Higher rank Brill—Noether theory and coherent systems open questions
DOI:
https://doi.org/10.22199/issn.0717-6279-5276Keywords:
vector bundles on curves, Brill-Noether theory, coherent systems, Clifford indicesAbstract
This article presents a list of open questions on higher rank Brill-Noether theory and coherent systems. Background material and appropriate references are included.
References
E. Arbarello, M. Cornalba, P. A. Griffiths, and J. Harris, Geometry of algebraic curves, vol. 1. New York: Springer, 1985. https://doi.org/10.1007/978-1-4757-5323-3
A. Bajravani and G. H. Hitching, “Brill–Noether loci on moduli spaces of symplectic bundles over curves”, Collectanea mathematica, vol. 72, no. 2, pp. 443–469, 2021. https://doi.org/10.1007/s13348-020-00300-7
B. Bakker and G. Farkas, “The Mercat conjecture for stable rank 2 vector bundles on generic curves”, American journal of mathematics, vol. 140, no. 5, pp. 1277–1295, 2018. https://doi.org/10.1353/ajm.2018.0031
E. Ballico, “Brill–noether theory for vector bundles on projective curves”, Mathematical proceedings of the Cambridge philosophical society, vol. 124, no. 3, pp. 483–499, 1998. https://doi.org/10.1017/s0305004198002539
E. Ballico, “Coherent systems with many sections on projective curves”, International journal of mathematics, vol. 17, no. 03, pp. 263–267, 2006. https://doi.org/10.1142/s0129167x06003497
E. Ballico, “Stable coherent systems on integral projective curves: an asymptotic existence theorem”, International journal of pure and applied mathematics, vol. 27, no. 2, pp. 203-214, 2006. [On line]. Available: https://bit.ly/3w4gGgl
E. Ballico and F. Prantil, “Coherent systems on singular genus one curves”, International journal of contemporary mathematical sciences, vol. 2, pp. 1527–1543, 2007. https://doi.org/10.12988/ijcms.2007.07160
A. Bertram and B. Feinberg, “On stable rank two bundles with canonical determinant and many sections”, in Algebraic geometry, P. E. Newstead, Ed. New York: CRC, 1998, pp. 259-270
U. N. Bhosle, “Brill-Noether theory on nodal curves”, International journal of mathematics, vol. 18, no. 10, pp. 1133–1150, 2007. https://doi.org/10.1142/s0129167x07004461
U. N. Bhosle, “Coherent systems on a nodal curve”, in Moduli spaces and vector bundles, L. Brambila-Paz, S. B. Bradlow, O. García-Prada and S. Ramanan, Eds. Cambridge: Cambridge University, 2009, pp. 437-455. https://doi.org/10.1017/CBO9781139107037.015
U. N. Bhosle, “Coherent systems on a nodal curve of genus one”, Mathematische nachrichten, vol. 284, no. 14-15, pp. 1829–1845, 2011. https://doi.org/10.1002/mana.200910133
U. N. Bhosle, L. Brambila-Paz and P. E. Newstead. “On coherent systems of type (n, d, n+1) on Petri curves”, 2007, arXiv: 0712.2215v1.
U. N. Bhosle, L. Brambila-Paz, and P. E. Newstead, “On linear series and a conjecture of D. C. Butler”, International journal of mathematics, vol. 26, no. 02, 1550007, 2015. https://doi.org/10.1142/s0129167x1550007x
U. N. Bhosle and A. J. Parameswaran, “Picard bundles and Brill–Noether loci in the compactified jacobian of a nodal curve”, International mathematics research notices, vol. 2014, no. 15, pp. 4241–4290, 2013. https://doi.org/10.1093/imrn/rnt069
U. N. Bhosle and S. K. Singh, “Brill-Noether loci and generated torsionfree sheaves over nodal and cuspidal curves”, Manuscripta mathematica, vol. 141, no. 1-2, pp. 241–271, 2013. https://doi.org/10.1007/s00229-012-0571-0
S. B. Bradlow and O. García-Prada, “An application of coherent systems to a brill-noether problem”, Journal für die reine und angewandte mathematik (Crelles journal), vol. 2002, no. 551, pp. 123-143, 2002. https://doi.org/10.1515/crll.2002.079
S. B. Bradlow, O. García-Prada, V. Muñoz, and P. E. Newstead, “Coherent systems and Brill–Noether theory”, International journal of mathematics, vol. 14, no. 07, pp. 683–733, 2003. https://doi.org/10.1142/s0129167x03002009
S. B. Bradlow, O. García-Prada, V. Mercat, V. Muñoz, and P. E. Newstead, “On the geometry of moduli spaces of coherent systems on algebraic curves”, International journal of mathematics, vol. 18, no. 04, pp. 411–453, 2007. https://doi.org/10.1142/s0129167x07004151
S. B. Bradlow, O. García-Prada, V. Mercat, V. Muñoz, and P. E. Newstead, “Moduli spaces of coherent systems of small slope on algebraic curves”, Communications in algebra, vol. 37, no. 8, pp. 2649–2678, 2009. https://doi.org/10.1080/00927870902747464
L. Brambila-Paz, “Non-emptiness of moduli spaces of coherent systems”, International journal of mathematics, vol. 19, no. 07, pp. 777–799, 2008. https://doi.org/10.1142/s0129167x0800487x
L. Brambila-Paz, I. Grzegorczyk and P. E. Newstead, “Geography of Brill-Noether loci for small slopes”, 1995, arXiv: alg-geom/9511003v1.
