Prym map and second Gaussian map for Prym-canonical line bundles (Q390759)

Let \({\mathcal R}_g\) the moduli space which parametrizes isomorphism classes of pairs \((C,A)\), where \(C\) is a smooth curve of genus \(g\) and \(A\) is a torsion point of order \(2\); or equivalently, isomorphism classes of unmarried double coverings \(\tilde C\to C\). The Prym map, \(\text{Pr}_g:{\mathcal R}_g\to {\mathcal A}_{g-1}\), associates to a point \((C,A)\) the isomorphism class of the connected component of zero of the kernel of the norm map \(J\tilde C\to J C\) with its principal polarization.NEWLINENEWLINEThe first main result of the paper is that the second fundamental form of the Prym map, \(\text{Pr}_g\), lifts the second Gaussian map, \(\mu_A\), of the Prym-canonical bundle. More precisely, if \(I_2\) is the kernel of the multiplication map \(S^2 H^0(K_C\otimes A)\to H^0(K_C^{\otimes 2})\), then the second Gaussian map associated to the Prym canonical bundle \(K_C\otimes A\), \(\mu_A:I_2\to H^0(K_C^{\otimes 4})\) is the composite of the second fundamental form of the Prym map, \(I_2\to S^2 H^0(K_C^{\otimes 2})\) and the multiplication map. The second main result is the surjectivity of \(\mu_A\) for \((C,A)\) generic and \(g\geq 20\).