Brill-Noether theory of curves on $\mathbb{P}^1 \times \mathbb{P}^1$: tropical and classical approach

TL;DR

This paper compares tropical and classical methods for computing the gonality sequence of smooth curves on  imes , leveraging recent tropical Brill--Noether theory and classical algebraic geometry extensions.

Contribution

It introduces two approaches—tropical and classical—for determining the gonality sequence of curves on  imes , integrating modern tropical techniques with classical algebraic geometry.

Findings

Tropical approach uses Brill--Noether theory on tropical curves.

Classical approach extends Hartshorne's work to  imes .

Both methods effectively compute the gonality sequence.

Abstract

The gonality sequence $(d_r)_{r\geq1}$ of a smooth algebraic curve comprises the minimal degrees $d_r$ of linear systems of rank $r$. We explain two approaches to compute the gonality sequence of smooth curves in $\mathbb{P}^1 \times \mathbb{P}^1$: a tropical and a classical approach. The tropical approach uses the recently developed Brill--Noether theory on tropical curves and Baker's specialization of linear systems from curves to metric graphs. The classical one extends the work of Hartshorne on plane curves to curves on $\mathbb{P}^1 \times \mathbb{P}^1$.