Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2024-12-25T06:26:22.608Z Has data issue: false hasContentIssue false
  • English
  • Français

(t, ℓ)-STABILITY AND COHERENT SYSTEMS

Part of: Surfaces and higher-dimensional varieties Curves

Published online by Cambridge University Press:  09 October 2019

L. BRAMBILA-PAZ  and
O. MATA-GUTIÉRREZ
L. BRAMBILA-PAZ
Affiliation:
CIMAT, Mineral de Valenciana S/N, Apdo. Postal 402, C.P. 36240. Guanajuato, Gto, Mexico e-mail: lebp@cimat.mx
O. MATA-GUTIÉRREZ
Affiliation:
Departamento de Matemáticas, CUCEI, Universidad de Guadalajara, Av. Revolución 1500, C.P. 44430, Guadalajara, Jalisco, Mexico e-mails: osbaldo.mata@academico.udg.mx, osbaldo@cimat.mx
Get access Rights & Permissions [Opens in a new window]

Abstract

Let X be a non-singular irreducible complex projective curve of genus g ≥ 2. The concept of stability of coherent systems over X depends on a positive real parameter α, given then a (finite) family of moduli spaces of coherent systems. We use (t, ℓ)-stability to prove the existence of coherent systems over X that are α-stable for all allowed α > 0.

MSC classification

Primary: 14H60: Vector bundles on curves and their moduli
Secondary: 14J60: Vector bundles on surfaces and higher-dimensional varieties, and their moduli
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 62 , Issue 3 , September 2020 , pp. 661 - 672
Copyright
© Glasgow Mathematical Journal Trust 2019

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Arbarello, E., Cornalba, M. and Griffiths, P.A., Geometry of algebraic curves, Vol. I, Grundlehren der Mathematischen Wissenschaften, vol. 268 (1985), (Springer, Heidelberg, 1985).CrossRefGoogle Scholar
Ballico, E., Coherent systems with many sections on projective curves. (English summary) Internat. J. Math. 17(3) (2006), 263–267.CrossRefGoogle Scholar
Bhosle, U. N., Brambila-Paz, L. Newstead, P. E., On coherent systems of type (n, d, n+1) on Petri curves, Manuscripta Math. 126 (2008), 409–441.CrossRefGoogle Scholar
Bhosle, U. N., Brambila-Paz, L. Newstead, P. E., On linear systems and a conjecture of D. C. Butler, Internat. J. Math. 26 (2015), 1550007, 18 p. doi:10.1142/S0129167X1550007X.Google Scholar
Bradlow, S. O. García-Prada, An application of coherent systems to a Brill–Noether problem, J Reine Angew. Math. 551 (2002), 123–143.Google Scholar
Bradlow, S., García-Prada, O., Mercat, V., Muñoz, V. Newstead, P. E., On the geometry of moduli spaces of coherent systems on algebraic curves, Internat. J. Math. 18 (2007), 411–453.CrossRefGoogle Scholar
Bradlow, S., García-Prada, O., Muñoz, V. Newstead, P. E., Coherent systems and Brill–Noether Theory, Internat. J. Math. 14 (2003), 683–733.Google Scholar
Brambila-Paz, L., Non-emptiness of moduli spaces of coherent systems, Internat. J. Math. 18(7) (2008), 777–799.CrossRefGoogle Scholar
Brambila-Paz, L., Grzegorczyk, I. and Newstead, P. E., Geography of Brill–Noether loci for small slopes, J. Alg. Geo. 6 (1997), 645–669.Google Scholar
Brambila-Paz, L. and Lange, H., A stratification of the moduli space of vector bundles on curves, J. Reine Angew. Math 494 (1988), 173–187.Google Scholar
Brambila-Paz, L., Mata-Gutiérrez, O., Ortega, A. and Newstead, P. E., On generated coherent systems and a conjecture of D. C. Butler, Internat. J. Math. 30(5) (2019). doi:10.1142/S0129167X19500241.CrossRefGoogle Scholar
Brambila-Paz, L., Mercat, V., Newstead, P. E. and Ongay, F., Nonemptiness of Brill–Noether loci, Internat. J. Math. 11 (2000), 737–760.Google Scholar
Butler, D. C., Birational maps of moduli of Brill–Noether pairs, preprint, arXiv:alg-geom/9705009.Google Scholar
Grothendieck, A. and Dieudonné, J., Éléments de géométrie algébrique, III, Pub. Math. IHES 17 France (1963).Google Scholar
Grzegorczyk, I. and Teixidor i Bigas, M., Brill–Noether Theory for stable bundles, in Moduli spaces and vector bundles. London Mathematical Society Note Series, vol. 359 (2009), 29–50.CrossRefGoogle Scholar
King, A. and Newstead, P. E., Moduli of Brill–Noether pairs on algebraic curves, Internat. J. Math. 6 (1995), 733–748.Google Scholar
Lange, H. and Narasihman, M. S., Maximal subbundles of rank two vector bundles on curves, Math. Ann. 266 (1983), 55–72.CrossRefGoogle Scholar
Lange, H. and Newstead, P. E., Clifford’s theorem for coherent systems, Archiv der Mathematik 90(3) (2008), 209–216.CrossRefGoogle Scholar
Mata-Gutiérrez, O. and Neumann, F., Geometry of moduli stacks of (k, ℓ)-stable vector bundles over an algebraic curve, J. Geo. Phys. 111 (2017), 54–70. doi:10.1016/j.geomphys.2016.10.003.Google Scholar
Narasimhan, M. S. and Ramanan, S., Deformation of the moduli space of vector bundles over an algebraic curve, Ann. of Math. 101(3) (1975), 391–417.CrossRefGoogle Scholar
Narasimhan, M. S. and Ramanan, S., Geometry of Hecke cycles - I, in C. P. Ramanujam - a tribute , Tata Institute of Fundamental Research Studies in Mathematics, vol. 8 (Springer-Verlag, Berlin-New York, 1978), pp. 291–345.Google Scholar
Newstead, P. E., Existence of a-stable coherent systems on algebraic curves, Clay Math. Proc. 14 (2011), 121–139.Google Scholar
Nitsure, N., Construction of Hilbert and Quot schemes, in Fundamental algebraic geometry – Grothendieck’s FGA explained (Fantachi, B. et al., Editor), Mathematical Surveys and Monographs, vol. 123, Part 2 (American Mathematical Society, France, 2005).Google Scholar
Raghavendra, N. and Vishwanath, P. A., Moduli of pairs and generalized theta divisors, Tohoku Math. J. 46 (1994), 321–340.CrossRefGoogle Scholar
Russo, B. and Teixidor i Bigas, M., On a conjecture of Lange, J. Alg. Geo. 8 (1999), 483–496.Google Scholar
Teixidor i Bigas, M., Existence of coherent systems II, Int. J. Math. 19, 449 (2008), 1269–1283.CrossRefGoogle Scholar
Teixidor i Bigas, M., Brill–Noether Theory for stable bundles, Duke Math. J. 62 (1991), 385–400.Google Scholar