A necessary and sufficient condition for Riemann's singularity theorem to hold on a Prym theta divisor

Abstract

Let $(P,\Xi)$ be the naturally polarized model of the Prym variety associated to the étale double cover $\pi : \tilde C\rightarrow C$ of smooth connected curves defined over an algebraically closed field k of characteristic $\ne 2$, where genus(C) = $g \ge 3$, Pic$^{(2g-2)}(\tilde C) \supset P = \{\mathcal L \in {\rm Pic}^{(2g-2)}(\tilde C) : {\rm Nm}(\mathcal L) = \omega_C$ and $h^0(\tilde C,\mathcal L)$ is even\} is the Prym variety, and $P \supset \Xi = \{\mathcal L \in P: h^0(\tilde C,\mathcal L) >0 \}$ is the Prym theta divisor with its reduced scheme structure. If $\mathcal L$ is any point on $\Xi$, we prove that ‘Riemann's singularity theorem holds at $\mathcal L$’, i.e. mult$_{\mathcal L}(\Xi) = (1/2)h^0(\tilde C,\mathcal L)$, if and only if $\mathcal L$ cannot be expressed as $\pi^*(\mathcal M)(B)$ where $B \ge 0$ is an effective divisor on $\tilde C$, and $\mathcal M$ is a line bundle on C with $h^0(C,\mathcal M) >(1/2)h^0(\tilde C,\mathcal L)$. This completely characterizes points of $\Xi$ where the tangent cone is the set theoretic restriction of the tangent cone of $\tilde {\Theta}$, hence also those points on $\Xi$ where Mumford's Pfaffian equation defines the tangent cone to $\Xi$.