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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Geometry of the Severi variety

Authors: Steven Diaz and Joe Harris
Journal: Trans. Amer. Math. Soc. 309 (1988), 1-34
MSC: Primary 14H10
MathSciNet review: 957060
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Abstract: This paper is concerned with the geometry of the Severi variety $ W$ parametrizing plane curves of given degree and genus, and specifically with the relations among various divisor classes on $ W$. Two types of divisor classes on $ W$ are described: those that come from the intrinsic geometry of the curves parametrized, and those characterized by extrinsic properties such as the presence of cusps, tacnodes, hyperflexes, etc. The goal of the paper is to express the classes of the extrinsically defined divisors in terms of the intrinsic ones; this, along with other calculations such as the determination of the canonical class of $ W$, is carried out by using various enumerative techniques. One corollary is that the variety of nodal curves of given degree and genus in the plane is affine.

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Article copyright: © Copyright 1988 American Mathematical Society