The positive dimensional fibres of the Prym map

Abstract

Let $C$ be an irreducible complex smooth curve of genus $g ; \pi: \bar{C} \rightarrow C$ a connected unramified double covering of $C$. The Prym variety associated to the covering is, by definition, the component of the origin of the kernel of the norm $\operatorname{map} P(\bar{C}, C)=\operatorname{Ker}\left(\mathrm{Nm}_\pi\right)^0 \subset J \bar{C}$, that is, a principally polarized abelian variety (p.p.a.v.) of dimension $g(\bar{C})-g=g-1$. One defines the Prym map $P_g: \mathscr{R}_g \rightarrow \mathscr{A}_{g-1},(\bar{C} \stackrel{\pi}{\rightarrow} C) \mapsto P(\bar{C}, C)$, where $\mathscr{R}_g$ is the coarse moduli space of the coverings $\pi$ as above and $\mathscr{A}_{g-1}$ stands for the coarse moduli space of p.p.a.v.'s of dimension $g-1$. It is well known that this map is generically injective for $g \geq 7$. On the other hand, this map is never injective. The coarse moduli space $\mathscr{R} \mathscr{H}_g$ of and the fibres of the restriction of $P_g$ to $\mathscr{R} \mathscr{H}_g$ have positive dimension. Let $\mathscr{R}_g$ be the coarse moduli space of the unramified double coverings $\pi: \bar{C} \rightarrow C$ such that $C$ is a smooth bi-elliptic curve of genus $g$. This variety has $[(g+1) / 2]+2$ irreducible components: $\mathscr{R} \mathscr{B}_g=\left(\bigcup_{t=0}^{[(g-1) / 2]} \mathscr{R}_{g, t}\right) \cup \mathscr{R}_{\mathscr{B}_g^{\prime}}$. In this note the author characterizes the fibres of positive dimension of the Prym map. Theorem. Assume $g \geq 13$. A fibre of $P_g$ is positivedimensional at $(\bar{C}, C)$ if and only if $C$ is either hyperelliptic or $(\bar{C}, C) \in \bigcup_{t \geq 1} \mathscr{R}_{g, t}$.