L. Brambila-Paz and H. Lange, “A stratification of the moduli space of vector bundles on curves”, Journal für die reine und angewandte mathematik (Crelles journal), vol. 1998, no. 494, pp. 173–187, 1998. https://doi.org/10.1515/crll.1998.005
L. Brambila-Paz and O. Mata-Gutiérrez, “(t, ℓ)-stability and coherent systems”, Glasgow mathematical journal, vol. 62, no. 3, pp. 661–672, 2019. https://doi.org/10.1017/s0017089519000405
L. Brambila-Paz, O. Mata-Gutiérrez, P. E. Newstead, and A. Ortega, “Generated coherent systems and a conjecture of D. C. Butler”, International journal of mathematics, vol. 30, no. 05, 1950024, 2019. https://doi.org/10.1142/S0129167X19500241
L. Brambila-Paz, V. Mercat, P. E. Newstead, and F. Ongay, “Nonemptiness of brill–noether loci”, International journal of mathematics, vol. 11, no. 06, pp. 737–760, 2000. https://doi.org/10.1142/s0129167x00000350
L. Brambila-Paz and A. Ortega, “Brill-Noether bundles and coherent systems on special curves”, in Moduli spaces and vector bundles, L. Brambila-Paz, S. B. Bradlow, O. García-Prada and S. Ramanan, Eds. Cambridge: Cambridge University, 2009, pp. 456-472. https://doi.org/10.1017/CBO9781139107037.016
L. Brambila-Paz and A. Ortega, “Estabilidad de sistemas coherentes”. Aportaciones matemáticas: Comunicaciones, vol. 40, pp. 15-26, 2009.
L. Brambila-Paz and H. Torres-López, “On chow stability for algebraic curves”, Manuscripta mathematica, vol. 151, no. 3-4, pp. 289–304, 2016. https://doi.org/10.1007/s00229-016-0843-1
S. Brivio and F. F. Favale, “On kernel bundles over reducible curves with a node”, International journal of mathematics, vol. 31, no. 07, 2050054, 2020. https://doi.org/10.1142/s0129167x20500548
S. Brivio and F. F. Favale, “Coherent systems on curves of compact type”, Journal of geometry and physics, vol. 158, 103850, 2020. https://doi.org/10.1016/j.geomphys.2020.103850
S. Brivio and F. Favale, “Coherent systems and BGN extensions on nodal reducible curves”, 2021, arXiv: 2104.06883v1.
D. C. Butler, “Normal generation of vector bundles over a curve”, Journal of differential geometry, vol. 39, no. 1, pp. 1-34, 1994. https://doi.org/10.4310/jdg/1214454673
D. C. Butler, “Birational maps of moduli of Brill-Noether pairs”, 1997, arXiv:alg-geom/9705009v1.
A. Castorena, A. López Martín, and M. Teixidor i Bigas, “Petri map for vector bundles near good bundles”, Journal of pure and applied algebra, vol. 222, no. 7, pp. 1692–1703, 2018. https://doi.org/10.1016/j.jpaa.2017.07.018
A. Castorena, E. C. Mistretta and H. Torres-López, “On linear stability and syzygy stability for rank 2 linear series”, 2020, arXiv: 2001.03609v1.
