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Genera of Brill-Noether curves and staircase paths in Young tableaux


Authors: Melody Chan, Alberto López Martín, Nathan Pflueger and Montserrat Teixidor i Bigas
Journal: Trans. Amer. Math. Soc. 370 (2018), 3405-3439
MSC (2010): Primary 05A15, 14H51
DOI: https://doi.org/10.1090/tran/7044
Published electronically: December 27, 2017
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Abstract: In this paper, we compute the genus of the variety of linear series of rank $ r$ and degree $ d$ on a general curve of genus $ g$, with ramification at least $ \alpha $ and $ \beta $ at two given points, when that variety is 1-dimensional. Our proof uses degenerations and limit linear series along with an analysis of random staircase paths in Young tableaux, and produces an explicit scheme-theoretic description of the limit linear series of fixed rank and degree on a generic chain of elliptic curves when that scheme is itself a curve.


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  • [1] A. C. Aitken, The monomial expansion of determinantal symmetric functions, Proc. Roy. Soc. Edinburgh. Sect. A. 61 (1943), 300-310. MR 0008065
  • [2] E. Arbarello, M. Cornalba, P. A. Griffiths, and J. Harris, Geometry of algebraic curves. Vol. I, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 267, Springer-Verlag, New York, 1985. MR 770932
  • [3] Abel Castorena, Alberto López Martín, and Montserrat Teixidor i Bigas, Invariants of the Brill-Noether curve, Adv. Geom. 17 (2017), no. 1, 39-52. MR 3652231
  • [4] Abel Castorena and Montserrat Teixidor i Bigas, Divisorial components of the Petri locus for pencils, J. Pure Appl. Algebra 212 (2008), no. 6, 1500-1508. MR 2391662, https://doi.org/10.1016/j.jpaa.2007.10.021
  • [5] David Eisenbud and Joe Harris, Limit linear series: basic theory, Invent. Math. 85 (1986), no. 2, 337-371. MR 846932, https://doi.org/10.1007/BF01389094
  • [6] David Eisenbud and Joe Harris, The Kodaira dimension of the moduli space of curves of genus $ \geq23$, Invent. Math. 90 (1987), no. 2, 359-387. MR 910206, https://doi.org/10.1007/BF01388710
  • [7] Gavril Farkas, Rational maps between moduli spaces of curves and Gieseker-Petri divisors, J. Algebraic Geom. 19 (2010), no. 2, 243-284. MR 2580676, https://doi.org/10.1090/S1056-3911-09-00510-4
  • [8] Gavril Farkas and Nicola Tarasca, Pointed Castelnuovo numbers, Math. Res. Lett. 23 (2016), no. 2, 389-404. MR 3512891, https://doi.org/10.4310/MRL.2016.v23.n2.a5
  • [9] J. S. Frame, G. de B. Robinson, and R. M. Thrall, The hook graphs of the symmetric groups, Canadian J. Math. 6 (1954), 316-324. MR 0062127
  • [10] William Fulton, Intersection theory, 2nd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 2, Springer-Verlag, Berlin, 1998. MR 1644323
  • [11] W. Fulton and R. Lazarsfeld, On the connectedness of degeneracy loci and special divisors, Acta Math. 146 (1981), no. 3-4, 271-283. MR 611386, https://doi.org/10.1007/BF02392466
  • [12] D. Gieseker, Stable curves and special divisors: Petri's conjecture, Invent. Math. 66 (1982), no. 2, 251-275. MR 656623, https://doi.org/10.1007/BF01389394
  • [13] Phillip Griffiths and Joseph Harris, On the variety of special linear systems on a general algebraic curve, Duke Math. J. 47 (1980), no. 1, 233-272. MR 563378
  • [14] George R. Kempf, Curves of $ g^1_d$'s, Compositio Math. 55 (1985), no. 2, 157-162. MR 795712
  • [15] John Murray and Brian Osserman, Linked determinantal loci and limit linear series, Proc. Amer. Math. Soc. 144 (2016), no. 6, 2399-2410. MR 3477056, https://doi.org/10.1090/proc/12965
  • [16] Angela Ortega, The Brill-Noether curve and Prym-Tyurin varieties, Math. Ann. 356 (2013), no. 3, 809-817. MR 3063897, https://doi.org/10.1007/s00208-012-0870-5
