On the moduli of stable sheaves on some nonreduced projective schemes
Author:
Michi-aki Inaba
Journal:
J. Algebraic Geom. 13 (2004), 1-27
DOI:
https://doi.org/10.1090/S1056-3911-03-00333-3
Published electronically:
August 26, 2003
MathSciNet review:
2008714
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Additional Information
Abstract: We study the moduli space of stable sheaves on a projective scheme whose structure sheaf has a nilpotent ideal with some property. We introduce a stratification on this moduli space. Each stratum is the moduli space of some extensions of sheaves. This stratification is described on a curve with multiple structure and on a double plane, and the structure of each stratum is studied. In the case of a curve with multiple structure, we also study a local structure of the moduli space of stable sheaves.
altmanA. Altman and S. Kleiman, Compactifying the Picard scheme, Adv. in Math. 35 (1980), no. 1, 50-112.
faltingsG. Faltings, Moduli-stacks for bundles on semistable curves, Math. Ann. 304 (1996) 489-515.
giesecker-liD. Gieseker and J. Li, Irreducibility of moduli of rank-$2$ vector bundles on algebraic surfaces, J. Differential Geometry 40 (1994), 23-104.
grothendieckA. Grothendieck, Éléments de géométrie algébrique, Chaps. I,II,III,IV, Inst. Hautes Études Sci. Publ. Math. Nos. 4,8,11,17,20,24,28,32 (1960-1967).
inabaM. Inaba, 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.
kawai-yoshiokaT. Kawai and K. Yoshioka, String partition functions and infinite products, Adv. Theor. Math. Phys. 4 (2000), no. 2, 397-485.
langeH. Lange, Universal families of extensions, J. Algebra 83 (1983), 101-112.
langerA. Langer, Semistable sheaves in positive characteristic, (to appear).
lepotier1J. Le Potier, Faisceaux semi-stables de dimension 1 sur le plan projectif, Rev. Roumaine Math. Pures Appl. 38 (1993) no. 7-8, 635-678.
lepotier2J. Le Potier, Faisceaux semi-stables et systèmes cohérents, Vector bundles in algebraic geometry, 179-239, London Math. Soc. Lecture Note Ser. 208, Cambridge Univ. Press, Cambridge, (1995).
lepotier3J. Le Potier, Systèmes cohérents et structures de niveau, Asterisque 214 (1993).
maruyama1M. Maruyama, Moduli of stable sheaves, II, J. Math. Kyoto Univ. 18 (1978), no. 3, 557-614.
maruyama2M. Maruyama, Construction of moduli spaces of stable sheaves via Simpson’s idea, Lecture Notes in Pure and Appl. Math. 179. Dekker, New York, (1996).
mumford 1D. Mumford, Geometric invariant theory, Springer-Verlag, Berlin Heidelberg New York (1965).
mumford 2D. Mumford, Lectures on curves on an algebraic surface, Annals of Math. Studies No. 59, Princeton Univ. Press, (1966).
nagarajD. 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.
nusler-trautmannT. Nüßler and G. Trautmann, Multiple Koszul structres on lines and instanton bundles, Internat. J. Math. 5 (1994), no. 3, 373-388.
oda-seshadriT. Oda and C. S. Seshadri, Compactification of the generalized Jacobian variety. Trans. Am. Math. Soc. 253 (1979) 1-90.
ossC. Okonek, M. Schneider and H. Spindler, Vector bundles on complex projective spaces, Birkhäuser, Boston (1980).
seshadri1C. S. Seshadri, Geometric reductivity over arbitrary base, Advances in Math. 26 (1977), no. 3, 225-274.
seshadri2C. S. Seshadri, Fibrés vectoriels sur les courbes algébriques, Asterisque 96. Société Mathématique de France, Paris, (1982).
simpsonC. T. Simpson, Moduli of representations of the fundamental group of a smooth projective variety I, Inst. Hautes Études Sci Publ. Math. No. 79 (1994), 47-129.
soberonS. Soberon-Chavez, Rank $2$ vector bundles over a complex quadric surface, Quart. J. Math. Oxford Ser. (2), 36 (1985), 159-172.
