long strand of linear syzygies, etc