A Riemann theorem for the theta divisors on moduli spaces of stable fibre bundles over a curve
Let \(C\) be a smooth projective curve of genus \(g\geq 2\). The moduli space \({\mathcal U}(r,g-1)\) (respectively \({\mathcal S}{\mathcal U}(r,g-1))\) of stable vector bundles on \(C\) of rank \(r\) and degree \(g-1\) (respectively of rank \(r\), degree \(g-1\) and fixed determinant) admits the so called generalized theta divisor \(\Theta_ r\) (respectively \(\Theta_ r'\)) defined as \(\Theta_ r=\{E\in{\mathcal U}(r,g-1)\mid h^ 0(C,E)\geq 1\}\) (respectively \(\Theta_ r'\) is the restriction of \(\Theta _ r\) to \({\mathcal S}{\mathcal U}(r,g-1))\). The main theorem of the paper is the following generalization of Riemann's singularity theorem: The multiplicity of \(\Theta_ r\) at \(E\) equals the dimension of the space of global sections of \(E\). If \(r\geq 2\) the analogous result holds for \(\Theta_ r'\). --- As a corollary it is shown that the divisor \(\Theta_ r'\) is a normal irreducible variety. For any \(E\in{\mathcal U}(r,g-1)\) denote by \(\Theta_ E\) the reduced subscheme of the Jacobian \(J(C)\) with support \(\{L\in J(C)\mid h^ 0(C,E\otimes L)\geq 1\}\). \(\Theta_ E\) is a divisor on \(J(C)\), if \(r=2\) and if \(r\geq 3\) for generic \(E\). It is shown that \(\Theta_ 2'\) admits a unique point \(E\) of multiplicity 3, and that its associated divisor \(\Theta_ E\) on \(J(C)\) is the image of the difference map \(C^ 2\to J(C)\), \((p,q)\mapsto{\mathcal O}_ C(p-q)\). Finally let \(E\in{\mathcal U}(2,g- 1)\) with \(\text{det} E=K_ C\). The following singularity theorem is proven for the divisor \(\Theta_ E\) on \(J(C)\): \(\Theta_ E\) is singular at \(L\) if and only if either \(h^ 0(C,E\otimes L)\geq 2\) or \(h^ 0(C,E\otimes L)=1\), \(E\otimes L\) is an extension of \(K_ C(-D)\) with \(L^ 2(D)\) (where \(D\) is a nonzero section of \(E^*\otimes L^{-1}\otimes K_ C)\) and \(h^ 0(C,L^ 2(D))=1\).
scientific article; zbMATH DE number 58550
scientific article