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. 



Fiber of the Prym map in dim 2
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


Verra, Alessandro: *[The Fibre of the Prym Map in Genus Three](http://gdz.sub.uni-goettingen.de/dms/load/img/?PID=GDZPPN002329107)*, Mathematische Annalen **276** (1986), 433-448.


