Number of moduli of irreducible families of plane curves with nodes and cusps

Details Number of moduli of irreducible families of plane curves with nodes and cusps Number of moduli of irreducible families of plane curves with nodes and cusps

2007-04-04

arxiv.org/abs/0704.0618v1

Collect. Math. 57,3 (2006), 319 - 346

Mathematics

Algebraic Geometry

22 pages

Scientific paper

Consider the family S of irreducible plane curves of degree n with d nodes and k cusps as singularities. Let W be an irreducible component of S. We consider the natural rational map from W to the moduli space of curves of genus g=(n-1)(n-2)/2-d-k. We define the "number of moduli of W" as the dimension of the image of W with respect to this map. If W has the expected dimension equal to 3n+g-1-k, then the number of moduli of W is at most equal to the min(3g-3, 3g-3+\rho-k), dove \rho is the Brill-Neother number of the linear series of degree n and dimension 2 on a smooth curve of genus g. We say that W has the expected number of moduli if the equality holds. In this paper we construct examples of families of irreducible plane curves with nodes and cusps as singularities having expected number of moduli and with non-positive Brill-Noether number.

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