Nagoya Mathematical Journal

On the moduli of stable sheaves on a reducible projective scheme and examples on a reducible quadric surface

Michi-Aki Inaba

Full-text: Open access


We study the moduli space of stable sheaves on a reducible projective scheme by use of a suitable stratification of the moduli space. Each stratum is the moduli space of "triples"', which is the main object investigated in this paper. As an application, we can see that the relative moduli space of rank two stable sheaves on quadric surfaces gives a nontrivial example of the relative moduli space which is not flat over the base space.

Article information

Nagoya Math. J., Volume 166 (2002), 135-181.

First available in Project Euclid: 27 April 2005

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 14D20: Algebraic moduli problems, moduli of vector bundles {For analytic moduli problems, see 32G13}
Secondary: 14J60: Vector bundles on surfaces and higher-dimensional varieties, and their moduli [See also 14D20, 14F05, 32Lxx]


Inaba, Michi-Aki. On the moduli of stable sheaves on a reducible projective scheme and examples on a reducible quadric surface. Nagoya Math. J. 166 (2002), 135--181.

Export citation


  • A. Altman and S. Kleiman, Compactifying the Picard scheme , Adv. in Math., 35 (1980), no. 1, 50--112.
  • G. Faltings, Moduli-stacks for bundles on semistable curves , Math. Ann., 304 (1996), 489--515.
  • D. Gieseker and J. Li, Irreducibility of moduli of rank-$2$ vector bundles on Algebraic surfaces , J. Differential Geom., 40 (1994), 23--104.
  • A. Grothendieck, Éléments de géométrie algébrique, Chaps. I,II,III,IV, Inst. Hautes Études Sci. Publ. Math. No. 4,8,11,17,20,24,28,32 (1960--1967).
  • M. Inaba, Moduli of parabolic stable sheaves on a projective scheme , J. Math. Kyoto Univ., 40-1 (2000), 119--136.
  • --------, On the moduli of stable sheaves on some nonreduced projective schemes, (to appear in J. Alg. Geometry) .
  • A. Langer, Semistable sheaves in positive characteristic, (to appear) .
  • M. Maruyama, Moduli of stable sheaves, II , J. Math. Kyoto Univ., 18 (1978), no. 3, 557--614.
  • --------, Construction of moduli spaces of stable sheaves via Simpson's idea , Lecture Notes in Pure and Appl. Math. 179, Dekker, New York, (1996).
  • D. Mumford, Geometric invariant theory, Springer-Verlag, Berlin Heidelberg New York (1965).
  • D. S. Nagaraj and C. S. Seshadri, Degenerations of the moduli spaces of vector bundles on curves, I , Proc. Indian Acad. Sci., 107 (1997), no. 2, 101--137.
  • T. Oda and C. S. Seshadri, Compactification of the generalized Jacobian variety , Trans. Amer. Math. Soc., 253 (1979), 1--90.
  • C. Okonek, M. Schneider, H. Spindler, Vector bundles on complex projective spaces, Birkhäuser, Boston (1980).
  • A. N. Rudakov, A description of Chern classes of semistable sheaves on a quadric surface , J. Reine Angew. Math., 453 (1994), 113--135.
  • C. S. Seshadri, Geometric reductivity over arbitrary base , Adv. in Math., 26 (1977), no. 3, 225--274.
  • --------, Fibrés vectoriels sur les courbes algébriques, Astérisque, 96, Société Mathématique de France, Paris, 1982.
  • C. T. Simpson, Moduli of representations of the fundamental group of a smooth projective variety I , Inst. Hautes Études Sci., 79 (1994), 47--129.
  • S. Soberon-Chavez, Rank $2$ vector bundles over a complex quadric surface , Quart. J. Math. Oxford (2), 36 (1985), 159--172.
  • K. Yokogawa, Compactification of moduli of parabolic sheaves and moduli of parabolic Higgs sheaves , J. Math. Kyoto Univ., 33 (1993), no. 2, 451--504.
  • H. Xia, Degenerations of moduli of stable bundles over algebraic curves , Compos. Math., 98 (1995), 305--330.