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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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