Hierarchical structure of graded Betti numbers in the quadratic strand

TL;DR

This paper establishes a hierarchical structure for the quadratic strand of graded Betti numbers in projective varieties, extending classical bounds and linking syzygies to geometric containment conditions.

Contribution

It introduces a comprehensive hierarchy for linear syzygies, extends classical bounds to all linear syzygies, and relates syzygy behavior to geometric containment in varieties.

Findings

Sharp upper bounds for 2_{p,1}(X) depending on degree

Stratification of the quadratic strand into finite hierarchies

Vanishing of 2_{p,1}(X) detects containment in minimal degree varieties

Abstract

The classical results, initiated by Castelnuovo and Fano and later refined by Eisenbud and Harris, provide several upper bounds on the number of quadrics defining a nondegenerate projective variety. Recently, it has been revealed that these bounds extend naturally to certain linear syzygies, suggesting the presence of a hierarchical structure governing the quadratic strand of graded Betti numbers. In this article, we establish such a hierarchy in full generality. We first prove sharp upper bounds for $\beta_{p,1}(X)$ depending on the degree of a projective variety $X$, extending the classical quadratic bounds to all linear syzygies and identifying the extremal varieties in each range. We then introduce geometric conditions that describe how containment of $X$ in low-degree varieties influences syzygies, and we show that these conditions stratify the quadratic strand into a finite sequence of hierarchies. This leads to a complete description of all possible extremal behavior. We also prove a generalized $K_{p,1}$-theorem, demonstrating that the vanishing of $\beta_{p,1}(X)$ detects containment in a variety of minimal degree at each hierarchy.