Year: 2026
Type: preprint
Abstract: Let $C$ be a smooth projective curve of genus $g$, and let $T_{n,L}$ denote the tautological line bundle on its $n$-th symmetric product $C_n$. We prove effective vanishing criteria for Koszul cohomology $K_{p,q}(C_n,T_{n,B};T_{n,L})$, extending two classical theorems of Mark Green. The main ingredient is a weight-one vanishing theorem proved by induction on $n$, using a filtration of the kernel bundle of $T_{n,L}$ to reduce the required cohomological vanishings to lower symmetric products. As a consequence, when $g\geq1$ and $n\geq2$, $T_{n,L}$ satisfies property $(N_p)$ for $\mathrm{deg}(L)\geq(n+1)g+p$. This bound is sharp for elliptic symmetric squares and also gives evidence for a folklore conjecture on adjoint syzygies.
Source: arXiv (Cornell University)
Authors: Niu, Wenbo, Wei, Peter Yi
Institution: University of Arkansas at Fayetteville
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7221823295