On Hurwitz--Severi numbers

Abstract

For a point $p\in C{P}^2$ and a triple $(g,d,\ell )$ of non-negative integers we define a {\em Hurwitz--Severi number} ${\mathfrak{H}}_{g,d,\ell }$ as the number of generic irreducible plane curves of genus $g$ and degree $d+\ell$ having an $\ell$-fold node at $p$ and at most ordinary nodes as singularities at the other points, such that the projection of the curve from $p$ has a prescribed set of local and remote tangents and lines passing through nodes. In the cases $d+\ell \ge g+2$ and $d+2\ell \ge g+2>d+\ell$ we express the Hurwitz--Severi numbers via appropriate ordinary Hurwitz numbers. The remaining case $d+2\ell <g+2$ is still widely open.

Version 2: title changed and some explanations added

References in corpus (1)

• An algebro-geometric proof of Witten's conjecture