On the gonality sequence of an algebraic curve

TL;DR

This paper investigates the gonality sequence of algebraic curves, focusing on cases where the typical slope inequality does not hold, by providing examples involving extremal curves, K3-surface curves, and complete intersections.

Contribution

It introduces new examples of algebraic curves where the gonality sequence violates the slope inequality, expanding understanding of gonality behavior.

Findings

Examples of extremal curves in projective space

Curves on general K3-surfaces with unusual gonality sequences

Complete intersection curves in projective 3-space

Abstract

For any smooth irreducible projective curve $X$, the gonality sequence $\{d_r \;| \; r \in \mathbb N \}$ is a strictly increasing sequence of positive integer invariants of $X$. In most known cases $d_{r+1}$ is not much bigger than $d_r$. In our terminology this means the numbers $d_r$ satisfy the slope inequality. It is the aim of this paper to study cases when this is not true. We give examples for this of extremal curves in $\PP^r$, for curves on a general K3-surface in $\PP^r$ and for complete intersections in $\PP^3$.