On the Semigroup of Graph Gonality Sequences

The \(r\)th gonality of a graph is the smallest degree of a divisor on the graph with rank \(r\). The gonality sequence of a graph is a tropical analogue of the gonality sequence of an algebraic curve. We show that the set of truncated gonality sequences of graphs forms a semigroup under addition. U...

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Published in:arXiv.org 2023-06
Main Authors: Fessler, Austin, Jensen, David, Kelsey, Elizabeth, Owen, Noah
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Algebra
Graphs
Semigroups
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Jensen, David
Kelsey, Elizabeth
Owen, Noah
description The \(r\)th gonality of a graph is the smallest degree of a divisor on the graph with rank \(r\). The gonality sequence of a graph is a tropical analogue of the gonality sequence of an algebraic curve. We show that the set of truncated gonality sequences of graphs forms a semigroup under addition. Using this, we study which triples \((x,y,z)\) can be the first 3 terms of a graph gonality sequence. We show that nearly every such triple with \(z \geq \frac{3}{2}x+2\) is the first three terms of a graph gonality sequence, and also exhibit triples where the ratio \(\frac{z}{x}\) is an arbitrary rational number between 1 and 3. In the final section, we study algebraic curves whose \(r\)th and \((r+1)\)st gonality differ by 1, and posit several questions about graphs with this property.
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Graphs
Semigroups
title On the Semigroup of Graph Gonality Sequences
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