Timeline for families of genus four curves with only hyperelliptic reduction
Current License: CC BY-SA 3.0
9 events
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| Jun 15, 2020 at 7:27 | history | edited | CommunityBot |
Commonmark migration
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| Feb 21, 2013 at 3:51 | comment | added | xuehang | 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. | |
| Feb 20, 2013 at 22:24 | comment | added | David Lehavi | My skepticism here is because I tried several similar arguments a few years back, and they all collapsed (it was years back - so I don't remember if I tried the Petri locus, and I don't remember if the degenerations I used were the same): - Is the Petri locus is ample in $\overline{\mathcal{M}_4}$ or in $\mathcal{M}_4$ ? - I edited my response above as there is a gap in my argument (you have to verify that the limits of the universal dualizing systems are the same ones as in the Hurwitz schemes: a) this looks like a lot of work, and b) I'm not sure it's correct. | |
| Feb 20, 2013 at 19:11 | comment | added | xuehang | 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... | |
| Feb 20, 2013 at 17:17 | history | edited | David Lehavi | CC BY-SA 3.0 |
found a gap in the claim
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| Feb 20, 2013 at 6:35 | comment | added | David Lehavi | Also, chances are thar if you csn complete the argument, my answer is wrong.... | |
| Feb 20, 2013 at 6:02 | comment | added | David Lehavi | I suspected thos is what you are after, but i dont see it , care to elaboraye. ? | |
| Feb 20, 2013 at 3:26 | comment | added | xuehang | 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? | |
| Feb 19, 2013 at 19:02 | history | answered | David Lehavi | CC BY-SA 3.0 |