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

TL;DR

This paper investigates the moduli of irreducible plane curves with nodes and cusps, establishing bounds and constructing examples with expected moduli and non-positive Brill-Noether number.

Contribution

It provides bounds on the number of moduli for families of such curves and constructs examples achieving the expected number of moduli with specific singularity conditions.

Findings

Bounds on the number of moduli for families of plane curves with nodes and cusps.

Construction of examples with expected moduli and non-positive Brill-Noether number.

Abstract

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.