User v m - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T12:26:53Z http://mathoverflow.net/feeds/user/6784 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/68503/has-anyone-studied-the-prym-map-for-double-covers-with-two-ramification-points/68528#68528 Answer by V M for Has anyone studied the Prym map for double covers with two ramification points? V M 2011-06-22T16:07:59Z 2011-06-22T16:07:59Z <p>Hi, this is actually a part of my Ph.D. thesis. I am going to discuss it in 6 months. Here you can find a preprint of the work with my advisor <a href="http://arxiv.org/abs/1010.4483" rel="nofollow">http://arxiv.org/abs/1010.4483</a>. It is not the final version, so there could be some minor mistakes.</p> <p>We have proved that, with two ramification point, the Prym map is generically injective when g is greater or equal than 6 and that it is dominant for g=4. </p> <p>To complete the proof of the Generic Torelli Theorem, we have also studied a partial extension of the Prym map to admissible coverings. </p> <p>Anyway, I am still working on this topic and the thesis will contain other results about this.</p> http://mathoverflow.net/questions/40473/effective-theta-characteristics Effective theta characteristics V M 2010-09-29T14:18:02Z 2010-12-15T04:47:50Z <p>Let $C$ be a complex smooth projective curve of genus $g$ and let $N$ be the number of effective theta-characteristics of $C$, or equivalently, the number of points of order two on the theta divisor of $J(C)$.</p> <p>It is known that, if $C$ is generic, $N$ is the number of the odd theta characteristics. Mumford proves that on a principally polarized abelian variety the theta divisor cannot contain all the points of order two. It follows that $N&lt;2^{2g}$.</p> <p>Given an arbitrary curve $C$, is it known a upper bound for $N$ depending on $g$?</p> http://mathoverflow.net/questions/28022/differential-of-the-torelli-morphism-at-the-boundary Differential of the Torelli morphism at the boundary V M 2010-06-13T11:22:23Z 2010-09-28T11:59:20Z <p>Let consider the Torelli morphism $T:\mathcal{M}_g \rightarrow \mathcal{A}_g$, from the moduli space of curves of genus $g$ to the moduli space of principal polarized abelian varieties of dimension $g$, that maps a curve to its Jacobian. The differential of $T$ at a point $[C]$ is the natural map $$H^1(C, T_C) \rightarrow Sym^2H^1(C, \mathcal{O}_C).$$ I know that $T$ can be extended to a map $$T:\bar{\mathcal{M}_g} \rightarrow \bar{\mathcal{A}_g}$$ from the Deligne-Mumford compactification of $\mathcal{M}_g$ to some compactification of $\mathcal{A}_g$.</p> <p>I would like to know if there is a way to describe the differential of $T$ at a point representing a nodal curve. More specifically, how can we describe the deformations space of a semi-abelian variety and in particular of a generalized Jacobian variety? Over $\mathbb{C}$, by computing the period matrix, one can show that the differential of $T$ has maximal rank at each point representing a nodal curve with non-hyperelliptic normalization. I'm wondering if, perhaps, there is a more algebraic way to see it.</p> http://mathoverflow.net/questions/40473/effective-theta-characteristics/40476#40476 Comment by V M V M 2010-10-02T14:46:21Z 2010-10-02T14:46:21Z I didn't know about it. I'm not sure I understand your answer. Do you mean that Scorza correspondence gives an estimate on the maximal number of effective theta characteristics? http://mathoverflow.net/questions/40473/effective-theta-characteristics/40476#40476 Comment by V M V M 2010-09-29T14:38:11Z 2010-09-29T14:38:11Z Yes, I mean upper bound. Thank you. Now I've modified the question and it is more precise. I take a look to your link. http://mathoverflow.net/questions/28022/differential-of-the-torelli-morphism-at-the-boundary Comment by V M V M 2010-07-23T13:30:14Z 2010-07-23T13:30:14Z I'm sorry for my late reply. I think you are right, though that book is not really clear to me (I think I will need some more time to understand it). In the complex case I'm thinking to the Voronoi compactification of the moduli space of abelian variety. I know this is not canonical but this is not a real problem for my purposes. I'm wondering what is the right analogous in the non-complex case.