  • [17] Brian Osserman, A limit linear series moduli scheme, Ann. Inst. Fourier (Grenoble) 56 (2006), no. 4, 1165-1205 (English, with English and French summaries). MR 2266887
  • [18] Brian Osserman, A simple characteristic-free proof of the Brill-Noether theorem, Bull. Braz. Math. Soc. (N.S.) 45 (2014), no. 4, 807-818. MR 3296194, https://doi.org/10.1007/s00574-014-0076-4
  • [19] B. Osserman,
    Limit linear series moduli stacks in higher rank,
    arxiv 1405.2937, 2014.
  • [20] B. Osserman,
    Limit linear series,
    draft monograph, 2015.
  • [21] N.  Pflueger,
    On linear series with negative Brill-Noether number,
    arXiv:1311.5845
  • [22] Gian Pietro Pirola, Chern character of degeneracy loci and curves of special divisors, Ann. Mat. Pura Appl. (4) 142 (1985), 77-90 (1986) (English, with Italian summary). MR 839032, https://doi.org/10.1007/BF01766588
  • [23] Lukas Riegler and Christoph Neumann, Playing jeu de taquin on d-complete posets, Sém. Lothar. Combin. 74 (2015), Art. B74D, 17. MR 3543440
  • [24] Richard P. Stanley, Enumerative combinatorics. Vol. 2, Cambridge Studies in Advanced Mathematics, vol. 62, Cambridge University Press, Cambridge, 1999. With a foreword by Gian-Carlo Rota and appendix 1 by Sergey Fomin. MR 1676282
  • [25] Nicola Tarasca, Brill-Noether loci in codimension two, Compos. Math. 149 (2013), no. 9, 1535-1568. MR 3109733, https://doi.org/10.1112/S0010437X13007215
  • [26] Montserrat Teixidor i Bigas, Brill-Noether theory for stable vector bundles, Duke Math. J. 62 (1991), no. 2, 385-400. MR 1104529, https://doi.org/10.1215/S0012-7094-91-06215-0
  • [27] Montserrat Teixidor i Bigas, Rank two vector bundles with canonical determinant, Math. Nachr. 265 (2004), 100-106. MR 2033069, https://doi.org/10.1002/mana.200310138
  • [28] Montserrat Teixidor i Bigas, Existence of vector bundles of rank two with sections, Adv. Geom. 5 (2005), no. 1, 37-47. MR 2110459, https://doi.org/10.1515/advg.2005.5.1.37
  • [29] Montserrat Teixidor i Bigas, Existence of coherent systems. II, Internat. J. Math. 19 (2008), no. 10, 1269-1283. MR 2466566, https://doi.org/10.1142/S0129167X08005126
  • [30] Montserrat Teixidor i Bigas, Existence of vector bundles of rank two with fixed determinant and sections, Proc. Japan Acad. Ser. A Math. Sci. 86 (2010), no. 7, 113-118. MR 2663652, https://doi.org/10.3792/pjaa.86.113
  • [31] Gerald E. Welters, A theorem of Gieseker-Petri type for Prym varieties, Ann. Sci. École Norm. Sup. (4) 18 (1985), no. 4, 671-683. MR 839690

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Additional Information

Melody Chan
Affiliation: Department of Mathematics, Brown University, Box 1917, Providence, Rhode Island 02912
Email: mtchan@math.brown.edu

Alberto López Martín
Affiliation: IMPA, Estrada Dona Castorina, 110, Rio de Janeiro, RJ 22460-902, Brazil
Email: alopez@impa.br

Nathan Pflueger
Affiliation: Department of Mathematics, Brown University, Box 1917, Providence, Rhode Island 02912
Address at time of publication: Department of Mathematics and Statistics, Amherst College, Amherst, Massachusetts 01002
Email: pflueger@math.brown.edu

Montserrat Teixidor i Bigas
Affiliation: Department of Mathematics, Tufts University, Medford, Massachusetts 02155
Email: montserrat.teixidoribigas@tufts.edu

DOI: https://doi.org/10.1090/tran/7044
Received by editor(s): July 22, 2015
Received by editor(s) in revised form: May 31, 2016, and August 8, 2016
Published electronically: December 27, 2017
Additional Notes: The first author was supported by NSF DMS Award 1204278
The second author was supported by CAPES-Brazil
Article copyright: © Copyright 2017 American Mathematical Society

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