ALGEBRAIC COMBINATORICS

no. 3
Brill–Noether theory of curves on ℙ 1 ×ℙ 1 : tropical and classical approaches
Cools, Filip1; D’Adderio, Michele2; Jensen, David3; Panizzut, Marta4
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.

The gonality sequence (d r ) r≥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 .

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DOI: 10.5802/alco.47
Keywords: gonality sequence, curves, bipartite graphs
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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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. https://alco.centre-mersenne.org/articles/10.5802/alco.47/

[1] Baker, Matthew Specialization of linear systems from curves to graphs, Algebra Number Theory, Volume 2 (2008) no. 6, pp. 613-653 (With an appendix by Brian Conrad) | DOI | MR | Zbl

[2] Baker, Matthew; Jensen, David Degeneration of linear series from the tropical point of view and applications, Nonarchimedean and tropical geometry (Simons Symposia), Springer, 2016, pp. 365-433 | DOI | Zbl

[3] Baker, Matthew; Norine, Serguei Riemann–Roch and Abel–Jacobi theory on a finite graph, Adv. Math., Volume 215 (2007) no. 2, pp. 766-788 | DOI | MR | Zbl

[4] Castryck, Wouter; Cools, Filip Linear pencils encoded in the Newton polygon, Int. Math. Res. Not. (2017) no. 10, pp. 2998-3049 | MR | Zbl

[5] Ciliberto, Ciro Alcune applicazioni di un classico procedimento di Castelnuovo, Semin. Geom., Univ. Studi Bologna, Volume 1982-1983 (1984) no. 10, pp. 17-43 | Zbl

[6] Cools, Filip; Draisma, Jan; Payne, Sam; Robeva, Elina A tropical proof of the Brill–Noether theorem, Adv. Math., Volume 230 (2012) no. 2, pp. 759-776 | DOI | MR | Zbl

[7] Cools, Filip; Panizzut, Marta The gonality sequence of complete graphs, Electron. J. Comb., Volume 24 (2017) no. 4, Paper no. P4.1, 20 pages | MR | Zbl

[8] Cox, David A.; Little, John B.; Schenck, Henry K. Toric varieties, Graduate Studies in Mathematics, 124, American Mathematical Society, 2011, xxiv+841 pages | MR | Zbl

[9] D’Adderio, Michele; Le Borgne, Yvan The sandpile model on K m,n and the rank of its configurations, Sémin. Lothar. Comb., Volume 77 (2016), Paper no. B77h, 48 pages | MR | Zbl

[10] Gathmann, Andreas; Kerber, Michael A Riemann–Roch theorem in tropical geometry, Math. Z., Volume 259 (2008) no. 1, pp. 217-230 | DOI | MR | Zbl

[11] Griffiths, Phillip; Harris, Joseph On the variety of special linear systems on a general algebraic curve, Duke Math. J., Volume 47 (1980) no. 1, pp. 233-272 | MR | Zbl

[12] Hartshorne, Robin Generalized divisors on Gorenstein curves and a theorem of Noether, J. Math. Kyoto Univ., Volume 26 (1986) no. 3, pp. 375-386 | DOI | MR | Zbl

[13] Hladký, Jan; Král’, Daniel; Norine, Serguei Rank of divisors on tropical curves, J. Comb. Theory, Ser. A, Volume 120 (2013) no. 7, pp. 1521-1538 | DOI | MR | Zbl

[14] Jensen, David; Ranganathan, Druv Brill–Noether theory for curves of a fixed gonality (2017) (https://arxiv.org/abs/1701.06579)

[15] Lange, Herbert; Martens, Gerriet On the gonality sequence of an algebraic curve, Manuscr. Math., Volume 137 (2012) no. 3-4, pp. 457-473 | DOI | MR | Zbl

[16] Lange, Herbert; Newstead, Peter E. Clifford indices for vector bundles on curves, Affine flag manifolds and principal bundles (Trends in Mathematics), Birkhäuser, 2010, pp. 165-202 | DOI | Zbl

[17] Luo, Ye Rank-determining sets of metric graphs, J. Comb. Theory, Ser. A, Volume 118 (2011) no. 6, pp. 1775-1793 | DOI | MR | Zbl

[18] Mikhalkin, Grigory; Zharkov, Ilia Tropical curves, their Jacobians and theta functions, Curves and abelian varieties (Contemporary Mathematics), Volume 465, American Mathematical Society, 2008, pp. 203-230 | DOI | MR | Zbl

[19] Noether, Max Zur Grundlegung der Theorie der algebraischen Raumcurven, Verl. König. Akad. Wiss., 1883

[20] Pflueger, Nathan Brill–Noether varieties of k-gonal curves, Adv. Math., Volume 312 (2017), pp. 46-63 | DOI | MR | Zbl

[21] Zariski, Oscar Introduction to the problem of minimal models in the theory of algebraic surfaces, Publications of the Mathematical Society of Japan, 4, Mathematical Society of Japan, 1958, vii+89 pages | MR | Zbl

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