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Plane Curves With Nodes

Published online by Cambridge University Press:  20 November 2018

Robert Treger
Robert Treger*
Affiliation:
Pennsylvania State University, Delaware County Campus, Media, Pennsylvania
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A smooth algebraic curve is birationally equivalent to a nodal plane curve. One of the main problems in the theory of plane curves is to describe the situation of nodes of an irreducible nodal plane curve (see [16, Art. 45], [10], [7, Book IV, Chapter I, §5], [12, p. 584], and [3]).

Let n denote the degree of a nodal curve and d the number of nodes. The case (AZ, d) — (6,9) has been analyzed by Halphen [10]. It follows from Lemma 3.5 and Proposition 3.6 that this is an exceptional case. The case d ≦n(n + 3)/6, d ≦(n — 1)(n — 2)/2, and (n, d) ≠ (6,9) was investigated by Arbarello and Cornalba [3]. We present a simpler proof (Corollary 3.8).

We consider the main case which is particularly important due to its applications to the moduli variety of curves, compare [19, Chapter VIII, Section 4]. Let Vn,d be the variety of irreducible curves of degree n with d nodes and no other singularities such that each curve of Vn,d can be degenerated into n lines in general position (see [17]).

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 41 , Issue 2 , 01 April 1989 , pp. 193 - 212
Copyright
Copyright © Canadian Mathematical Society 1989

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