A. Castorena and H. Torres-López, “Linear stability and stability of Syzygy Bundles”, International journal of mathematics, vol. 29, no. 11, 1850080, 2018. https://doi.org/10.1142/s0129167x18500805
K. Cook-Powell and D. Jensen, “Components of Brill-Noether loci for curves of fixed gonality”, 2019, arXiv: 1907.08366v1.
E. Cotterill, A. Alonso Gonzalo, and N. Zhang, “The strong maximal rank conjecture and higher rank Brill–noether theory”, Journal of the London mathematical society, vol. 104, no. 1, pp. 169–205, 2021. https://doi.org/10.1112/jlms.12427
G. Farkas, D. Jensen and S. Payne, The non-abelian Brill-Noether divisor on M₁₃ and the Kodaira dimension of Ṝ₁₃, 2021, arXiv: 2110.09553.
G. Farkas and A. Ortega, “The maximal rank conjecture and rank two Brill-Noether theory”, Pure and applied mathematics quarterly, vol. 7, no. 4, pp. 1265–1296, 2011. https://doi.org/10.4310/pamq.2011.v7.n4.a9
G. Farkas and A. Ortega, “Higher rank brill–noether theory on sections of K3 surfaces”, International journal of mathematics, vol. 23, no. 07, 1250075, 2012. https://doi.org/10.1142/s0129167x12500759
G. Farkas and M. Popa, “Effective divisors on Mg, curves on K3 surfaces, and the slope conjecture”, Journal of algebraic geometry, vol. 14, pp. 241-267, 2005. [On line]. Available: https://bit.ly/3MMOQex
S. Feyzbakhsh and C. Li, “Higher rank Clifford Indices of curves on a K3 surface”, Selecta mathematica, vol. 27, no. 3, 2021. https://doi.org/10.1007/s00029-021-00664-z
F. Ghione, “Un problème du type Brill-Noether pour les fibrés vectoriels”, in Algebraic Geometry - Open Problems, C. Ciliberto, F. Ghione, and F. Orecchia, Eds. Heidelberg: Springer, 1983, pp. 197–209. https://doi.org/10.1007/bfb0061644
I. Grzegorczyk, V. Mercat and P. E. Newstead, “Stable bundles of rank 2 with four sections”, International journal of mathematics, vol. 22, no. 12, pp. 1743–1762, 2011. https://doi.org/10.1142/s0129167x11007434
I. Grzegorczyk and P. E. Newstead, “On coherent systems with fixed determinant”, International journal of mathematics, vol. 25, no. 05, 1450045, 2014. https://doi.org/10.1142/s0129167x14500451
D. H. Ruipérez and C. T. Prieto, “Fourier-Mukai transforms for coherent systems on elliptic curves”, Journal of the London mathematical society, vol. 77, no. 1, pp. 15–32, 2007. https://doi.org/10.1112/jlms/jdm089
A. Hirschowitz, “Problèmes de Brill-Noether en rang supérieur”, Comptes rendus des séances de l'Académie des sciences. Série 1, Mathématique, vol. 307, no. 4, pp. 153-156, 1988. [On line]. Available: https://bit.ly/3CVglOI
G. H. Hitching, M. Hoff, and P. E. Newstead, “Nonemptiness and smoothness of twisted Brill–Noether loci”, Annali di matematica pura ed applicata, vol. 200, no. 2, pp. 685–709, 2020. https://doi.org/10.1007/s10231-020-01009-x
