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http://dx.doi.org/10.4134/CKMS.c180515

LIMITS OF TRIVIAL BUNDLES ON CURVES  

Ballico, Edoardo (Department of Mathematics University of Trento)
Publication Information
Communications of the Korean Mathematical Society / v.35, no.1, 2020 , pp. 43-61 More about this Journal
Abstract
We extend the work of A. Beauville on rank 2 vector bundles on a smooth curve in several directions. We give families of examples with large dimension, add new existence and non-existence results and prove the existence of indecomposable limits with arbitrary rank. To construct the large dimensional families we use the examples of limits of rank 2 trivial bundles on ℙ2 and ℙ3 due to C. Banica. We also consider a more flexible notion: limits of trivial bundles on nearby curves.
Keywords
Vector bundle; rank 2 vector bundle; vector bundles on curves; degenerations of a vector bundle; limits of trivial bundles;
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Times Cited By KSCI : 1  (Citation Analysis)
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1 E. Arbarello, A. Bruno, G. Farkas, and G. Sacca, Explicit Brill-Noether-Petri general curves, Comment. Math. Helv. 91 (2016), no. 3, 477-491. https://doi.org/10.4171/CMH/392   DOI
2 E. Arbarello and M. Cornalba, Footnotes to a paper of Beniamino Segre, Math. Ann. 256 (1981), no. 3, 341-362. https://doi.org/10.1007/BF01679702   DOI
3 E. Arbarello, M. Cornalba, and P. A. Griffiths, Geometry of algebraic curves. Volume II, Grundlehren der Mathematischen Wissenschaften, 268, Springer, Heidelberg, 2011. https://doi.org/10.1007/978-3-540-69392-5
4 M. Atiyah, On the Krull-Schmidt theorem with application to sheaves, Bull. Soc. Math. France 84 (1956), 307-317.   DOI
5 M. Atiyah, Vector bundles over an elliptic curve, Proc. London Math. Soc. (3) 7 (1957), 414-452. https://doi.org/10.1112/plms/s3-7.1.414   DOI
6 E. Ballico, A remark on linear series on general k-gonal curves, Boll. Un. Mat. Ital. A (7) 3 (1989), no. 2, 195-197.
7 C. Banica, Topologisch triviale holomorphe Vektorbundel auf $P^n$ (C), J. Reine Angew. Math. 344 (1983), 102-119.
8 A. Beauville, Limits of the trivial bundle on a curve, Epijournal Geom. Algebrique 2 (2018), Art. 9, 6 pp.
9 M. Coppens and G. Martens, Linear series on a general k-gonal curve, Abh. Math. Sem. Univ. Hamburg 69 (1999), 347-371. https://doi.org/10.1007/BF02940885   DOI
10 S. Casalaina-Martin, D. Jensen, and R. Laza, The geometry of the ball quotient model of the moduli space of genus four curves, in Compact moduli spaces and vector bundles, 107-136, Contemp. Math., 564, Amer. Math. Soc., Providence, RI. https://doi.org/10.1090/conm/564/11153
11 M. Coppens and G. Martens, Linear series on 4-gonal curves, Math. Nachr. 213 (2000), 35-55. https://doi.org/10.1002/(SICI)1522-2616(200005)213:1<35::AID-MANA35>3.0.CO;2-Z   DOI
12 D. Eisenbud and J. Harris, Limit linear series: basic theory, Invent. Math. 85 (1986), no. 2, 337-371. https://doi.org/10.1007/BF01389094   DOI
13 J. Harris and D. Mumford, On the Kodaira dimension of the moduli space of curves, Invent. Math. 67 (1982), no. 1, 23-86. https://doi.org/10.1007/BF01393371   DOI
14 M. Fedorchuk, The final log canonical model of the moduli space of stable curves of genus 4, Int. Math. Res. Not. IMRN 2012 (2012), no. 24, 5650-5672. https://doi.org/10.1093/imrn/rnr242   DOI
15 W. Fulton, Hurwitz schemes and irreducibility of moduli of algebraic curves, Ann. of Math. (2) 90 (1969), 542-575. https://doi.org/10.2307/1970748   DOI
16 J. Harris, Theta-characteristics on algebraic curves, Trans. Amer. Math. Soc. 271 (1982), no. 2, 611-638. https://doi.org/10.2307/1998901   DOI
17 R. Hartshorne, Algebraic Geometry, Springer-Verlag, New York, 1977.
18 H. Kim, Chow stability of canonical genus 4 curves, Bull. Korean Math. Soc. 50 (2013), no. 3, 1029-1040. https://doi.org/10.4134/BKMS.2013.50.3.1029   DOI
19 H. Lange, Zur Klassifikation von Regelmannigfaltigkeiten, Math. Ann. 262 (1983), no. 4, 447-459. https://doi.org/10.1007/BF01456060   DOI
20 H. Lange, Some geometrical aspects of vector bundles on curves, in Topics in algebraic geometry (Guanajuato, 1989), 53-74, Aportaciones Mat. Notas Investigacion, 5, Soc. Mat. Mexicana, Mexico, 1992.
21 H. Lange and M. S. Narasimhan, Maximal subbundles of rank two vector bundles on curves, Math. Ann. 266 (1983), no. 1, 55-72. https://doi.org/10.1007/BF01458704   DOI
22 G. Martens and F.-O. Schreyer, Line bundles and syzygies of trigonal curves, Abh. Math. Sem. Univ. Hamburg 56 (1986), 169-189. https://doi.org/10.1007/BF02941515   DOI
23 M. Teixidor i Bigas, Half-canonical series on algebraic curves, Trans. Amer. Math. Soc. 302 (1987), no. 1, 99-115. https://doi.org/10.2307/2000899   DOI
24 J. Rojas and I. Vainsencher, Canonical curves in ${\mathbb{P}^3}$, Proc. London Math. Soc. (3) 85 (2002), no. 2, 333-366. https://doi.org/10.1112/S0024611502013503   DOI
25 U. Schafft, Nichtsepariertheit instabiler Rang-2-Vektorbundel auf $P_2$, J. Reine Angew. Math. 338 (1983), 136-143. https://doi.org/10.1515/crll.1983.338.136   DOI
26 S. A. Strmme, Deforming vector bundles on the projective plane, Math. Ann. 263 (1983), no. 3, 385-397. https://doi.org/10.1007/BF01457140   DOI
27 M. Teixidor i Bigas, The divisor of curves with a vanishing theta-null, Compositio Math. 66 (1988), no. 1, 15-22.