Gonality of curves whose normalizations are one or two copies of $\mathbb P^1$

TL;DR

This paper investigates the gonality of certain singular curves over complex numbers, specifically those with normalizations as one or two copies of the projective line, establishing bounds and generic equality conditions.

Contribution

It proves that the classical gonality bound applies to these curves with normalization as one or two P^1's, and shows that the bound is generically sharp.

Findings

The bound $ ext{gon}(C) \,\leq\, \lfloor\frac{g(C)+3}{2}\rfloor$ holds for these curves.

Equality in the gonality bound is achieved generically.

The results apply to nodal and binary curves with normalization as P^1.

Abstract

We study the gonality of curves $C$ over $\mathbb C$ whose normalization is composed of one or two copies of $\mathbb P^1$. In the first case, $C$ is a nodal curve with $g(C)$ nodes, and in the second case $C$ is a so-called binary curve. In any case we show that the usual bound $\mathrm{gon}(C)\leq\lfloor\frac{g(C)+3}{2}\rfloor$ holds if $g(C)\geq 2$, with equality holding generically.