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

Chan, Melody; López Martín, Alberto; Pflueger, Nathan; Teixidor i Bigas, Montserrat

doi:10.1090/tran/7044

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
MathSciNet review: 3766853
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Abstract: In this paper, we compute the genus of the variety of linear series of rank and degree on a general curve of genus , 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.

[1] A. C. Aitken, The monomial expansion of determinantal symmetric functions, Proc. Roy. Soc. Edinburgh Sect. A 61 (1943), 300–310. MR 8065
[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, https://doi.org/10.1515/advgeom-2016-0027
[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 ≥23, 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, Canad. J. Math. 6 (1954), 316–324. MR 62127, https://doi.org/10.4153/cjm-1954-030-1
[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 𝑔¹_{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/S0002-9939-2015-12965-6
[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 ([2015-2018]), 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