Simple vector bundles on plane degenerations of an elliptic curve

Bodnarchuk, Lesya; Drozd, Yuriy; Greuel, Gert-Martin

doi:10.1090/S0002-9947-2011-05354-7

Simple vector bundles on plane degenerations of an elliptic curve
HTML articles powered by AMS MathViewer

by Lesya Bodnarchuk, Yuriy Drozd and Gert-Martin Greuel PDF

Trans. Amer. Math. Soc. 364 (2012), 137-174 Request permission

Abstract:

In 1957 Atiyah classified simple and indecomposable vector bundles on an elliptic curve. In this article we generalize his classification by describing the simple vector bundles on all reduced plane cubic curves. Our main result states that a simple vector bundle on such a curve is completely determined by its rank, multidegree and determinant. Our approach, based on the representation theory of boxes, also yields an explicit description of the corresponding universal families of simple vector bundles.

References

Allen B. Altman and Steven L. Kleiman, Compactifying the Picard scheme. II, Amer. J. Math. 101 (1979), no. 1, 10–41. MR 527824, DOI 10.2307/2373937
M. F. Atiyah, Vector bundles over an elliptic curve, Proc. London Math. Soc. (3) 7 (1957), 414–452. MR 131423, DOI 10.1112/plms/s3-7.1.414
Claudio Bartocci, Ugo Bruzzo, Daniel Hernández Ruipérez, and José M. Muñoz Porras, Relatively stable bundles over elliptic fibrations, Math. Nachr. 238 (2002), 23–36. MR 1900809, DOI 10.1002/1522-2616(200205)238:1<23::AID-MANA23>3.3.CO;2-R
W. Barth, C. Peters, and A. Van de Ven, Compact complex surfaces, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 4, Springer-Verlag, Berlin, 1984. MR 749574, DOI 10.1007/978-3-642-96754-2
Lesya Bodnarchuk, Igor Burban, Yuriy Drozd, and Gert-Martin Greuel, Vector bundles and torsion free sheaves on degenerations of elliptic curves, Global aspects of complex geometry, Springer, Berlin, 2006, pp. 83–128. MR 2264108, DOI 10.1007/3-540-35480-8_{3}
Lesya Yu. Bodnarchuk and Yuriy A. Drozd, Stable vector bundles over cuspidal cubics, Cent. Eur. J. Math. 1 (2003), no. 4, 650–660. MR 2040656, DOI 10.2478/BF02475185
Bodnarchuk, L., Drozd, Yu. A.: One class of wild but brick-tame matrix problems preprint MPIM2009-18, arXiv: math.RT /0903.4374v2.
W. L. Burt and M. C. R. Butler, Almost split sequences for bocses, Representations of finite-dimensional algebras (Tsukuba, 1990) CMS Conf. Proc., vol. 11, Amer. Math. Soc., Providence, RI, 1991, pp. 89–121. MR 1143847
Igor Burban and Yurij Drozd, Coherent sheaves on rational curves with simple double points and transversal intersections, Duke Math. J. 121 (2004), no. 2, 189–229. MR 2034641, DOI 10.1215/S0012-7094-04-12121-9
Burban, I., Drozd, Yu.: Tilting on non-commutative rational projective curves, arXiv:0905.1231 (to appear in Mat. Annalen).
I. Burban, Yu. Drozd, and G.-M. Greuel, Vector bundles on singular projective curves, Applications of algebraic geometry to coding theory, physics and computation (Eilat, 2001) NATO Sci. Ser. II Math. Phys. Chem., vol. 36, Kluwer Acad. Publ., Dordrecht, 2001, pp. 1–15. MR 1866891
Igor Burban and Bernd Kreussler, Fourier-Mukai transforms and semi-stable sheaves on nodal Weierstraßcubics, J. Reine Angew. Math. 584 (2005), 45–82. MR 2155085, DOI 10.1515/crll.2005.2005.584.45
