Hostname: page-component-6dbcb7884d-g628x Total loading time: 0 Render date: 2025-02-13T23:59:15.688Z Has data issue: false hasContentIssue false
  • English
  • Français

Brill-Noether theory for curves of a fixed gonality

Part of: Curves

Published online by Cambridge University Press:  08 January 2021

David Jensen  and
Dhruv Ranganathan
David Jensen
Affiliation:
Department of Mathematics, University of Kentucky, 719 Patterson Office Tower, Lexington, KY40506, USA; E-mail: dave.jensen@uky.edu.
Dhruv Ranganathan
Affiliation:
Department of Mathematics, University of Kentucky, 719 Patterson Office Tower, Lexington, KY40506, USA; E-mail: dave.jensen@uky.edu. Department of Pure Mathematics and Mathematical Statistics, Wilberforce Road, University of Cambridge, CambridgeCB2 1TP, UK; E-mail: dr508@cam.ac.uk.

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We prove a generalisation of the Brill-Noether theorem for the variety of special divisors $W^r_d(C)$ on a general curve C of prescribed gonality. Our main theorem gives a closed formula for the dimension of $W^r_d(C)$ . We build on previous work of Pflueger, who used an analysis of the tropical divisor theory of special chains of cycles to give upper bounds on the dimensions of Brill-Noether varieties on such curves. We prove his conjecture, that this upper bound is achieved for a general curve. Our methods introduce logarithmic stable maps as a systematic tool in Brill-Noether theory. A precise relation between the divisor theory on chains of cycles and the corresponding tropical maps theory is exploited to prove new regeneration theorems for linear series with negative Brill-Noether number. The strategy involves blending an analysis of obstruction theories for logarithmic stable maps with the geometry of Berkovich curves. To show the utility of these methods, we provide a short new derivation of lifting for special divisors on a chain of cycles with generic edge lengths, proved using different techniques by Cartwright, Jensen, and Payne. A crucial technical result is a new realisability theorem for tropical stable maps in obstructed geometries, generalising a well-known theorem of Speyer on genus $1$ curves to arbitrary genus.

MSC classification

Primary: 14H51: Special divisors (gonality, Brill-Noether theory)
Secondary: 14T05: Tropical geometry
Type
Algebraic and Complex Geometry
Information
Forum of Mathematics, Pi , Volume 9 , 2021 , e1
Check for updates
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s), 2020. Published by Cambridge University Press

