On Hurwitz--Severi numbers

TL;DR

This paper introduces Hurwitz--Severi numbers, counts certain plane curves with specified singularities and tangency conditions, and relates them to classical Hurwitz numbers in specific cases, leaving some cases open.

Contribution

It defines Hurwitz--Severi numbers for plane curves with prescribed singularities and tangencies, and expresses them in terms of Hurwitz numbers in certain parameter ranges.

Findings

Hurwitz--Severi numbers are expressed via Hurwitz numbers for specific cases.

The case $d+2\ell<g+2$ remains open.

Provides a framework linking curve counts to classical enumerative invariants.

Abstract

For a point $p\in CP^2$ 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.