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Hyperelliptic Gorenstein curves. (English) Zbl 0940.14018
The 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.

MSC:
14H10 Families, moduli of curves (algebraic)
14H52 Elliptic curves
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[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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