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Toward moduli of singular varieties. (English) Zbl 0666.14003

Summary: The paper under review is part of the author’s thesis devoted to the study of a problem of the theory of moduli of algebraic varieties. Main result:
Theorem. Let us consider families of polarized varieties \((V,X)\) with fixed Hilbert polynomial. Then the coarse moduli space of moduli of such objects exists as a separable algebraic space of finite type in the following cases:
(i) irregular, normal, non-ruled surfaces;
(ii) non-ruled surfaces with \(q>0\) or with \(p_ g>0\) and only with rational singularities;
(iii) non-ruled surfaces with only minimal singularities;
(iv) irregular, normal, non-uniruled, 3-dimensional varieties with only isolated singularities;
(v) non-uniruled 3-dimensional varieties with only minimal singularities;
(vi) 3-dimensional varieties with \(p_ g>0\) and only rational singularities;
(vii) non-uniruled smooth varieties.

MSC:

14D20 Algebraic moduli problems, moduli of vector bundles
14J10 Families, moduli, classification: algebraic theory
14J17 Singularities of surfaces or higher-dimensional varieties
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References:

[1] Artin, M. : On isolated rational singularities of surfaces . Amer. J. Math. 88 (1966) 129-136. · Zbl 0142.18602
[2] Artin, M. : Algebraic approximation of structures over complete local rings . Publ. Math. IHES 36 (1969) 23-58. · Zbl 0181.48802
[3] Artin, M. : Versal deformations and algebraic stacks . Inv. Math. 27 (1974) 165-189. · Zbl 0317.14001
[4] Buchweitz, R.-O. , Greuel, G. - M.: The Milnor number and deformations of complex curve singularities . Inv. Math. 58 (1980) 241-281. · Zbl 0458.32014
[5] Davis, E.D. : Geometric interpretation of seminormality . Proc. A.M.S. 68 (1978) 1-5. · Zbl 0386.13012
[6] Elkik, R. : Singularités rationelles et déformations . Inv. Math. 47 (1978) 139-147. · Zbl 0363.14002
[7] Fujiki, A. : Deformation of uni-ruled manifolds . Publ. Res. Inst. Math. Sci. 17 (1981) 687-702. · Zbl 0509.32011
[8] Grauert, H. , Riemenschneider, O. : Verschwindungssätze für analytische Kohomologiegruppen auf Komplexen Räumen . Inv. Math. 11 (1970) 263-292. · Zbl 0202.07602
[9] Grothendieck, A. : Eléments de géométrie algébrique , Publ. Math. IHES 4, 8, 11, 17, 20, 24, 28, 32. |
[10] Grothendieck, A. : Cohomologie Locale des Faisceaux Cohérents et Théorèmes de Lefschetz Locaux et Globaux . North Holland (1968).
[11] Grothendieck, A. : Technique de descente et théorèmes d’existence en géométrie algébrique. IV. Les schémas de Hilbert. Sem. Bourbaki #221. · Zbl 0238.14014
[12] Grothendieck, A. : Technique de descente et théorèmes d’existence en géométric algébrique. VI . Les schémas de Picard, #236. · Zbl 0238.14014
[13] Grothendieck, A. : Local Cohomology, LN41 . Springer (1967). · Zbl 0185.49202
[14] Hartshorne, R. : Complete intersections and connectedness . Amer. J. Math. 84 (1962) 497-508. · Zbl 0108.16602
[15] Karras, U. : Deformations of cusp singularities , Proc. Symp Pure Math, XXX Several Complex Variables (1970) 37-44. · Zbl 0352.14007
[16] Koizumi, S. : On specializations of the Albanese and Picard varieties . Mem. Coll. Sci. Univ. Kyoto, Ser. A 32 (1960) 371-382. · Zbl 0107.14802
[17] Kollár, J. , Matsusaka, T. : Riemann-Roch type inequalities . Amer. J. Math. 105 (1983) 229-252. · Zbl 0538.14006
[18] Levine, M. : Deformations of uni-ruled varieties . Duke Math. J. 48 (1981) 467-473. · Zbl 0485.14011
[19] Levine, M. : Some examples from the deformation theory of ruled varieties . Amer. J. Math. 103 (1981) 997-1020. · Zbl 0494.14002
[20] Lieberman, D. , Mumford, D. : Matsusaka’s big Theorem , Proc. Symp. Pure Math 29 (1973) 513-530. · Zbl 0321.14004
[21] Matsumura, H. : Commutative Algebra , 2nd edition. Benjamin/Cummings (1980). · Zbl 0441.13001
[22] Matsusaka, T. : On canonically polarized varieties . II. Amer. J. Math. 92 (1970) 283-292. · Zbl 0195.22802
[23] Matsusaka, T. : Algebraic deformations of polarized varieties . Nagoya Math J. 31 (1968) 185-245. · Zbl 0167.49501
[24] Matsusaka, T. : On polarized varieties of dimension 3. I-II-III . Amer. J. Math. 101 (1979) 212-232, 102 (1980) 357-376, 103 (1981) 449-454. · Zbl 0493.14021
[25] Matsusaka, T. : On deformations of polarized normal varieties . To appear. · Zbl 0167.49501
[26] Matsusaka, T. : Polarized varieties with a given Hilbert polynomial . Amer. J. Math. 94 (1972) 1027-1077. · Zbl 0256.14004
[27] Matsusaka, T. , Mumford, D. : Two fundamental theorems on deformations of polarized varieties . Amer. J. Math. 86 (1964) 668-684. · Zbl 0128.15505
[28] Mumford, D. : Stability of projective varieties . L’Enseignement Math. 23 (1977) 39-110. · Zbl 0363.14003
[29] Mumford, D. , Fogarthy, J. : Geometric Invariant Theory , 2nd Edition. Springer (1982). · Zbl 0504.14008
[30] Popp, H. : Moduli Theory and Classification Theory of Algebraic Varieties , LN 620. Springer (1977). · Zbl 0359.14005
[31] Reid, M. : Canonical 3-folds , Proc. Conf. Alg. Geom. Angers (1979). A. Beauville (ed.), Sijthoff and Nordhoff, 273-310. · Zbl 0451.14014
[32] Saito, K. : Einfach-elliptische Singularitäten . Inv. Math. 23 (1974) 289-325. · Zbl 0296.14019
[33] Sally, J.D. : On the associated graded ring of a local Cohen-Macaulay ring . J. Math. Kyoto Univ. 17 (1977) 19-21. · Zbl 0353.13017
[34] Schenzel, P. : Über die freien Auflösungen extramaler Cohen-Macaulay-Ringe . J. Alg. 64 (1980) 93-101. · Zbl 0449.13008
[35] Shepherd-Barron, N.I. : Some questions on singularities in 2 and 3 dimensions. Thesis , Univ. of Warwick (1980), unpublished.
[36] Serre, J.-P. : Groupes Algébriques et corps de classes Hermann (1959). · Zbl 0097.35604
[37] Shah, J. : Insignificant limit singularities of surfaces and their mixed Hodge structure . Ann. of Math. 109 (1979) 497-536. · Zbl 0414.14022
[38] Traverso, C. : Seminormality and Picard Group . Ann. Sc. Norm. Pisa 25 (1970) 585-595. · Zbl 0205.50501
[39] Xambó, S. : On projective varieties of minimal degree . Collectanea Math. 32 (1981) 149-163. · Zbl 0501.14020
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