We consider the Prym map from the space of double coverings of a curve of genus g with r branch points to the moduli space of abelian varieties. We prove that 𝒫:ℛg,r→𝒜δg−1+r/2 is generically injective if We also show that a very general Prym variety of dimension at least 4 is not isogenous to a Jacobian.

Given a curve of genus 3 with an unramified double cover, we give an explicit description of the associated Prym variety. We also describe how an unramified double cover of a non-hyperelliptic genus 3 curve can be mapped into the Jacobian of a curve of genus 2 over its field of definition and how this can be used to perform Chabauty- and Brauer–Manin-type calculations for curves of genus 5 with an fixed-point-free involution. As an application, we determine the rational points on a smooth plane quartic and give examples of curves of genus 3 and 5 violating the Hasse principle. The methods are, in principle, applicable to any genus 3 curve with a double cover. We also show how these constructions can be used to design smooth plane quartics with specific arithmetic properties. As an example, we give a smooth plane quartic with all 28 bitangents defined over . By specialization, this also gives examples over .

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 $\Xi$.

In this paper, we show that the Chern classes ck of the de Rham bundle ${\cal H}_{{\rm d}R}$ defined on any ‘good’ toroidal compactification $\bar{\cal A}_g$ of the moduli space of Abelian varieties of dimension g are zero in the rational Chow ring of $\bar{\cal A}_g$, for g = 4, 5 and k>0.