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Register a new userGlobal Prym-Torelli for double coverings ramified in at least 6 points
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We prove that the ramified Prym map $mathcal P_{g, r}$ which sends a covering $pi:Dlongrightarrow C$ ramified in $r$ points to the Prym variety $P(pi):=text{Ker}(text{Nm}_{pi})$ is an embedding for all $rge 6$ and for all $g(C)>0$. Moreover, by studying the restriction to the locus of coverings of hyperelliptic curves, we show that $mathcal P_{g, 2}$ and $mathcal P_{g, 4}$ have positive dimensional fibers.
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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 $rge max {6,frac 23(g+2) }$. 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 previously by Nagaraj and Ramanan, and also by Bardelli, Ciliberto and Verra where the degree of the map is $3$. The paper closes with an appendix by Alessandro Verra with an independent result, the rationality of the moduli space of coverings with $g=2,r=6$, whose proof is self-contained.
We study the Prym varieties arising from etale cyclic coverings of degree 7 over a curve of genus 2. These Prym varieties are products of Jacobians JY x JY of genus 3 curves Y with polarization type D=(1,1,1,1,1,7). We describe the fibers of the Prym map between the moduli space of such coverings and the moduli space of abelian sixfolds with polarization type D, admitting an automorphism of order 7.
A mapping class group of an oriented manifold is a quotient of its diffeomorphism group by the isotopies. In the published version of Mapping class group and a global Torelli theorem for hyperkahler manifolds I made an error based on a wrong quotation of Dennis Sullivans famous paper Infinitesimal computations in topology. I claimed that the natural homomorphism from the mapping class group to the group of automorphims of cohomology of a simply connected Kahler manifold has finite kernel. In a recent preprint arXiv:1907.05693, Matthias Kreck and Yang Su produced counterexamples to this statement. Here I correct this error and other related errors, observing that the results of Mapping class group and a global Torelli theorem remain true after an appropriate change of terminology.
It was conjectured in cite{Namikawa_ExtendedTorelli} that the Torelli map $M_gto A_g$ associating to a curve its jacobian extends to a regular map from the Deligne-Mumford moduli space of stable curves $bar{M}_g$ to the (normalization of the) Igusa blowup $bar{A}_g^{rm cent}$. A counterexample in genus $g=9$ was found in cite{AlexeevBrunyate}. Here, we prove that the extended map is regular for all $gle8$, thus completely solving the problem in every genus.
We prove an analogue of Kirchhoffs matrix tree theorem for computing the volume of the tropical Prym variety for double covers of metric graphs. We interpret the formula in terms of a semi-canonical decomposition of the tropical Prym variety, via a careful study of the tropical Abel-Prym map. In particular, we show that the map is harmonic, determine its degree at every cell of the decomposition, and prove that its global degree is $2^{g-1}$. Along the way, we use the Ihara zeta function to provide a new proof of the analogous result for finite graphs. As a counterpart, the appendix by Sebastian Casalaina-Martin shows that the degree of the algebraic Abel-Prym map is $2^{g-1}$ as well.
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