Hyperelliptic Gorenstein curves.

*(English)*Zbl 0940.14018The subject of this paper is the moduli space of pointed singular hyperelliptic curves of genus \(g\geq 2\).

It is proved that every such curve can be realized as a curve of degree \(2g+2\) on a rational cone in \({\mathbb P}^{g+2}\), not passing through its vertex, its \(g^1_2\) being cut on the curve by the lines of the cone. Moreover every morphism between two such curves is induced by an automorphism of \({\mathbb P}^{g+2}\) itself, which carries the cone to itself.

Using this idea, it is proved that the classes of isomorphism of such curves possessing a two-branched point of singularity degree \(\delta \) form a family of dimension \(2g-2\delta\), while the family of those equipped with an unibranched Weierstrass point has dimension \(2g-2\delta -1\) (the dimension of the family with a non-Weierstrass point has dimension \(2g\)).

The result is first proved in any characteristic \(p\neq 2\); the case \(p=2\) is then explicitly described.

It is proved that every such curve can be realized as a curve of degree \(2g+2\) on a rational cone in \({\mathbb P}^{g+2}\), not passing through its vertex, its \(g^1_2\) being cut on the curve by the lines of the cone. Moreover every morphism between two such curves is induced by an automorphism of \({\mathbb P}^{g+2}\) itself, which carries the cone to itself.

Using this idea, it is proved that the classes of isomorphism of such curves possessing a two-branched point of singularity degree \(\delta \) form a family of dimension \(2g-2\delta\), while the family of those equipped with an unibranched Weierstrass point has dimension \(2g-2\delta -1\) (the dimension of the family with a non-Weierstrass point has dimension \(2g\)).

The result is first proved in any characteristic \(p\neq 2\); the case \(p=2\) is then explicitly described.

Reviewer: Alessandro Gimigliano (Bologna)

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\textit{K.-O. Stöhr}, J. Pure Appl. Algebra 135, No. 1, 93--105 (1999; Zbl 0940.14018)

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##### References:

[1] | de Carvalho, C.F.; Stöhr, K.-O., Higher order differentials and Weierstrass points on Gorenstein curves, Manuscripta math., 85, 361-380, (1994) · Zbl 0839.14019 |

[2] | Eisenbud, D.; Harris, J.; Koh, J.; Stillmann, M., Clifford’s theorem for singular curves, Amer. J. math., 110, 532-539, (1988) |

[3] | Hartshorne, R., Algebraic geometry, (1977), Springer New York · Zbl 0367.14001 |

[4] | M. Homma, Singular hyperelliptic curves, Preprint. · Zbl 0940.14019 |

[5] | da Rosa, R.M., Curvas trigonais de Gorenstein com invariante de maroni igual a zero, () |

[6] | Rosenlicht, M., Equivalence relations on algebraic curves, Ann. of math., 56, 169-171, (1952) · Zbl 0047.14503 |

[7] | Serre, J.-P., Groupes algébriques et corps de classes, (1959), Hermann Paris · Zbl 0097.35604 |

[8] | Stöhr, K.-O., On the poles of regular differentials of singular curves, Bol. soc. bras. mat., 24, 105-136, (1993) · Zbl 0788.14020 |

[9] | Tate, J., The arithmetic of elliptic curves, Invent. math., 23, 179-206, (1974) · Zbl 0296.14018 |

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