Linear subspaces of the intersection of two quadrics via Kuznetsov component
Yanjie Li, Shizhuo Zhang
TL;DR
The paper develops a derived-categorical bridge between linear subspaces of a smooth intersection of two quadrics Y in \mathbb{P}^{2g+1} and vector bundles on the associated hyperelliptic curve C of genus g, via the Kuznetsov component Ku(Y) and the projection functor to D^b(C). For each subspace V of dimension l and integers m in the range 2g-3-l \le m \le 2g-3, it constructs a vector bundle \mathcal{F}_{m,V} on C of rank 2^{g-1-l}, related by a rotation autoequivalence and extensions, and shows injection of the Hilbert scheme of l-subspaces into moduli spaces of stable bundles whenever the bundles are stable. This yields a unified, categorical perspective on classical correspondences (Desale–Ramanan, Reid) by identifying V \mapsto [\mathcal{F}_{m,V}] as a closed embedding into moduli spaces such as Pic^d(C) and SU_C(2,h_m), with a notable instance for g=3 where Y embeds into SU_C^s(4,h). The work also outlines Brill-Noether questions for these projected objects and proposes a program to extend the moduli-interpretation to broader intersections of quadrics and their Hitchin-type fibrations.
Abstract
Let $Q_i(i=1,2)$ be $2g$ dimensional quadrics in $\mathbb{P}^{2g+1}$ and let $Y$ be the smooth intersection $Q_1\cap Q_2$. We associate the linear subspace in $Y$ with vector bundles on the hyperelliptic curve $C$ of genus $g$ by the left adjoint functor of $Φ:D^b(C)\rightarrow D^b(Y)$. As an application, we give a different proof of the classification of line bundles and stable bundles of rank $2$ on hyperelliptic curves given by Desale and Ramanan. When $g=3$, we show that the projection functor induces a closed embedding $α:Y\rightarrow SU^s_C(4,h)$ into the moduli space of stable bundles on $C$ of rank $4$ of fixed determinant.