Hierarchical structure of graded Betti numbers in the quadratic strand

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 $β{p,1}(X)$ 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 $β_{p,1}(X)$ detects containment in a variety of minimal degree at each hierarchy.