The fibre of the Prym map in genus three (Q1073123)

Let \(\bar {\mathcal R}_ 3\) be the Prym moduli space for allowable double covers of stable genus three curves. Let \(\bar P : \bar {\mathcal R}_ 3 \to {\mathcal A}_ 2\) be the Prym map, the main result of this paper is the following description of the fibre of \(\bar P:\) Let S be a principally polarized abelian surface, \(s\in {\mathcal A}_ 2\) its moduli point, \(\Theta\) \(\subset S\) a symmetric theta divisor. Assume S general (i.e. \(Aut(\Theta)={\mathbb{Z}}/2{\mathbb{Z}})\) then \(\bar P^{-1}(s)\) is obtained from the Siegel modular quartic threefold V [cf. \textit{G. Van der Geer}, Math. Ann. 260, 317-350 (1982; Zbl 0473.14017)], after a sequence of two blowing up's \(\sigma_ 1\) and \(\sigma_ 2 : \sigma_ 1 : \hat V\to V\) blows up V in a point v, while \(\sigma_ 2\) is centered along a curve Q isomorphic to \(\Theta\) /Aut(\(\Theta)\). The exceptional divisor of \(\sigma =\sigma_ 2\cdot \sigma_ 1\) is the union of two surfaces \(E_ 1=\sigma^{-1}(v)\) and \(E_ 2\). The points of \(E_ 1\) are the moduli points for double covers \(\pi : \tilde C\to C\) such that \(\tilde C\) is hyperelliptic. The points of \(E_ 2\) are the moduli points for double covers \(\pi : \tilde C\to C\) such that both \(\tilde C\) and C are elliptic tails. Notice that V is a projective model of the Satake compactification of the moduli space of principally polarized abelian surfaces endowed with a level two structure.