xiaH. Xia, Degenerations of moduli of stable bundles over algebraic curves, Compos. Math. 98 (1995) 305-330.
altmanA. Altman and S. Kleiman, Compactifying the Picard scheme, Adv. in Math. 35 (1980), no. 1, 50-112.
faltingsG. Faltings, Moduli-stacks for bundles on semistable curves, Math. Ann. 304 (1996) 489-515.
giesecker-liD. Gieseker and J. Li, Irreducibility of moduli of rank-$2$ vector bundles on algebraic surfaces, J. Differential Geometry 40 (1994), 23-104.
grothendieckA. Grothendieck, Éléments de géométrie algébrique, Chaps. I,II,III,IV, Inst. Hautes Études Sci. Publ. Math. Nos. 4,8,11,17,20,24,28,32 (1960-1967).
inabaM. Inaba, 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.
kawai-yoshiokaT. Kawai and K. Yoshioka, String partition functions and infinite products, Adv. Theor. Math. Phys. 4 (2000), no. 2, 397-485.
langeH. Lange, Universal families of extensions, J. Algebra 83 (1983), 101-112.
langerA. Langer, Semistable sheaves in positive characteristic, (to appear).
lepotier1J. Le Potier, Faisceaux semi-stables de dimension 1 sur le plan projectif, Rev. Roumaine Math. Pures Appl. 38 (1993) no. 7-8, 635-678.
lepotier2J. Le Potier, Faisceaux semi-stables et systèmes cohérents, Vector bundles in algebraic geometry, 179-239, London Math. Soc. Lecture Note Ser. 208, Cambridge Univ. Press, Cambridge, (1995).
lepotier3J. Le Potier, Systèmes cohérents et structures de niveau, Asterisque 214 (1993).
maruyama1M. Maruyama, Moduli of stable sheaves, II, J. Math. Kyoto Univ. 18 (1978), no. 3, 557-614.
maruyama2M. Maruyama, Construction of moduli spaces of stable sheaves via Simpson’s idea, Lecture Notes in Pure and Appl. Math. 179. Dekker, New York, (1996).
mumford 1D. Mumford, Geometric invariant theory, Springer-Verlag, Berlin Heidelberg New York (1965).
mumford 2D. Mumford, Lectures on curves on an algebraic surface, Annals of Math. Studies No. 59, Princeton Univ. Press, (1966).
nagarajD. 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.
nusler-trautmannT. Nüßler and G. Trautmann, Multiple Koszul structres on lines and instanton bundles, Internat. J. Math. 5 (1994), no. 3, 373-388.
oda-seshadriT. Oda and C. S. Seshadri, Compactification of the generalized Jacobian variety. Trans. Am. Math. Soc. 253 (1979) 1-90.
ossC. Okonek, M. Schneider and H. Spindler, Vector bundles on complex projective spaces, Birkhäuser, Boston (1980).
seshadri1C. S. Seshadri, Geometric reductivity over arbitrary base, Advances in Math. 26 (1977), no. 3, 225-274.
seshadri2C. S. Seshadri, Fibrés vectoriels sur les courbes algébriques, Asterisque 96. Société Mathématique de France, Paris, (1982).
simpsonC. T. Simpson, Moduli of representations of the fundamental group of a smooth projective variety I, Inst. Hautes Études Sci Publ. Math. No. 79 (1994), 47-129.
soberonS. Soberon-Chavez, Rank $2$ vector bundles over a complex quadric surface, Quart. J. Math. Oxford Ser. (2), 36 (1985), 159-172.
xiaH. Xia, Degenerations of moduli of stable bundles over algebraic curves, Compos. Math. 98 (1995) 305-330.
Additional Information
Michi-aki Inaba
Affiliation:
Department of Mathematics, Faculty of Science, Kyoto University, Kyoto 606-8502, Japan
Address at time of publication:
Graduate School of Mathematics, Kyushu University 33, Fukuoka 812-8581, Japan
Email:
inaba@kusm.kyoto-u.ac.jp, inaba@math.kyushu-u.ac.jp
Received by editor(s):
June 27, 2000
Received by editor(s) in revised form:
October 21, 2001
Published electronically:
August 26, 2003
