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تسجيل مستخدم جديدThe trigonal construction in the ramified case
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To every double cover ramified in two points of a general trigonal curve of genus g, one can associate an etale double cover of a tetragonal curve of genus g+1. We show that the corresponding Prym varieties are canonically isomorphic as principally polarized abelian varieties.
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In the local, unramified case the determinantal functions associated to the group-ring of a finite group satisfy Galois descent. This note examines the obstructions to Galois determinantal descent in the ramified case.
The aim of these notes is to generalize Laumons construction [18] of automorphic sheaves corresponding to local systems on a smooth, projective curve $C$ to the case of local systems with indecomposable unipotent ramification at a finite set of point
s. To this end we need an extension of the notion of parabolic structure on vector bundles to coherent sheaves. Once we have defined this, a lot of arguments from the article On the geometric Langlands conjecture by Frenkel, Gaitsgory and Vilonen [10] carry over to our situation. We show that our sheaves descend to the moduli space of parabolic bundles if the rank is $leq 3$ and that the general case can be deduced form a generalization of the vanishing conjecture of [10].
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 generica
lly 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.
Using Galois Theory, we construct explicitly absolutely simple (principally polarized) Prym varieties that are not isomorphic to jacobians of curves even if we ignore the polarizations. Our approach is based on the previous papers math/0610138 [math.AG] and math/0605028 [math.AG] .
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 study
ing 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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