most recent 30 from http://mathoverflow.net 2013-05-21T09:32:59Z http://mathoverflow.net/feeds/user/17673 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/92946/families-of-genus-four-curves-with-only-hyperelliptic-reduction xuehang 2012-04-03T00:31:07Z 2013-02-20T17:17:59Z

<p>Is it possible to construct a nonisotrivial family of genus four curves $X \rightarrow S$, with the following properties:</p> <p>(1) $S$ is a complete curve;</p> <p>(2) All the fibers are smooth;</p> <p>(3) The generic fiber lies on a singular quadratic in $\mathbb{P}^3$.</p> <p>It is not possible to construct such a family if we replace (3) by "all the fibers...". But the problem is that there could be smooth hyperelliptic reductions.</p>

http://mathoverflow.net/questions/109243/genus-two-curve-with-special-automorphisms xuehang 2012-10-09T17:37:11Z 2012-10-09T18:18:58Z

<p>Is it possible to find a genus two curve $C$ (over the field of complex numbers) with an endomorphism $\phi: C \to C$, such that $\phi$ has no fixed points and $\phi$ does not take any point to its hyperelliptic involution?</p>

http://mathoverflow.net/questions/98746/flat-morphism-between-regular-local-rings xuehang 2012-06-03T23:57:07Z 2012-06-03T23:57:07Z

<p>Suppose $f: A \rightarrow B$ is a local homomorphism of local rings. Assume that $A$ and $B$ are noetherian, regular and $\mathrm{Spec} B \rightarrow \mathrm{Spec} A$ is quasi-finite. Is is necessary that $f$ is flat?</p>

http://mathoverflow.net/questions/97907/isomorphism-of-line-bundles-over-mathrmspec-mathbbz xuehang 2012-05-25T04:01:12Z 2012-05-25T08:14:23Z

<p>Suppose $X$ is a scheme, the structure morphism $X \rightarrow \mathrm{Spec}\mathbb{Z}$ smooth and surjective. Assume further that $H^0(X \times \mathrm{Spec} \mathbb{C}, \mathcal{O}^\times) = \mathbb{C}^\times$. Then is it necessary that $H^0(X, \mathcal{O}^\times) = \pm 1$ ?</p>

http://mathoverflow.net/questions/94090/genus-four-curve-with-3p-mathfrakg1-3 xuehang 2012-04-15T05:21:42Z 2012-04-15T18:24:45Z

<p>Let $M_4$ be the moduli space of genus four curves. Let $\Sigma \subset M_4$ be the locus such that for $X \in \Sigma$, there is a point $p$ on $X$, with $\dim H^0(X, \mathcal{O}(3p)) =2$. What is the codimension of this locus in $M_4$?</p>

http://mathoverflow.net/questions/92946/families-of-genus-four-curves-with-only-hyperelliptic-reduction/122341#122341 xuehang 2013-02-21T03:51:24Z 2013-02-21T03:51:24Z

At least I think the closure of the Petri locus is ample. There are two reasons. One is that its divisor class is $34 \lambda$ where $\lambda$ is the hodge bundle. The other is that the complement in $M_4$ of the closure of Petri locus parametrizes smooth $(3,3)$-curves on $\mathbf{P}^1 \times \mathbf{P}^1$, which is affine.

http://mathoverflow.net/questions/92946/families-of-genus-four-curves-with-only-hyperelliptic-reduction/122341#122341 xuehang 2013-02-20T19:11:52Z 2013-02-20T19:11:52Z

The nonexistence of a surface in $M_4$ was my original motivation of this question. My idea is that the closure of the Petri locus (the locus of curves which lies on the singular cone) is an ample divisor in $M_4$. If we had a surface in $M_4$, then it must intersect with the Petri locus. This will (after possibly desingularization of the intersection) give a family of curves with the property described in the question. Previously I thought I need this nonexistence for my paper, but now I realize that I don't need this anymore...

http://mathoverflow.net/questions/92946/families-of-genus-four-curves-with-only-hyperelliptic-reduction/122341#122341 xuehang 2013-02-20T03:26:59Z 2013-02-20T03:26:59Z

Thank you very much for your answer. One consequence of this nonexistence is that there is no projective surface in the $M_4$ (fine moduli). Otherwise the intersection of this surface and the closure Petri locus will give a family like this. Is this right?

http://mathoverflow.net/questions/109243/genus-two-curve-with-special-automorphisms/109246#109246 xuehang 2012-10-10T02:10:16Z 2012-10-10T02:10:16Z

thank you for your comment. Can you give a little more detail?

http://mathoverflow.net/questions/92946/families-of-genus-four-curves-with-only-hyperelliptic-reduction xuehang 2012-04-08T02:33:27Z 2012-04-08T02:33:27Z

Does this construction give any information of the dualzing sheaf of the family?

http://mathoverflow.net/questions/92946/families-of-genus-four-curves-with-only-hyperelliptic-reduction xuehang 2012-04-04T15:05:14Z 2012-04-04T15:05:14Z

Thanks for the comment. Concerning the genus three curves, there is such a family like that. I'm not sure of the genus four family. But that seems to be easier than the original question. A naive idea to the original question would be take a complete surface in $M_4$, and the intersection with the Petri locus gives the desired family. But I've just found out yesterday that we don't know if such a complete surface exists.