# ALGEBRAIC COMBINATORICS

Brill–Noether theory of curves on ${ℙ}^{1}×{ℙ}^{1}$: tropical and classical approaches
Algebraic Combinatorics, Volume 2 (2019) no. 3, p. 323-341

The gonality sequence ${\left({d}_{r}\right)}_{r\ge 1}$ 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 ${ℙ}^{1}×{ℙ}^{1}$: 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 [1]. The classical one extends the work [12] of Hartshorne on plane curves to curves on ${ℙ}^{1}×{ℙ}^{1}$.

Revised : 2018-10-01
Accepted : 2018-11-03
Published online : 2019-06-06
DOI : https://doi.org/10.5802/alco.47
Keywords: gonality sequence, curves, bipartite graphs
@article{ALCO_2019__2_3_323_0,
author = {Cools, Filip and D'Adderio, Michele and Jensen, David and Panizzut, Marta},
title = {Brill--Noether theory of curves on $\protect \mathbb{P}^1 \times \protect \mathbb{P}^1$: tropical and classical approaches},
journal = {Algebraic Combinatorics},
publisher = {MathOA foundation},
volume = {2},
number = {3},
year = {2019},
pages = {323-341},
doi = {10.5802/alco.47},
language = {en},
url = {https://alco.centre-mersenne.org/item/ALCO_2019__2_3_323_0}
}

Brill–Noether theory of curves on $\protect \mathbb{P}^1 \times \protect \mathbb{P}^1$: tropical and classical approaches. Algebraic Combinatorics, Volume 2 (2019) no. 3, pp. 323-341. doi : 10.5802/alco.47. https://alco.centre-mersenne.org/item/ALCO_2019__2_3_323_0/

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