References

Abramovich, D., Caporaso, L. and Payne, S., ‘The tropicalization of the moduli space of curves’, Ann. Sci. Éc. Norm. Supér. 48 (2015), 765–809.CrossRefGoogle Scholar
Abramovich, D. and Chen, Q., ‘Stable logarithmic maps to Deligne-Faltings pairs II’, Asian J. Math. 18 (2014), 465–488.CrossRefGoogle Scholar
Abramovich, D., Chen, Q., Gillam, D., Huang, Y., Olsson, M., Satriano, M. and Sun, S., ‘Logarithmic geometry and moduli’, in Handbook of moduli. Vol. I, 1–61, Adv. Lect. Math. (ALM), 24, Int. Press, Somerville, MA, 2013.Google Scholar
Abramovich, D., Chen, Q., Marcus, S., Ulirsch, M. and Wise, J., ‘Skeletons and fans of logarithmic structures’, in Nonarchimedean and Tropical Geometry . Proceedings of the Simons Symposium (Springer, 2016), 287–336.CrossRefGoogle Scholar
Abramovich, D. and Wise, J., ‘Birational invariance in logarithmic Gromov-Witten theory’, Comput. Math. 154 (2018), 595–620.Google Scholar
Amini, O., Baker, M., Brugallé, E. and Rabinoff, J., ‘Lifting harmonic morphisms. I: metrized complexes and Berkovich skeleta’, Res. Math. Sci. 2 (2015), 67.CrossRefGoogle Scholar
Amini, O., Baker, M., Brugallé, E. and Rabinoff, J., ‘Lifting harmonic morphisms. II: tropical curves and metrized complexes’, Algebra Number Theory 9 (2015), 267–315.CrossRefGoogle Scholar
Baker, M., ‘Specialization of linear systems from curves to graphs’, Algebra Number Theory 2 (2008), 613–653.CrossRefGoogle Scholar
Baker, M. and Jensen, D., ‘Degeneration of linear series from the tropical point of view and applications’, in Nonarchimedean and Tropical Geometry . Proceedings of the Simons Symposium (Springer, 2016), 365–433.CrossRefGoogle Scholar
Baker, M. and Norine, S., ‘Riemann-Roch and Abel-Jacobi theory on a finite graph’, Adv. Math. 215 (2007), 766–788.CrossRefGoogle Scholar
Baker, M., Payne, S. and Rabinoff, J., ‘On the structure of non-Archimedean analytic curves’, in Tropical and Non-Archimedean Geometry, vol. 605 of Contemporary Mathematics (American Mathematical Society, Providence, RI, 2013), 93–121.CrossRefGoogle Scholar
Baker, M., Payne, S. and Rabinoff, J., ‘Nonarchimedean geometry, tropicalization, and metrics on curves’, Algebr. Geom. 3 (2016), 63–105.CrossRefGoogle Scholar
Ballico, E. and Keem, C., ‘On linear series on general $k$ -gonal projective curves’, Proc. Amer. Math. Soc. 124 (1996), 7–9.CrossRefGoogle Scholar
Cartwright, D., ‘Lifting matroid divisors on tropical curves’, Res. Math. Sci. 2 (2015), 1–24.CrossRefGoogle Scholar
Cartwright, D., Jensen, D. and Payne, S., ‘Lifting divisors on a generic chain of loops’, Can. Math. Bull. 58 (2015), 250–262.CrossRefGoogle Scholar
Cavalieri, R., Markwig, H. and Ranganathan, D., ‘Tropicalizing the space of admissible covers’, Math. Ann. 364 (2016), 1275–1313.CrossRefGoogle Scholar
Chen, Q., ‘Stable logarithmic maps to Deligne-Faltings pairs I’, Ann. Math. 180 (2014), 341–392.CrossRefGoogle Scholar
Cheung, M.-W., Fantini, L., Park, J. and Ulirsch, M., ‘Faithful realizability of tropical curves’, Int. Math. Res. Not. 5 (2015), 4706–4727.Google Scholar
Clifford, W. K., ‘On the classification of loci’, Philos. Trans. R. S. Lond. 169 (1878), 663–681.Google Scholar
Cook-Powell, K. and Jensen, D., ‘Components of Brill-Noether loci for curves of fixed gonality’, (2019), arXiv:1907.08366v1.CrossRefGoogle Scholar
Cools, F., Draisma, J., Payne, S. and Robeva, E., ‘A tropical proof of the Brill-Noether theorem’, Adv. Math. 230 (2012), 759–776.CrossRefGoogle Scholar
Coppens, M. and Martens, G., ‘Linear series on a general k-gonal curve’, in Abhandlungen aus dem mathematischen Seminar der Universität Hamburg [Treatises from the mathematical seminar of the University of Hamburg], vol. 69 (Springer, 1999), 347–371.Google Scholar
Coppens, M. and Martens, G., ‘On the varieties of special divisors’, Indag. Math. 13 (2002), 29–45.CrossRefGoogle Scholar
Cox, D. A., Little, J. B. and Schenck, H. K., ‘Toric varieties’, Grad. Stud. Math. 124 (2011), 575.Google Scholar
Cox, D. A., ‘The functor of a smooth toric variety’, Tohoku Math. J. (2) 47 (1995), no. 2, 251–262.CrossRefGoogle Scholar
Edidin, D., ‘Brill-Noether theory in codimension-two’, J. Algebr. Geom. 2 (1993), 25–67.Google Scholar