M. Hoff, “A note on syzygies and normal generation for trigonal curves”, 2021, arXiv: 2108.06106v2.
A. D. King and P. E. Newstead, “Moduli of Brill-Noether pairs on algebraic curves”, International journal of mathematics, vol. 06, no. 05, pp. 733–748, 1995. https://doi.org/10.1142/s0129167x95000316
H. Lange, V. Mercat, and P. E. Newstead, “On an example of Mukai”, Glasgow mathematical journal, vol. 54, no. 2, pp. 261–271, 2011. https://doi.org/10.1017/s0017089511000577
H. Lange and P. E. Newstead, “Coherent systems of genus 0”, International journal of mathematics, vol. 15, no. 04, pp. 409–424, 2004. https://doi.org/10.1142/s0129167x04002326
H. Lange and P. E. Newstead, “Coherent systems on elliptic curves”, International journal of mathematics, vol. 16, no. 07, pp. 787–805, 2005. https://doi.org/10.1142/s0129167x05003090
H. Lange and P. E. Newstead, “Coherent systems of genus 0 II: Existence results for K ≥ 3”, International journal of mathematics, vol. 18, no. 04, pp. 363–393, 2007. https://doi.org/10.1142/s0129167x07004072
H. Lange and P. E. Newstead, “Coherent systems of genus 0 III: Computation of flips for K = 1”, International journal of mathematics, vol. 19, no. 09, pp. 1103–1119, 2008. https://doi.org/10.1142/s0129167x08005047
H. Lange and P. E. Newstead, “Clifford’s theorem for Coherent Systems”, Archiv der mathematik, vol. 90, no. 3, pp. 209–216, 2008. https://doi.org/10.1007/s00013-007-2534-3
H. Lange and P. E. Newstead, “Hodge polynomials and birational types of moduli spaces of coherent systems on elliptic curves”, Manuscripta mathematica, vol. 130, no. 1, pp. 1–19, 2009. https://doi.org/10.1007/s00229-009-0276-1
H. Lange and P. E. Newstead, “Clifford indices for vector bundles on curves”, in Affine flag manifolds and principal bundles, A. Schmitt, Ed. Basel: Birkhäuser, 2010, pp. 165-202. https://doi.org/10.1007/978-3-0346-0288-4_6
H. Lange and P. E. Newstead, “Generation of vector bundles computing clifford indices”, Archiv der mathematik, vol. 94, no. 6, pp. 529–537, 2010. https://doi.org/10.1007/s00013-010-0126-0
H. Lange and P. Newstead, “Further examples of stable bundles of rank 2 with 4 sections”, Pure and applied mathematics quarterly, vol. 7, no. 4, pp. 1517–1528, 2011. https://doi.org/10.4310/pamq.2011.v7.n4.a20
H. Lange and P. E. Newstead, “Bundles computing clifford indices on trigonal curves”, Archiv der mathematik, vol. 101, no. 1, pp. 21–31, 2013. https://doi.org/10.1007/S00013-013-0540-1
H. Lange and P. E. Newstead, “Vector bundles of rank 2 computing Clifford indices”, Communications in algebra, vol. 41, no. 6, pp. 2317–2345, 2013. https://doi.org/10.1080/00927872.2012.658532
H. Lange and P. E. Newstead, “On bundles of rank 3 computing Clifford indices”, Kyoto journal of mathematics, vol. 53, no. 1, pp. 25-54, 2013. https://doi.org/10.1215/21562261-1966062
H. Lange and P. E. Newstead, “Bundles of rank 3 on curves of Clifford index 3”, Journal of symbolic computation, vol. 57, pp. 3–18, 2013. https://doi.org/10.1016/j.jsc.2013.05.002
H. Lange and P. E. Newstead, “Higher rank BN-theory for curves of genus 4”, Communications in algebra, vol. 45, no. 9, pp. 3948–3966, 2016. https://doi.org/10.1080/00927872.2016.1251938
H. Lange and P. E. Newstead, “Higher rank BN-theory for curves of genus 5”, Revista matemática complutense, vol. 29, no. 3, pp. 691–717, 2016. https://doi.org/10.1007/s13163-016-0203-4
H. Lange and P. E. Newstead, “Higher rank BN-theory for curves of genus 6”, International journal of mathematics, vol. 29, no. 02, p. 1850014, 2018. https://doi.org/10.1142/S0129167X18500143
H. Lange, P. E. Newstead, and S. S. Park, “Nonemptiness of brill–Noether loci in M(2,k),” Communications in algebra, vol. 44, no. 2, pp. 746–767, 2015. https://doi.org/10.1080/00927872.2014.990020
H. Lange, P. E. Newstead, and V. Strehl, “Nonemptiness of brill–Noether loci in M(2,L),” International journal of mathematics, vol. 26, no. 13, p. 1550108, 2015. https://doi.org/10.1142/s0129167x15501086
