# Fiber of the Prym map in dim 2

This must be very classical, but I can't find a reference.

Is there an explicit description of the (generic?) fibers of the Prym map $\mathcal{R}_3 \to \mathcal{A}_2$?

By this I mean the map that to any double etale cover of a genus 3 curve associates the corresponding Prym variety.

• If (A,Theta) is a 2 dimensional principally polarized abelian variety, the linear series |2Theta| defines a 2:1 map from A to the associated quartic Kummer surface in P^3. To a general plane section of that surface is thus associated a double cover of a quartic plane curve of genus three, whose Prym variety, if memory serves, is A. This gives a rational map from the dual P^3 to the fiber of the Prym map over A. As linked below, Verra defines a regular extension of that map and computes its fibers. This case is already discussed by Wirtinger, Untersuchungen uber Thetafunctionen, on page 113, Feb 23, 2017 at 19:03
• Ah yes, by Masiewicki’s criterion, (Trans. Am. Math. Soc., 222 (1976), 221-240.) any curve in the series |2Theta| is the Abel Prym model of a curve with involution having A as Prym variety. Feb 23, 2017 at 19:16
• In fact, since the map from A to its Kummer surface takes each point x to -x, translation by a point of order two is a fiber preserving isomorhism of A. Thus each genus 3 hyperplane section curve and its double cover lie in an orbit of isomoprphic double covers parametrized by points of order two on A. Hence the rational parametrization of the fiber of Prym by the dual P^3 has degree 16. So, for what it's worth, this is the rough classical story as I recall it, that Verra renders extremely fine and precise. Feb 23, 2017 at 20:13

The fibres of the extended Prym map $\overline{P} \colon \overline{\mathcal{R}}_3 \to \mathcal{A}_2$ are studied in detail in the paper