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, Tome 2 (2019) no. 3, pp. 323-341.

Résumé

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 $ℙ^{1} \times ℙ^{1}$ .

Reçu le : 2017-09-22
Révisé le : 2018-10-01
Accepté le : 2018-11-03
Publié le : 2019-06-06

MR Zbl

DOI : 10.5802/alco.47

Mots clés : gonality sequence, curves, bipartite graphs

Affiliations des auteurs :

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

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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},
     journal = {Algebraic Combinatorics},
     pages = {323--341},
     publisher = {MathOA foundation},
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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, Tome 2 (2019) no. 3, pp. 323-341. doi : 10.5802/alco.47. http://www.numdam.org/articles/10.5802/alco.47/

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