Brill–Noether theory of curves on $\protect \mathbb{P}^1 \times \protect \mathbb{P}^1$: tropical and classical approaches

Cools, Filip; D’Adderio, Michele; Jensen, David; Panizzut, Marta

doi:10.5802/alco.47

Brill–Noether theory of curves on

ℙ^{1} \times ℙ^{1}

: tropical and classical approaches

Cools, Filip ¹; D’Adderio, Michele ²; Jensen, David ³; Panizzut, Marta ⁴

¹ KU Leuven Department of Mathematics Celestijnenlaan 200B B-3001 Heverle Belgium
² Université Libre de Bruxelles (ULB) Département de Mathématique Boulevard du Triomphe B-1050 Bruxelles Belgium
³ University of Kentucky Department of Mathematics 719 Patterson Office Tower Lexington KY 40506-0027, USA
⁴ TU Berlin Institut für Mathematik Straße des 17. Juni 136 10623 Berlin Germany

Algebraic Combinatorics, Volume 2 (2019) no. 3, pp. 323-341

Abstract

The gonality sequence ${(d_{r})}_{r \geq 1}$ 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 $ℙ^{1} \times ℙ^{1}$ : 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 $ℙ^{1} \times ℙ^{1}$ .

Received: 2017-09-22
Revised: 2018-10-01
Accepted: 2018-11-03
Published online: 2019-06-06

MR Zbl

DOI: 10.5802/alco.47

Keywords: gonality sequence, curves, bipartite graphs

Author's affiliations:

Cools, Filip ¹; D’Adderio, Michele ²; Jensen, David ³; Panizzut, Marta ⁴

¹ KU Leuven Department of Mathematics Celestijnenlaan 200B B-3001 Heverle Belgium
² Université Libre de Bruxelles (ULB) Département de Mathématique Boulevard du Triomphe B-1050 Bruxelles Belgium
³ University of Kentucky Department of Mathematics 719 Patterson Office Tower Lexington KY 40506-0027, USA
⁴ TU Berlin Institut für Mathematik Straße des 17. Juni 136 10623 Berlin Germany

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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     title = {Brill{\textendash}Noether theory of curves on $\protect \mathbb{P}^1 \times \protect \mathbb{P}^1$: tropical and classical approaches},
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Cools, Filip; D’Adderio, Michele; Jensen, David; Panizzut, Marta. 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

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