Igor Burban and Bernd Kreußler, Derived categories of irreducible projective curves of arithmetic genus one, Compos. Math. 142 (2006), no. 5, 1231–1262. MR 2264663, DOI 10.1112/S0010437X06002090
Burban, I., Kreußler, B.: Vector bundles on cubic curves and Yang-Baxter equations. arxiv: math.AG/0708.1685v2.
Usha Bhosle, Generalised parabolic bundles and applications to torsionfree sheaves on nodal curves, Ark. Mat. 30 (1992), no. 2, 187–215. MR 1289750, DOI 10.1007/BF02384869
Usha N. Bhosle, Generalized parabolic bundles and applications. II, Proc. Indian Acad. Sci. Math. Sci. 106 (1996), no. 4, 403–420. MR 1425615, DOI 10.1007/BF02837696
Usha N. Bhosle, Vector bundles on curves with many components, Proc. London Math. Soc. (3) 79 (1999), no. 1, 81–106. MR 1687547, DOI 10.1112/S0024611599011855
Bodnarchuk, L.: Simple vector bundles on degenerations of elliptic curves of type II, III and IV. Ph.D. thesis, Kaiserslautern (2007) http://kluedo.ub.uni-kl.de/volltexte/2008/2281/.
V. M. Bondarenko, Representations of bundles of semichained sets and their applications, Algebra i Analiz 3 (1991), no. 5, 38–61 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 3 (1992), no. 5, 973–996. MR 1186235
Burban, I.: Stable vector bundles on a rational curve with one simple node. Ukrainian Mathematical Journal 5, (2003).
W. W. Crawley-Boevey, On tame algebras and bocses, Proc. London Math. Soc. (3) 56 (1988), no. 3, 451–483. MR 931510, DOI 10.1112/plms/s3-56.3.451
W. W. Crawley-Boevey, Functorial filtrations. II. Clans and the Gel′fand problem, J. London Math. Soc. (2) 40 (1989), no. 1, 9–30. MR 1028911, DOI 10.1112/jlms/s2-40.1.9
W. W. Crawley-Boevey, Matrix problems and Drozd’s theorem, Topics in algebra, Part 1 (Warsaw, 1988) Banach Center Publ., vol. 26, PWN, Warsaw, 1990, pp. 199–222. MR 1171233
Yuri A. Drozd and Gert-Martin Greuel, Tame and wild projective curves and classification of vector bundles, J. Algebra 246 (2001), no. 1, 1–54. MR 1872612, DOI 10.1006/jabr.2001.8934
Ju. A. Drozd, Tame and wild matrix problems, Representations and quadratic forms (Russian), Akad. Nauk Ukrain. SSR, Inst. Mat., Kiev, 1979, pp. 39–74, 154 (Russian). MR 600111
Yu. A. Drozd, Matrix problems, small reduction and representations of a class of mixed Lie groups, Representations of algebras and related topics (Kyoto, 1990) London Math. Soc. Lecture Note Ser., vol. 168, Cambridge Univ. Press, Cambridge, 1992, pp. 225–249. MR 1211482
Yuriy A. Drozd, Reduction algorithm and representations of boxes and algebras, C. R. Math. Acad. Sci. Soc. R. Can. 23 (2001), no. 4, 97–125. MR 1869054
Yuriy A. Drozd, Semi-continuity for derived categories, Algebr. Represent. Theory 8 (2005), no. 2, 239–248. MR 2162284, DOI 10.1007/s10468-005-0967-6
Robert Friedman, John W. Morgan, and Edward Witten, Vector bundles over elliptic fibrations, J. Algebraic Geom. 8 (1999), no. 2, 279–401. MR 1675162
Hernández Ruipérez, D., López Martin, A.C., Sánchez Gómez, D.S., Prieto, C.T.: Moduli spaces of semistable sheaves on singular genus one curves. Int. Math. Res. Notices 11, 4428–4462 (2009), DOI 10.1093/imrn/rnp094, arXiv:math.AG/0806.2034.
I. M. Gel′fand and V. A. Ponomarev, Indecomposable representations of the Lorentz group, Uspehi Mat. Nauk 23 (1968), no. 2 (140), 3–60 (Russian). MR 0229751