E. Larson, H. K. Larson and I. Vogt, “Global Brill-Noether theory over the Hurwitz space”, 2020, arXiv: 2008.10765v2.
H. K. Larson, “Refined brill–noether theory for all trigonal curves”, European journal of mathematics, vol. 7, no. 4, pp. 1524–1536, 2021. https://doi.org/10.1007/s40879-021-00493-6
R. Lazarsfeld, “Some applications of the theory of positive vector bundles”, in Complete intersections, S. Greco and R. Strano, Eds. Berlin: Springer, 1984, pp. 29-61. https://doi.org/10.1007/BFb0099356
M. Lelli-Chiesa, “Stability of rank-3 Lazarsfeld-Mukai Bundles on K3 surfaces”, Proceedings of the London mathematical society, vol. 107, no. 2, pp. 451–479, 2013. https://doi.org/10.1112/plms/pds087
C. Li, “On stability conditions for the quintic threefold”, Inventiones mathematicae, vol. 218, no. 1, pp. 301–340, 2019. https://doi.org/10.1007/s00222-019-00888-z
V. Mercat, “Le problème de brill-noether pour des fibrés stables de petite pente”, Journal für die reine und angewandte Mathematik (Crelles journal), vol. 1999, no. 506, pp. 1–41, 1999. https://doi.org/10.1515/crll.1999.506.1
V. Mercat, “Le problème de brill-noether et le théorème de teixidor”, Manuscripta mathematica, vol. 98, no. 1, pp. 75–85, 1999. https://doi.org/10.1007/s002290050126
V. Mercat, “Fibrés stables de Pente 2”, Bulletin of the London mathematical society, vol. 33, no. 5, pp. 535–542, 2001. https://doi.org/10.1112/s0024609301008256
V. Mercat, “Clifford's theorem and higher rank vector bundles”, International journal of mathematics, vol. 13, no. 07, pp. 785–796, 2002. https://doi.org/10.1142/s0129167x02001484
E. C. Mistretta and L. Stoppino, “Linear series on curves: Stability and Clifford index”, International journal of mathematics, vol. 23, no. 12, 1250121, 2012. https://doi.org/10.1142/s0129167x12501212
S. Mukai, “Vector bundles and Brill-Noether theory”, in Current topics in complex algebraic geometry, H. Clemens and J. Kollár, Eds. Cambridge: Cambridge University, 1995, pp. 145-158.
S. Mukai, “Non-abelian Brill-Noether theory and Fano 3-folds”, 1997, arXiv: alg-geom/9704015v1
V. Muñoz, “Hodge polynomials of the moduli spaces of rank 3 pairs”, Geometriae dedicata, vol. 136, no. 1, pp. 17–46, 2008. https://doi.org/10.1007/s10711-008-9272-y
V. Muñoz, D. Ortega, and M.-J. Vázquez-Gallo, “Hodge polynomials of the moduli spaces of pairs”, International journal of mathematics, vol. 18, no. 06, pp. 695–721, 2007. https://doi.org/10.1142/s0129167x07004266
V. Muñoz, D. Ortega, and M.-J. Vázquez-Gallo, “Hodge polynomials of the moduli spaces of triples of rank (2, 2)”, The quarterly journal of mathematics, vol. 60, no. 2, pp. 235–272, 2008. https://doi.org/10.1093/qmath/han007
P. E. Newstead, “Existence of α-stable coherent systems on algebraic curves”, in Grassmannians, moduli spaces and vector bundles, D. A. Ellwood and E. Previato Eds. Providence: AMS, 2011, pp. 121-140.
