Singular principal bundles on reducible nodal curves

Muñoz Castañeda, Ángel; Schmitt, Alexander

doi:10.1090/tran/8464

Singular principal bundles on reducible nodal curves
HTML articles powered by AMS MathViewer

by Ángel Luis Muñoz Castañeda and Alexander H. W. Schmitt PDF

Trans. Amer. Math. Soc. 374 (2021), 8639-8660 Request permission

Abstract:

Studying degenerations of moduli spaces of semistable principal bundles on smooth curves leads to the problem of studying moduli spaces on singular curves. In this note, we will see that the moduli spaces of $\delta$-semistable pseudo bundles on a nodal curve become, for large values of $\delta$, the moduli spaces of semistable singular principal bundles. The latter are reasonable candidates for degenerations and a potential basis for further developments as on irreducible nodal curves. In particular, we find a notion of semistability for principal bundles on reducible nodal curves. The understanding of the asymptotic behavior of $\delta$-semistability rests on a lemma from geometric invariant theory. The results allow for the construction of a universal moduli space of semistable singular principal bundles over the moduli space of stable curves. Due to recent work of Wilson, this universal moduli space has a close relation to the sheaf of algebras of conformal blocks.

References

V. Balaji, Torsors on semistable curves and degenerations, Proc. Indian Acad. Sci. (to appear).
E. Ballico, Stable sheaves on reduced projective curves, Ann. Mat. Pura Appl. (4) 175 (1998), 375–393. MR 1748234, DOI 10.1007/BF01783694
Arnaud Beauville and Yves Laszlo, Conformal blocks and generalized theta functions, Comm. Math. Phys. 164 (1994), no. 2, 385–419. MR 1289330
P. Belkale and A. Gibney, On finite generation of the section ring of the determinant of cohomology line bundle, Trans. Amer. Math. Soc. 371 (2019), no. 10, 7199–7242. MR 3939575, DOI 10.1090/tran/7564
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, Vector bundles on curves with many components, Proc. London Math. Soc. (3) 79 (1999), no. 1, 81–106. MR 1687547, DOI 10.1112/S0024611599011855
Usha N. Bhosle, Tensor fields and singular principal bundles, Int. Math. Res. Not. 57 (2004), 3057–3077. MR 2098029, DOI 10.1155/S1073792804133114
Lucia Caporaso, A compactification of the universal Picard variety over the moduli space of stable curves, J. Amer. Math. Soc. 7 (1994), no. 3, 589–660. MR 1254134, DOI 10.1090/S0894-0347-1994-1254134-8
P. R. Cook, Local and global aspects of the module theory of singular curves. PhD thesis, Liverpool, 1993, 129 pp.
P. Deligne and D. Mumford, The irreducibility of the space of curves of given genus, Inst. Hautes Études Sci. Publ. Math. 36 (1969), 75–109. MR 262240
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, Lectures on moduli of curves, Tata Institute of Fundamental Research Lectures on Mathematics and Physics, vol. 69, Published for the Tata Institute of Fundamental Research, Bombay by Springer-Verlag, Berlin-New York, 1982. MR 691308
D. Gieseker, A degeneration of the moduli space of stable bundles, J. Differential Geom. 19 (1984), no. 1, 173–206. MR 739786
T. L. Gómez, A. Langer, A. H. W. Schmitt, and I. Sols, Moduli spaces for principal bundles in arbitrary characteristic, Adv. Math. 219 (2008), no. 4, 1177–1245. MR 2450609, DOI 10.1016/j.aim.2008.05.015
George R. Kempf, Instability in invariant theory, Ann. of Math. (2) 108 (1978), no. 2, 299–316. MR 506989, DOI 10.2307/1971168
Shrawan Kumar, M. S. Narasimhan, and A. Ramanathan, Infinite Grassmannians and moduli spaces of $G$-bundles, Math. Ann. 300 (1994), no. 1, 41–75. MR 1289830, DOI 10.1007/BF01450475
