Resumen de Number of moduli of irreducible families of plane curves with nodes and cusps
Let $\Sigma^n_{k,d} \subset \mathbb{P}^{n(n+3)/2}$ be the family of irreducible plane curves of degree $n$ with $d$ nodes and $k$ cusps as singularities. Let $\Sigma \in \Sigma^n_{k,d}$ be an irreducible component. We consider the natural rational map $\Pi_\Sigma : \Sigma - - \rightarrow \mathcal{M}_g$, from to the moduli space of curves of genus $g = {n-1\choose 2}-d-k$. We define \textit{the number of moduli of} $\Sigma$ as the dimension $dim(\Pi_\Sigma(\Sigma))$. If $\Sigma$ has the expected dimension equal to $3n + g - 1 - k$, then $dim(\Pi_\Sigma(\Sigma)) \leq min(dim(\mathcal{M}_g), dim(\mathcal{M}_g) + \rho - k)$, (1) where $\rho$ := $\rho(2, g, n) = 3n-2g-6$ 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 $\Sigma$ has the expected number of moduli if the equality holds in (1). 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 nonpositive Brill-Noether number.
