Sharp bounds for higher linear syzygies and classifications of projective varieties

TL;DR

This paper establishes sharp upper bounds for higher linear syzygies of projective varieties, generalizing classical results, and classifies extremal cases using a new framework based on Partial Elimination Ideals and inner projections.

Contribution

It introduces a novel approach combining PEIs and inner projections to bound and classify syzygies of projective varieties, extending classical geometric results.

Findings

Derived fundamental inequalities relating Betti numbers and inner projections.

Established sharp upper bounds for higher linear syzygies based on codimension.

Classified extremal and next-to-extremal cases for projective varieties.

Abstract

In the present paper, we consider upper bounds of higher linear syzygies i.e. graded Betti numbers in the first linear strand of the minimal free resolutions of projective varieties in arbitrary characteristic. For this purpose, we first remind `Partial Elimination Ideals (PEIs)' theory and introduce a new framework in which one can study the syzygies of embedded projective schemes well using PEIs theory and the reduction method via inner projections. Next we establish fundamental inequalities which govern the relations between the graded Betti numbers in the first linear strand of an algebraic set $X$ and those of its inner projection $X_q$. Using these results, we obtain some natural sharp upper bounds for higher linear syzygies of any nondegenerate projective variety in terms of the codimension with respect to its own embedding and classify what the extremal case and next-to-extremal case are. This is a generalization of Castelnuovo and Fano's results on the number of quadrics containing a given variety and another characterization of varieties of minimal degree and del Pezzo varieties from the viewpoint of `syzygies'. Note that our method could be also applied to get similar results for more general categories (e.g. connected in codimension one algebraic sets).