A Generalized RSK for Enumerating Linear Series on n-pointed Curves
Algebraic Combinatorics, Volume 6 (2023) no. 1, pp. 1-16.

We give a combinatorial proof of a recent geometric result of Farkas and Lian on linear series on curves with prescribed incidence conditions. The result states that the expected number of degree-d morphisms from a general genus g, n-marked curve C to r , sending the marked points on C to specified general points in r , is equal to (r+1) g for sufficiently large d. This computation may be rephrased as an intersection problem on Grassmannians, which has a natural combinatorial interpretation in terms of Young tableaux by the classical Littlewood-Richardson rule. We give a bijection, generalizing the well-known RSK correspondence, between the tableaux in question and the (r+1)-ary sequences of length g, and we explore our bijection’s combinatorial properties.

We also apply similar methods to give a combinatorial interpretation and proof of the fact that, in the modified setting in which r=1 and several marked points map to the same point in 1 , the number of morphisms is still 2 g for sufficiently large d.

Published online:
DOI: 10.5802/alco.250
Classification: 05E14, 05A05, 14N10
Keywords: Young tableaux, RSK algorithm, Schubert calculus, Brill-Noether theory
Gillespie, Maria 1; Reimer-Berg, Andrew 1

1 Colorado State University Department of Mathematics 1874 Campus Delivery. Fort Collins CO 80523 (USA)
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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Gillespie, Maria; Reimer-Berg, Andrew. A Generalized RSK for Enumerating Linear Series on $n$-pointed Curves. Algebraic Combinatorics, Volume 6 (2023) no. 1, pp. 1-16. doi : 10.5802/alco.250. https://alco.centre-mersenne.org/articles/10.5802/alco.250/

[1] Castelnuovo, Guido Numero delle involuzioni razionali giacenti sopra una curva di dato genere, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. (1889), p. 130–133 | Zbl

[2] Cela, Alessio; Lian, Carl Generalized Tevelev degrees of 1 (2021) | arXiv

[3] Cela, Alessio; Pandharipande, Rahul; Schmitt, Johannes Tevelev degrees and Hurwitz moduli spaces, Math. Proc. Cambridge Philos. Soc. (2021), pp. 1-32

[4] Chan, Melody; López Martín, Alberto; Pflueger, Nathan; Teixidor i Bigas, Montserrat Genera of Brill–Noether curves and staircase paths in Young tableaux, Trans. Amer. Math. Soc., Volume 370 (2018), pp. 3405-3439 | DOI | MR | Zbl

[5] Chan, Melody; Pflueger, Nathan Combinatorial relations on skew Schur and skew stable Grothendieck polynomials, Algebr. Comb., Volume 4 (2021) no. 1, pp. 175-188 | DOI | Numdam | MR | Zbl

[6] Chan, Melody; Pflueger, Nathan Euler characteristics of Brill–Noether varieties, Trans. Amer. Math. Soc., Volume 374 (2021), pp. 1513-1533 | DOI | MR | Zbl

[7] Eisenbud, David; Harris, Joe Divisors on general curves and cuspidal rational curves, Invent. Math., Volume 74 (1983), pp. 371-418 | DOI | MR | Zbl

[8] Eisenbud, David; Harris, Joe Limit linear series: basic theory, Invent. Math., Volume 85 (1986) no. 2, pp. 337-371 | DOI | MR | Zbl

[9] Farkas, Gavril; Lian, Carl Linear series on general curves with prescribed incidence conditions (2021) | arXiv

[10] Fulton, William Young tableaux, London Math Soc. Student Texts 35, Cambridge University Press, 1997

[11] Gillespie, Maria Variations on a Theme of Schubert Calculus, Recent Trends in Algebraic Combinatorics (Association for Women in Mathematics Series), Springer, 2019, pp. 115-158 (Edited by Hélène Barcelo, Gizem Karaali, and Rosa Orellana) | DOI | MR | Zbl

[12] 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

[13] Larson, Eric; Larson, Hannah; Vogt, Isabel Global Brill–Noether Theory over the Hurwitz Space (2020) | arXiv

[14] Lian, Carl Private communication, 2021

[15] Osserman, Brian The number of linear series on curves with given ramification, Int. Math. Res. Not. (2003) no. 47, pp. 2513-2527 | DOI | MR | Zbl

[16] Reiner, Victor; Shimozono, Mark Percentage-avoiding, northwest shapes and peelable tableaux, J. Combin. Theory Ser. A, Volume 82 (1998) no. 1, pp. 1-73 | DOI | MR | Zbl

[17] Stanley, Richard; Fomin, Sergey Enumerative Combinatorics: Volume 2., Cambridge University Press, 1999 | DOI

[18] Stanley, Richard P. GL (n,C) for combinatorialists, Surveys in combinatorics (Southampton, 1983) (London Math. Soc. Lecture Note Ser.), Volume 82, Cambridge Univ. Press, Cambridge (1983), pp. 187-199 | DOI | MR | Zbl

[19] Tevelev, Jenia Scattering amplitudes of stable curves (2020) | arXiv

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