P. E. Newstead, “Some examples of rank-2 Brill–Noether loci”, Revista matemática complutense, vol. 31, no. 1, pp. 201–215, 2017. https://doi.org/10.1007/s13163-017-0241-6
P. Newstead and M. Teixidor i Bigas, “Coherent systems on the projective line”, The quarterly journal of mathematics, vol. 72, no. 1-2, pp. 115–136, 2020. https://doi.org/10.1093/qmathj/haaa024
B. Osserman, “Brill–Noether loci with fixed determinant in rank 2”, International journal of mathematics, vol. 24, no. 13, 1350099, 2013. https://doi.org/10.1142/s0129167x13500997
B. Osserman, “Special determinants in higher-rank Brill–noether theory”, International journal of mathematics, vol. 24, no. 11, p. 1350084, 2013. https://doi.org/10.1142/s0129167x13500845
S. Pasotti and F. Prantil, “Holomorphic triples on elliptic curves”, Results in mathematics, vol. 50, no. 3-4, pp. 227–239, 2007. https://doi.org/10.1007/s00025-007-0248-2
S. Pasotti and F. Prantil, “Holomorphic triples of genus 0”, Central european journal of mathematics, vol. 6, no. 1, pp. 129–142, 2008. https://doi.org/10.2478/s11533-008-0008-x
N. Pflueger, “Brill–Noether varieties of k-gonal curves”, Advances in mathematics, vol. 312, pp. 46–63, 2017. https://doi.org/10.1016/j.aim.2017.01.027
R. Re, “Multiplication of sections and Clifford bounds for stable vector bundles on curves”, Communications in algebra, vol. 26, no. 6, pp. 1931–1944, 1998. https://doi.org/10.1080/00927879808826250
L. Roa-Leguizamón, “Segre invariant and a stratification of the moduli space of coherent systems”, International journal of mathematics, vol. 31, no. 14, p. 2050117, 2020. https://doi.org/10.1142/s0129167x20501177
B. Russo and M. Teixidor i Bigas, “On a conjecture of Lange”, Journal of algebraic geometry, vol. 8, pp. 483-496, 1999.
A. H. W. Schmitt, “A general notion of coherent systems”, Revista matemática Iberoamericana, 2022. https://doi.org/10.4171/rmi/1314
A. Schmitt, “Notes on coherent systems”, Revista de matemática: teoría y aplicaciones, vol. 28, no. 1, pp. 1–38, 2021. https://doi.org/10.15517/rmta.v28i1.42154
M. Teixidor i Bigas, “Brill-Noether theory for stable vector bundles”, Duke mathematical journal, vol. 62, no. 2, pp. 385-400, 1991. https://doi.org/10.1215/s0012-7094-91-06215-0
M. Teixidor i Bigas, “Moduli spaces of (semi)stable vector bundles on tree-like curves”, Mathematische annalen, vol. 290, no. 1, pp. 341–348, 1991. https://doi.org/10.1007/bf01459249
M. Teixidor i Bigas, “Moduli spaces of vector bundles on reducible curves”, American journal of mathematics, vol. 117, no. 1, pp. 125-139, 1995. https://doi.org/10.2307/2375038
M. Teixidor i Bigas, “Curves in Grassmannians”, Proceedings of the American mathematical society, vol. 126, no. 6, pp. 1597–1603, 1998. https://doi.org/10.1090/s0002-9939-98-04475-x
M. Teixidor i Bigas, “Rank two vector bundles with canonical determinant”, Mathematische nachrichten, vol. 265, no. 1, pp. 100–106, 2004. https://doi.org/10.1002/mana.200310138
M. Teixidor i Bigas, “Existence of coherent systems of rank two and dimension four”, Collectanea mathematica, vol. 58, no. 2, pp. 193-198, 2007.
M. Teixidor i Bigas, “Petri map for rank two bundles with canonical determinant”, Compositio mathematica, vol. 144, no. 3, pp. 705–720, 2008. https://doi.org/10.1112/s0010437x07003442
M. Teixidor i Bigas, “Existence of coherent systems II”, International journal of mathematics, vol. 19, no. 10, pp. 1269–1283, 2008. https://doi.org/10.1142/s0129167x08005126
M. Teixidor i Bigas, “Existence of vector bundles of rank two with fixed determinant and sections”, Proceedings of the Japan academy, Series A, mathematical sciences, vol. 86, no. 7, 2010. https://doi.org/10.3792/pjaa.86.113
M. Teixidor i Bigas, “Injectivity of the Petri map for twisted brill–noether loci”, Manuscripta mathematica, vol. 145, no. 3-4, pp. 389–397, 2014. https://doi.org/10.1007/s00229-014-0690-x
M. Thaddeus, “Stable pairs, linear systems and the Verlinde formula”, Inventiones mathematicae, vol. 117, no. 1, pp. 317–353, 1994. https://doi.org/10.1007/bf01232244
N. Zhang, “Towards the bertram–feinberg–mukai conjecture”, Journal of pure and applied algebra, vol. 220, no. 4, pp. 1588–1654, 2016. https://doi.org/10.1016/j.jpaa.2015.09.020
N. Zhang, “Expected dimensions of higher-rank Brill-Noether Loci”, Proceedings of the American mathematical society, vol. 145, no. 9, pp. 3735–3746, 2017. https://doi.org/10.1090/proc/13542
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