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Half-canonical series on algebraic curves. (English) Zbl 0627.14022
Let $${\mathcal M}^ r_ g$$ be the subloci of the moduli space $${\mathcal M}_ g$$ of curves of genus $$g$$ of those having a halfcanonical $$g^ s_{g- 1}$$ with $$s\geq r$$. The author gives the upper bound $$3g-3r+2$$ for the dimension of $${\mathcal M}^ r_ g$$ (which is sharp in the sense that for every r there is one g for which it is attained) and determines the codimension (in $${\mathcal M}_ g)$$ in the case $$r\leq 4$$. Also when $$r\leq 4$$ the author proves that the generic point in every component of $${\mathcal M}^ r_ g$$ has a unique halfcanonical $$g^ r_{g-1}.$$
The above results are obtained mainly by using deformation techniques developed by E. Arbarello and M. Cornalba [Comment. Math. Helv. 56, 1-38 (1981; Zbl 0505.14002) and Math. Ann. 256, 341-362 (1981; Zbl 0454.14023)].
Reviewer: A.Del Centina

##### MSC:
 14H10 Families, moduli of curves (algebraic) 14C20 Divisors, linear systems, invertible sheaves 14D15 Formal methods and deformations in algebraic geometry
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