A Gieseker type degeneration of moduli stacks of vector bundles on curves

Kausz, Ivan

doi:10.1090/S0002-9947-04-03618-9

A Gieseker type degeneration of moduli stacks of vector bundles on curves
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by Ivan Kausz PDF

Trans. Amer. Math. Soc. 357 (2005), 4897-4955

Abstract:

We construct a new degeneration of the moduli stack of vector bundles over a smooth curve when the curve degenerates to a singular curve which is irreducible with one double point. We prove that the total space of the degeneration is smooth and its special fibre is a divisor with normal crossings. Furthermore, we give a precise description of how the normalization of the special fibre of the degeneration is related to the moduli space of vector bundles over the desingularized curve.

References

Garrett Birkhoff and Morgan Ward, A characterization of Boolean algebras, Ann. of Math. (2) 40 (1939), 609–610. MR 9, DOI 10.2307/1968945
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
Arnaud Beauville and Yves Laszlo, Un lemme de descente, C. R. Acad. Sci. Paris Sér. I Math. 320 (1995), no. 3, 335–340 (French, with English and French summaries). MR 1320381
Lucien Szpiro (ed.), Séminaire sur les pinceaux arithmétiques: la conjecture de Mordell, Société Mathématique de France, Paris, 1985. Papers from the seminar held at the École Normale Supérieure, Paris, 1983–84; Astérisque No. 127 (1985) (1985). MR 801916
Gerd Faltings, Moduli-stacks for bundles on semistable curves, Math. Ann. 304 (1996), no. 3, 489–515. MR 1375622, DOI 10.1007/BF01446303
D. Gieseker, A degeneration of the moduli space of stable bundles, J. Differential Geom. 19 (1984), no. 1, 173–206. MR 739786
A. Grotendik and Ž. A. D′edonne, Elements of algebraic topology, Uspehi Mat. Nauk 27 (1972), no. 2(164), 135–148 (Russian). Translated from the French (Éléments de géométrie algébrique, I: Le langage des schemas, second edition, pp. 4–18, Springer, Berlin, 1971) by F. V. Širokov. MR 0432634
Peter Scherk, Bemerkungen zu einer Note von Besicovitch, J. London Math. Soc. 14 (1939), 185–192 (German). MR 29, DOI 10.1112/jlms/s1-14.3.185
Peter Scherk, Bemerkungen zu einer Note von Besicovitch, J. London Math. Soc. 14 (1939), 185–192 (German). MR 29, DOI 10.1112/jlms/s1-14.3.185
A. Grothendieck, Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. I, Inst. Hautes Études Sci. Publ. Math. 20 (1964), 259 (French). MR 173675
Revêtements étales et groupe fondamental, Lecture Notes in Mathematics, Vol. 224, Springer-Verlag, Berlin-New York, 1971 (French). Séminaire de Géométrie Algébrique du Bois Marie 1960–1961 (SGA 1); Dirigé par Alexandre Grothendieck. Augmenté de deux exposés de M. Raynaud. MR 0354651
Ivan Kausz, A modular compactification of the general linear group, Doc. Math. 5 (2000), 553–594. MR 1796449
I. Kausz: A canonical decomposition of generalised theta functions on the moduli stack of Gieseker vector bundles. Preprint (April 2003).
Finn F. Knudsen, The projectivity of the moduli space of stable curves. II. The stacks $M_{g,n}$, Math. Scand. 52 (1983), no. 2, 161–199. MR 702953, DOI 10.7146/math.scand.a-12001
Hideyuki Matsumura, Commutative ring theory, 2nd ed., Cambridge Studies in Advanced Mathematics, vol. 8, Cambridge University Press, Cambridge, 1989. Translated from the Japanese by M. Reid. MR 1011461
James S. Milne, Étale cohomology, Princeton Mathematical Series, No. 33, Princeton University Press, Princeton, N.J., 1980. MR 559531
Gérard Laumon and Laurent Moret-Bailly, Champs algébriques, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 39, Springer-Verlag, Berlin, 2000 (French). MR 1771927
M. S. Narasimhan and T. R. Ramadas, Factorisation of generalised theta functions. I, Invent. Math. 114 (1993), no. 3, 565–623. MR 1244913, DOI 10.1007/BF01232680
D. S. Nagaraj and C. S. Seshadri, Degenerations of the moduli spaces of vector bundles on curves. II. Generalized Gieseker moduli spaces, Proc. Indian Acad. Sci. Math. Sci. 109 (1999), no. 2, 165–201. MR 1687729, DOI 10.1007/BF02841533
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
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
C. S. Seshadri, Degenerations of the moduli spaces of vector bundles on curves, School on Algebraic Geometry (Trieste, 1999) ICTP Lect. Notes, vol. 1, Abdus Salam Int. Cent. Theoret. Phys., Trieste, 2000, pp. 205–265. MR 1795864
Christoph Sorger, La formule de Verlinde, Astérisque 237 (1996), Exp. No. 794, 3, 87–114 (French, with French summary). Séminaire Bourbaki, Vol. 1994/95. MR 1423621
Xiaotao Sun, Degeneration of moduli spaces and generalized theta functions, J. Algebraic Geom. 9 (2000), no. 3, 459–527. MR 1752012
Montserrat Teixidor i Bigas, Compactifications of moduli spaces of (semi)stable bundles on singular curves: two points of view, Collect. Math. 49 (1998), no. 2-3, 527–548. Dedicated to the memory of Fernando Serrano. MR 1677092