| Abstract |
A celebrated theorem of Mark Green from the 1980s describes the syzygies of a curve embedded by a line bundle of sufficiently large degree: if $L$ is a line bundle on a smooth projective curve $C$ of genus $g$ with $\deg(L)\geq 2g+1+p$, then $L$ satisfies property $N_p$; equivalently, the minimal free resolution of its homogeneous ideal is linear for the first $p$ steps. Since then, many efforts have been made to find analogues of Green’s theorem in broader settings. Two major directions are the study of linear syzygies of adjoint line bundles on higher-dimensional varieties, as predicted by syzygetic Fujita conjectures, and the study of higher-weight syzygies, encoded by properties $N_{d,p}$.
In joint work with Wenbo Niu, we address both directions by studying the syzygies of tautological line bundle on symmetric products of curves. Let $C$ be a smooth projective curve of genus $g$, let $L$ be a line bundle on $C$, and let $T_{k+1,L}$ be the tautological line bundle on the symmetric product $C_{k+1}$, obtained by descent from $L^{\boxtimes (k+1)}$ on $C^{k+1}$. We prove that for each integer $d$ with $0\leq d\leq k$, if $\deg(L)\geq dg+2g+1+p$, then $T_{k+1,L}$ satisfies property $N_{k+2-d,p}$. This yields a family of sharp results on higher syzygies of higher weight for symmetric products of curves in arbitrary dimension, and in particular recovers Green’s classical theorem for curves. We also prove a sharp numerical criterion for the nefness of certain divisors on $C_{k+1}$. Taken together, these results provide strong evidence for syzygetic Fujita conjecture on property $N_p$ adjoint linear series and may be viewed as a natural higher-dimensional analogue of Green’s theorem. |
