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Feb
21
comment families of genus four curves with only hyperelliptic reduction
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
comment families of genus four curves with only hyperelliptic reduction
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
comment families of genus four curves with only hyperelliptic reduction
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?
Oct
10
awarded  Scholar
Oct
10
comment genus two curve with special automorphisms
thank you for your comment. Can you give a little more detail?
Oct
10
accepted genus two curve with special automorphisms
Oct
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asked genus two curve with special automorphisms
Jun
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asked flat morphism between regular local rings
May
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asked isomorphism of line bundles over $\mathrm{Spec} \mathbb{Z}$
Apr
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asked genus four curve with $|3p|= \mathfrak{g}^1_3$
Apr
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comment families of genus four curves with only hyperelliptic reduction
Does this construction give any information of the dualzing sheaf of the family?
Apr
4
comment families of genus four curves with only hyperelliptic reduction
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.
Apr
4
awarded  Editor
Apr
4
revised families of genus four curves with only hyperelliptic reduction
added 14 characters in body
Apr
3
awarded  Student
Apr
3
asked families of genus four curves with only hyperelliptic reduction