Adrian Langer, Moduli spaces of principal bundles on singular varieties, Kyoto J. Math. 53 (2013), no. 1, 3–23. MR 3049305, DOI 10.1215/21562261-1966053
Christopher Manon, The algebra of conformal blocks, J. Eur. Math. Soc. (JEMS) 20 (2018), no. 11, 2685–2715. MR 3861806, DOI 10.4171/JEMS/822
A. L. Muñoz Castañeda, Principal $G$-bundles on nodal curves. PhD thesis, Berlin, 2017, ii+158 pp. Available at http://www.diss.fu-berlin.de/diss/content/below/index.xml.
Ángel Luis Muñoz Castañeda, On the moduli spaces of singular principal bundles on stable curves, Adv. Geom. 20 (2020), no. 4, 573–584. MR 4160290, DOI 10.1515/advgeom-2020-0003
A. L. Muñoz Castañeda, A compactification of the universal moduli space of principal $G$-bundles. Med. J. Math (to appear).
A. L. Muñoz Castañeda, Generalized parabolic structures over smooth curves with many components and principal bundles on reducible nodal curves. arXiv:1910.13403, 27 pp., 2019
Rahul Pandharipande, A compactification over $\overline {M}_g$ of the universal moduli space of slope-semistable vector bundles, J. Amer. Math. Soc. 9 (1996), no. 2, 425–471. MR 1308406, DOI 10.1090/S0894-0347-96-00173-7
A. Ramanathan, Moduli for principal bundles over algebraic curves. II, Proc. Indian Acad. Sci. Math. Sci. 106 (1996), no. 4, 421–449. MR 1425616, DOI 10.1007/BF02837697
Alexander H. W. Schmitt, Singular principal bundles over higher-dimensional manifolds and their moduli spaces, Int. Math. Res. Not. 23 (2002), 1183–1209. MR 1903952, DOI 10.1155/S1073792802107069
Alexander H. W. Schmitt, A closer look at semistability for singular principal bundles, Int. Math. Res. Not. 62 (2004), 3327–3366. MR 2097106, DOI 10.1155/S1073792804132984
Alexander Schmitt, A universal construction for moduli spaces of decorated vector bundles over curves, Transform. Groups 9 (2004), no. 2, 167–209. MR 2056535, DOI 10.1007/s00031-004-7010-6
Alexander Schmitt, The Hilbert compactification of the universal moduli space of semistable vector bundles over smooth curves, J. Differential Geom. 66 (2004), no. 2, 169–209. MR 2106123
Alexander H. W. Schmitt, Singular principal $G$-bundles on nodal curves, J. Eur. Math. Soc. (JEMS) 7 (2005), no. 2, 215–251. MR 2127994, DOI 10.4171/JEMS/27
Alexander H. W. Schmitt, Moduli spaces for semistable honest singular principal bundles on a nodal curve which are compatible with degeneration. A remark on U. N. Bhosle’s paper: “Tensor fields and singular principal bundles” [Int. Math. Res. Not. 2004, no. 57, 3057–3077; MR2098029], Int. Math. Res. Not. 23 (2005), 1427–1437. MR 2152237, DOI 10.1155/IMRN.2005.1427
Alexander Schmitt, Moduli for decorated tuples of sheaves and representation spaces for quivers, Proc. Indian Acad. Sci. Math. Sci. 115 (2005), no. 1, 15–49. MR 2120597, DOI 10.1007/BF02829837
Alexander H. W. Schmitt, Geometric invariant theory and decorated principal bundles, Zurich Lectures in Advanced Mathematics, European Mathematical Society (EMS), Zürich, 2008. MR 2437660, DOI 10.4171/065
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, Moduli spaces of torsion free sheaves on nodal curves and generalisations. I, Moduli spaces and vector bundles, London Math. Soc. Lecture Note Ser., vol. 359, Cambridge Univ. Press, Cambridge, 2009, pp. 484–505. MR 2543257
P. Solis, A complete degeneration of the moduli of $G$-bundles on a curve. arXiv:1311.6847, 22 pp., 2013
Montserrat Teixidor I. Bigas, Existence of coherent systems, Internat. J. Math. 19 (2008), no. 4, 449–454. MR 2416725, DOI 10.1142/S0129167X08004777
Wolmer V. Vasconcelos, Reflexive modules over Gorenstein rings, Proc. Amer. Math. Soc. 19 (1968), 1349–1355. MR 237480, DOI 10.1090/S0002-9939-1968-0237480-2
A. Wilson, Compactifications of moduli of $G$-bundles and conformal blocks. arXiv:2104.07549, 23 pp., 2021