Nets of quadrics and vector bundles on a double plane
Let \(p: X\to {\mathbb{P}}^ 2\) be a double cover whose branch locus is a connected nonsingular curve C of degree \(d=2b\). Denote by M the moduli space of semistable vector bundles F of rank 2n on X with i-action \(j: F\to i*F,\) \(i*j\circ f=id\), where i denotes the involution on X, such that (1) F carries an everywhere nondegenerate quadratic form with values in \({\mathcal O}_ X(-1)\), (2) the eigenbundle corresponding to the eigenvalue \(-1\) for the i-action on the restriction of F to \(p^{-1}(C)\) is isomorphic to \(p*\vartheta(1-b)\) where \(\vartheta\) is a fixed ineffective theta characteristic on C, (3) \(c_ 1(F)={\mathcal O}_ X(-n),\) \(c_ 2(F)=b^ 2+n(n-1)\) and (4) the restriction of F to the non- singular hyperelliptic curve \(p^{-1}(\ell)\) is semistable for a generic line \(\ell \subset {\mathbb{P}}^ 2\). On the other hand, let P be the subscheme of the Grassmannian \(G_{b-n}(H^ 0(\vartheta (1))^{\vee})\) consisting of \((b-n)\)-dimensional linear subspaces which are contained in the intersection of all quadrics in the net determined by \((C,\vartheta)\). The main result of the paper under review establishes that M is isomorphic to P. This generalises the relationship between a pencil of quadrics and bundles on the non-singular hyperelliptic curve associated to the pencil due to the author and \textit{S. Ramanan} [see the author, Compos. Math. 51, 15-40 (1984; Zbl 0539.14011)].
moduli space of semistable vector bundles
scientific article; zbMATH DE number 4004360