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On the Severi problem. (English) Zbl 0596.14017
Let \({\mathbb{P}}^ N\) be the projective space of plane curves of degree d and let \(V^{d,g}\) be the closure in \({\mathbb{P}}^ N\) of the locus of reduced, irreducible curves of genus g. F. Severi [”Vorlesungen über algebraische Geometrie” (1921), pp. 307-353] gave a proof that \(V^{d,g}\) is an irreducible variety. Apparently, Severi’s proof was incorrect [W. Fulton, Algebraic Geometry - open problems, Proc. Conf., Ravello/Italy 1982, Lecture Notes Math. 997, 146-155 (1983; Zbl 0514.14012)]. Here, a proof is given along the lines suggested by Severi. It is sufficient to show that every component of \(V^{d,g}\) contains a component of \(V^{d,g-1}\). The proof requires much complicated detail, but it is done by essentially classical techniques such as degeneration of varieties. It is a most impressive result.
Reviewer: J.W.P.Hirschfeld

MSC:
14H10 Families, moduli of curves (algebraic)
14N05 Projective techniques in algebraic geometry
14D15 Formal methods and deformations in algebraic geometry
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References:
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