Theta divisors and the geometry of tautological model
Let \(C\) be a smooth projective curve of genus \(g \geq 3\) over \({\mathbb C}\), and let \(E\) be a vector bundle of rank \(r\) and slope \(2g-1\) over \(C\). If \(E\) is general in moduli, then the set \[ \{ \ell \in \mathrm{Pic}^{-g} (C) : h^0 ( C, E \otimes \ell ) \geq 1 \} \] is the support of a divisor \(\Theta_E\), the \textit{theta divisor of \(E\)} (see Beauville-Narasimhan-Ramanan [\textit{A. Beauville} et al., J. Reine Angew. Math. 398, 169--179 (1989; Zbl 0666.14015)]). Such generalised theta divisors have been the subject of much interest; see for example [\textit{A. Beauville}, Adv. Stud. Pure Math. 45, 145--156 (2006; Zbl 1115.14025)]. In this paper, the author studies an interesting connection between \(\Theta_E\) and the geometry of the \textit{tautological model} of \(E\). The latter is the image \(P_E\) of the natural map \[ {\mathbb P}(E^*) \dashrightarrow {\mathbb P} ( H^0 ( C, E )^* ) \cong | {\mathcal O}_{{\mathbb P}(E^*)} (1) |^* . \] Let \({\mathcal V}_{g, g-2} ( P_E )\) be the variety of \((g-2)\)-planes which are \(g\)-secant to \(P_E\). The author shows that if \(C\) is a Petri curve and \(r > g-1\), then for \(E\) stable and general in moduli, \(\Theta_E\) is birational to an irreducible component of \({\mathcal V}_{g, g-2} ( P_E )\). Conversely, with some assumptions on \(E\), the existence of a suitable component of \({\mathcal V}_{g, g-2} ( P_E )\) implies that \(E\) admits a theta divisor. Another result which may have further applications is Proposition 8.3 (1), which gives a sufficient condition for \(E\) \textit{not} to admit a theta divisor.
scientific article; zbMATH DE number 6841312
scientific article