Hierarchical Structure of Graded Betti Numbers in the Quadratic Strand

Thời gian: 9h30, thứ 6 ngày 6 tháng 2 năm 2026

Địa điểm: Hội trường Hoàng Tụy, nhà A6, Viện Toán học

Tóm tắt : 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 talk, 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 and to identify the extremal varieties in each range.  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.