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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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##### References:
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