On the variety of plane curves of degree $d$ with $\delta$ nodes and $\kappa$ cusps

Abstract:

Let ${{\mathbf {P}}^N}$ be the projective space which parametrizes all plane curves of degree $d$ and $V(d,\delta ,\kappa )$ the subvariety of ${{\mathbf {P}}^N}$ consisting of all reduced and irreducible plane curves of degree $d$ with $\delta$ nodes and $\kappa$ cusps as their only singularities. In this paper we prove that $V(d,\delta ,\kappa )$ is irreducible if $\kappa \leqslant 3$, except possibly when $\kappa = 3$ and $d = 5$ or $6$.