Eisenbud, D. and Harris, J., ‘Limit linear series: basic theory’, Invent. Math. 85 (1986), 337–371.CrossRefGoogle Scholar
Farkas, G., ‘Regular components of moduli spaces of stable maps’, Proc. Amer. Math. Soc. 131 (2003), 2027–2036.CrossRefGoogle Scholar
Griffiths, P. and Harris, J., ‘On the variety of special linear systems on a general algebraic curve’, Duke Math. J . 47 (1980), 233–272.CrossRefGoogle Scholar
Gross, M. and Siebert, B., ‘Logarithmic Gromov-Witten invariants’, J. Amer. Math. Soc. 26 (2013), 451–510.CrossRefGoogle Scholar
Gubler, W., ‘Tropical varieties for non-Archimedean analytic spaces’, Invent. Math. 169 (2007), 321–376.CrossRefGoogle Scholar
Jensen, D. and Payne, S., ‘Tropical independence I: shapes of divisors and a proof of the Gieseker-Petri theorem’, Algebra Number Theory 8 (2014), 2043–2066.CrossRefGoogle Scholar
Jensen, D. and Payne, S., ‘Tropical independence, II: the maximal rank conjecture for quadrics’, Algebra Number Theory 10 (2016), 1601–1640.CrossRefGoogle Scholar
Jensen, D. and Payne, S., ‘Combinatorial and inductive methods for the tropical maximal rank conjecture’, J. Combin. Theory Ser. A 152 (2017), 138–158.CrossRefGoogle Scholar
Katz, E., ‘Lifting tropical curves in space and linear systems on graphs’, Adv. Math. 230 (2012), 853–875.CrossRefGoogle Scholar
Kempf, G., ‘Schubert methods with an application to algebraic curves’, Zuivere Wiskunde (1971), 1–18.Google Scholar
Kleiman, S. L. and Laksov, D., ‘On the existence of special divisors’, Amer. J. Math. 94 (1972), 431–436.CrossRefGoogle Scholar
Larson, H., ‘A refined Brill-Noether theory over Hurwitz spaces’, (2019), arXiv:1907.08597v2.CrossRefGoogle Scholar
Lazarsfeld, R., ‘Brill-Noether-Petri without degenerations’, J. Differ. Geom. 23 (1986), 299–307.CrossRefGoogle Scholar
Luo, Y. and Manjunath, M., ‘Smoothing of limit linear series of rank one on saturated metrized complexes of algebraic curves’, Can. J. Math . 70 (2018), 628–682.CrossRefGoogle Scholar
Mikhalkin, G., ‘Enumerative tropical geometry in $ {\mathbb{R}}^{2\prime }$ , J. Amer. Math. Soc . 18 (2005), 313–377.CrossRefGoogle Scholar
Nishinou, T., ‘Describing tropical curves via algebraic geometry’. 2015, arXiv:1503.06435.Google Scholar
Nishinou, T. and Siebert, B., ‘Toric degenerations of toric varieties and tropical curves’, Duke Math. J. 135 (2006), 1–51.CrossRefGoogle Scholar
Osserman, B. and Payne, S., ‘Lifting tropical intersections’, Doc. Math. 18 (2013), 121–175.Google Scholar
Pflueger, N., ‘On linear series with negative Brill-Noether number’. 2013, arXiv:1311.5845v1.Google Scholar
Pflueger, N., ‘Brill-Noether varieties of k-gonal curves’, Adv. Math. 312 (2017), 46–63.CrossRefGoogle Scholar
Pflueger, N., ‘Special divisors on marked chains of cycles’, J. Combin. Theory Ser. A 150 (2017), 182–207.CrossRefGoogle Scholar
Ranganathan, D., ‘Superabundant curves and the Artin fan’, Int. Math. Res. Not. 4 (2016), 1103–1115.Google Scholar
Ranganathan, D., ‘Skeletons of stable maps II: superabundant geometries’, Res. Math. Sci. 4 (2017), 11.CrossRefGoogle Scholar
Ranganathan, D., Santos-Parker, K. and Wise, J., ‘Moduli of stable maps in genus one and logarithmic geometry, II’, Algebra Number Theory 13 (2019), 1765–1805.CrossRefGoogle Scholar
Speyer, D. E., ‘Parameterizing tropical curves. I: curves of genus zero and one’, Algebra Number Theory 8 (2014), 963–998.CrossRefGoogle Scholar
Thuillier, A., ‘Géométrie toroïdale et géométrie analytique non Archimédienne. Application au type d’homotopie de certains schémas formels’ [Toroidal geometry and non-Archimedean analytical geometry. Application to the type of homotopy of certain formal patterns’], Manuscripta Math. 123 (2007), 381–451.CrossRefGoogle Scholar
Ulirsch, M., ‘Tropical compactification in log-regular varieties’, Math. Z. 280 (2015), 195–210.CrossRefGoogle Scholar
Ulirsch, M., ‘Non-Archimedean geometry of Artin fans’, Adv. Math. 345 (2019), 346–381.CrossRefGoogle Scholar
Vakil, R., ‘Murphy’s law in algebraic geometry: badly-behaved deformation spaces’, Invent. Math. 164 (2006), 569–590.CrossRefGoogle Scholar
Wise, J., ‘Local model of virtual fundamental cycle’. URL: http://mathoverflow.net/q/122086.Google Scholar