A note on theta divisors of stable bundles
Let \(C\) be a complex projective smooth curve of genus \(g \geq 3\). Let \(E\) be a vector bundle of rank \(r\) over \(C\) which is semistable and of integral slope \(m := \mathrm{deg}(E)/r\). For \(h := g - 1 - m\), consider the locus \[ \{ N \in \mathrm{Pic}^h (C) : h^0 (\mathrm{gr}(E) \otimes N) \geq 1 \} . \] If not the whole of \(\mathrm{Pic}^h(C)\), this locus is the support of a divisor \(\Theta_E \in |r \Theta_M |\), where \(\Theta_M\) is a translate of the Riemann theta divisor on \(\mathrm{Pic}^{g-1}(C)\). The author shows that if \(C\) is a Petri curve of genus \(g \geq 4\), and \(E\) is a general stable bundle, then \(\Theta_E\) is an integral divisor with singular locus of dimension \(g-4\). Moreover, in this case a line bundle \(N\) is a singular point of \(\Theta_E\) if and only if \(h^0 ( E \otimes N ) \geq 2\). Furthermore, if \(C\) is nonhyperelliptic of genus \(3\), then for general stable \(E\) the divisor \(\Theta_E\) is smooth and integral. The proof relies on the Petri condition and uses various facts on generalised Brill-Noether loci from the references \textit{S. B. Bradlow} et al. [Int. J. Math. 14, No. 7, 683--733 (2003; Zbl 1057.14041)], \textit{S. Casalaina-Martin} and \textit{M. Teixidor i Bigas} [Math. Nachr. 284, No. 14--15, 1846--1871 (2011; Zbl 1233.14025)], \textit{Y. Laszlo} [Duke Math. J. 64, No. 2, 333--347 (1991; Zbl 0753.14023)], \textit{V. Mercat} [Manuscr. Math. 98, No. 1, 75--85 (1999; Zbl 0976.14021)] and \textit{M. Teixidor i Bigas} [Math. Nachr. 196, 251--257 (1998; Zbl 0954.14024)] and \textit{M. Teixidor i Bigas} [Duke Math. J. 62, No. 2, 385--400 (1991; Zbl 0739.14006)].
scientific article; zbMATH DE number 6496474