Counter-examples of high Clifford index to Prym-Torelli (Q2913224)

The present article addresses two fundamental questions in the theory of abelian varieties: (1) determine the non-injectivity locus of the Prym map; (2) determine which multiples of the minimal class of a principally polarized abelian variety are representable by an irreducible curve.NEWLINENEWLINERegarding question (1), the authors generalize Donagi's tetragonal construction to certain 4-to-1 simply ramified covers of a smooth curve of any genus. As a consequence, they show the existence of curves \(X\) of arbitrarily large Clifford index such that the Prym map is not injective at any étale double cover of \(X\). The proof of this result provided by the authors is interesting in its own right because it gives explicit correspondences that induce the (generalized) tetragonal isomorphisms among the Pryms in question. In particular, this gives a new proof of Donagi's result.NEWLINENEWLINERegarding question (2), given a simply ramified \(n\)-sheeted cover of smooth projective curves \(X \to Y\) and an étale double cover \(\tilde{X} \to X\), the authors produce curves \(\tilde{C}_1\) and \(\tilde{C}_2\) with natural morphisms to the Prym \(P\) of the cover \(\tilde{X} \to X\). Moreover, the authors show that the image of \(\tilde{C}_i\) in \(P\) for \(i=1,2\) has the class NEWLINE\[NEWLINE2^{n-1}\frac{[\Theta_P]^{g_X-2}}{(g_X-2)!},NEWLINE\]NEWLINE where \(\Theta_P\) is the principal polarization of \(P\) and \(g_X\) is the genus of \(X\). Their proof proceeds by degenerating to the case when \(Y\) is an irreducible rational nodal curve.