Moduli of rank-2 vector bundles, theta divisors, and the geometry of curves in projective space
Let \(L\) be a line bundle of positive degree on a smooth projective curve \(C\) of genus \(g\). Let \({\mathcal M}_{2,L}\) denote the moduli space of semistable rank-2 vector bundles on \(C\) with determinant \(L\). The extension map \(\phi_ L:\mathbb{P}_ L = P(\text{Ext}^ 1(L,{\mathcal O}_ C)^*) \to {\mathcal M}_{2,L}\) is the rational map defined by associating to an extension \(0\to {\mathcal O}_ C \to E \to L\to 0\) the vector bundle \(E\). The main theorem of the paper gives explicitly a sequence of blow- ups \(\sigma: \widetilde{\mathbb{P}}_ L \to {\mathbb{P}}_ L\) which resolves the extension map \(\phi_ L\) into a morphism \(\widetilde{\phi}_ L = \phi_ L\circ \sigma\). The main application of the theorem concerns the spaces \(H^ 0({\mathcal M}_{2,L},{\mathcal O}(k\theta))\) where \(\theta\) denotes the theta divisor of \({\mathcal M}_{2,L}\), namely if \(\text{deg }L = 2g\) then there is a natural identification \[ H^ 0({\mathcal M}_{2,L},{\mathcal O}(k\theta)) = H^ 0(\mathbb{P}_ L,{\mathcal O}(kg) \otimes J_ C^{k(g-1)}) \] , where \(J_ C\) denotes the ideal of \(C\) in \(\mathbb{P}_ L\). Similarly for \(\text{deg }L = 2g - 1\). Other applications are a new simple proof that \(\text{Pic}({\mathcal M}_{2,L}) \simeq \mathbb{Z}\), that the class of \(\theta\) in \(\text{Pic}({\mathcal M}_{2,\omega_ C})\) is irreducible and moreover, that if \(C\) is not hyperelliptic, then \(\theta \subset {\mathcal M}_{2,L}\) is birationally very ample for odd \(\text{deg }L\).
line bundle on a smooth projective curve
moduli space of semistable rank-2 vector bundles
scientific article; zbMATH DE number 120208