On the gonality sequence of smooth curves: normalizations of singular curves in a quadric surface

TL;DR

This paper constructs examples of smooth curves with high genus where the expected slope inequality between gonality sequences fails, challenging previous assumptions about their behavior.

Contribution

It provides the first known examples of high-genus smooth curves where the gonality slope inequality does not hold.

Findings

Existence of smooth curves with genus ≥ 40805 where the slope inequality fails

Explicit constructions of such curves in a quadric surface

Counterexamples to previously assumed gonality sequence inequalities

Abstract

Let $C$ be a smooth curve of genus $g$. For each positive integer $r$ the $r$-gonality $d_r(C)$ of $C$ is the minimal integer $t$ such that there is $L\in {Pic}^t(C)$ with $h^0(C,L) =r+1$. In this paper for all $g\ge 40805$ we construct several examples of smooth curves $C$ of genus $g$ with $d_3(C)/3< d_4(C)/4$, i.e. for which a slope inequality fails.