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Generic injectivity of the Prym map for double ramified coverings. (arXiv:1708.06512v2 [math.AG] UPDATED)
来源于:arXiv
In this paper we consider the Prym map for double coverings of curves of
genus $g$ ramified at $r>0$ points. That is, the map associating to a double
ramified covering its Prym variety. The generic Torelli theorem states that the
Prym map is generically injective as soon as the dimension of the space of
coverings is less or equal to the dimension of the space of polarized abelian
varieties. We prove the generic injectivity of the Prym map in the cases of
double coverings of curves with: (a) $g=2$, $r=6$, and (b) $g= 5$, $r=2$. In
the first case the proof is constructive and can be extended to the range $r\ge
\max \{6,\frac 23(g+2) \}$. For (b) we study the fibre along the locus of the
intermediate Jacobians of cubic threefolds to conclude the generic injectivity.
This completes the work of Marcucci and Pirola who proved this theorem for all
the other cases, except for the bielliptic case $g=1$ (solved later by Marcucci
and the first author), and the case $g=3, r=4$ considered previo 查看全文>>