Lee Klingler and Lawrence S. Levy, Sweeping-similarity of matrices, Linear Algebra Appl. 75 (1986), 67–104. MR 825400, DOI 10.1016/0024-3795(86)90182-5
A. V. Roĭter and M. M. Kleiner, Representations of differential graded categories, Representations of algebras (Proc. Internat. Conf., Carleton Univ., Ottawa, Ont., 1974) Lecture Notes in Math., Vol. 488, Springer, Berlin, 1975, pp. 316–339. MR 0435145
Igor Moiseevich Krichever, Integration of nonlinear equations by the methods of algebraic geometry, Funkcional. Anal. i Priložen. 11 (1977), no. 1, 15–31, 96 (Russian). MR 0494262
Lenzing, H. and Meltzer, H.: Sheaves on a weighted projective line of genus one, and representations of a tubular algebra. Can. Math. Soc. Conf. Proc. 14, 313– 337 (1993).
Ana Cristina López-Martín, Simpson Jacobians of reducible curves, J. Reine Angew. Math. 582 (2005), 1–39. MR 2139709, DOI 10.1515/crll.2005.2005.582.1
Ana Cristina López-Martín, Relative Jacobians of elliptic fibrations with reducible fibers, J. Geom. Phys. 56 (2006), no. 3, 375–385. MR 2171891, DOI 10.1016/j.geomphys.2005.02.007
Ju. I. Manin, Matrix solitons and vector bundles over curves with singularities, Funktsional. Anal. i Prilozhen. 12 (1978), no. 4, 53–63 (Russian). MR 515629
Motohico Mulase, Algebraic theory of the KP equations, Perspectives in mathematical physics, Conf. Proc. Lecture Notes Math. Phys., III, Int. Press, Cambridge, MA, 1994, pp. 151–217. MR 1314667
Nazarova, L.A., Roiter, A.V.: Finitely generated modules over a dyad of two local Dedekind rings, and finite groups with an Abelian normal divisor of index $p$. Math. USSR Izv., 3, 65–86 (1969); translation from Izv. Akad. Nauk SSSR, Ser. Mat., 33, 65–89 (1969).
P. E. Newstead, Introduction to moduli problems and orbit spaces, Tata Institute of Fundamental Research Lectures on Mathematics and Physics, vol. 51, Tata Institute of Fundamental Research, Bombay; Narosa Publishing House, New Delhi, 1978. MR 546290
Ovsienko, S.: Bimodule and matrix problems. Euroconference Essen Computational Methods for Representations of Groups and Algebras, April 1 - 5, (1997).
A. Polishchuk, Classical Yang-Baxter equation and the $A_\infty$-constraint, Adv. Math. 168 (2002), no. 1, 56–95. MR 1907318, DOI 10.1006/aima.2001.2047
Polishchuk, A.: Massey products on cycles of projective lines and trigonometric solutions of the Yang-Baxter equations. arXiv:math/0612761v3.
A. V. Rojter, Matrix problems and representations of BOCSs, Representation theory, I (Proc. Workshop, Carleton Univ., Ottawa, Ont., 1979), Lecture Notes in Math., vol. 831, Springer, Berlin-New York, 1980, pp. 288–324. MR 607144
A. V. Roĭter, Matrix problems and representations of BOCSs, Representations and quadratic forms (Russian), Akad. Nauk Ukrain. SSR, Inst. Mat., Kiev, 1979, pp. 3–38, 154 (Russian). MR 600110
Paul Seidel and Richard Thomas, Braid group actions on derived categories of coherent sheaves, Duke Math. J. 108 (2001), no. 1, 37–108. MR 1831820, DOI 10.1215/S0012-7094-01-10812-0
C. S. Seshadri, Fibrés vectoriels sur les courbes algébriques, Astérisque, vol. 96, Société Mathématique de France, Paris, 1982 (French). Notes written by J.-M. Drezet from a course at the École Normale Supérieure, June 1980. MR 699278, DOI 10.1070/IM1969v003n01ABEH000748