Hostname: page-component-848d4c4894-hfldf Total loading time: 0 Render date: 2024-05-18T10:09:14.193Z Has data issue: false hasContentIssue false
  • English
  • Français

Kulikov surfaces form a connected component of the moduli space

Published online by Cambridge University Press:  11 January 2016

Tsz On Mario Chan  and
Stephen Coughlan
Tsz On Mario Chan
Affiliation:
Lehrstuhl Mathematik VIII, Mathematisches Institut der Universität Bayreuth, 95447 Bayreuth, Germany, mariochan@graduate.hku.hk
Stephen Coughlan
Affiliation:
Department of Mathematics and Statistics, University of Massachusetts, Amherst, Massachusetts 01003, USA, coughlan@math.umass.edu
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We show that the Kulikov surfaces form a connected component of the moduli space of surfaces of general type with pg = 0 and K2 = 6. We also give a new description for these surfaces, extending ideas of Inoue. Finally, we calculate the bicanonical degree of Kulikov surfaces and prove that they verify the Bloch conjecture.

Keywords

14J2914J1014J25
Type
Research Article
Information
Nagoya Mathematical Journal , Volume 210 , June 2013 , pp. 1 - 27
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2013

References

[AP] Alexeev, V. and Pardini, R., Explicit compactifications of moduli spaces of Campedelli and Burniat surfaces, preprint, arXiv:0901.4431 [math.AG]Google Scholar
[Ba] Barlow, R., Rational equivalence of zero cycles for some more surfaces with pg = 0, Invent. Math. 79 (1985), 303–308. MR 0778129. DOI 10.1007/BF01388975.Google Scholar
[BC1] Bauer, I. C. and Catanese, F., A volume maximizing canonical surface in 3-space, Comment. Math. Helv. 83 (2008), 387–406. MR 2390050. DOI 10.4171/CMH/129.CrossRefGoogle Scholar
[BC2] Bauer, I. C. and Catanese, F., Burniat surfaces, II: Secondary Burniat surfaces form three connected components of the moduli space, Invent. Math. 180 (2010), 559–588. MR 2609250. DOI 10.1007/s00222-010-0237-z.CrossRefGoogle Scholar
[BC3] Bauer, I. C. and Catanese, F., “Burniat surfaces, I: Fundamental groups and moduli of primary Burniat surfaces” in Classification of Algebraic Varieties, EMS Ser. Congr. Rep., Eur. Math. Soc., Zürich, 2011, 49–76. MR 2779467. DOI 10.4171/007-1/3.Google Scholar
[BC4] Bauer, I. C. and Catanese, F., Burniat surfaces, III: Deformations of automorphisms and extended Burniat surfaces, preprint, arXiv:1011.3770v1 [math.AG]Google Scholar
[BKL] Bloch, S., Kas, A., and Lieberman, D., Zero cycles on surfaces with pg = 0, Compos. Math. 33 (1976), 135–145. MR 0435073.Google Scholar
[BCP] Bosma, W., Cannon, J., and Playoust, C., “The Magma algebra system, I: The user language” in Computational Algebra and Number Theory (London, 1993), J. Symbolic Comput. 24 (1997), 235–265. MR 1484478. DOI 10.1006/jsco.1996.0125.Google Scholar
[Bu] Burniat, P., Sur les surfaces de genre P12 > 1, Ann. Mat. Pura Appl. (4) 71 (1966), 1–24. MR 0206810.Google Scholar
[C] Catanese, F., On the moduli spaces of surfaces of general type, J. Differential Geom. 19 (1984), 483–515. MR 0755236.Google Scholar
[IM] Inose, H. and Mizukami, M., Rational equivalence of 0-cycles on some surfaces of general type with pg = 0, Math. Ann. 244 (1979), 205–217. MR 0553252. DOI 10. 1007/BF01420343.Google Scholar
[I] Inoue, M., Some new surfaces of general type, Tokyo J. Math. 17 (1994), 295–319. MR 1305801. DOI 10.3836/tjm/1270127954.Google Scholar
[KSB] Kollár, J. and Shepherd-Barron, N. I., Threefolds and deformations of surface singularities, Invent. Math. 91 (1988), 299–338. MR 0922803. DOI 10.1007/BF01389370.Google Scholar
[K] Kulikov, V. S., Old examples and a new example of surfaces of general type with pg = 0 (in Russian), Izv. Ross. Akad. Nauk Ser. Mat. 68 (2004), no. 5, 123–170; English translation in Izv. Math. 68 (2004), no. 5, 965–1008. MR 2104852. DOI 10.1070/IM2004v068n05ABEH000505.Google Scholar
[MP1] Lopes, M. Mendes and Pardini, R., A connected component of the moduli space of surfaces with pg = 0, Topology 40 (2001), 977–991. MR 1860538. DOI 10.1016/S0040-9383(00)00004-5.Google Scholar
[MP2] Lopes, M. Mendes and Pardini, R., Surfaces of general type with pg = 0, K2 = 6 and non birational bicanonical map, Math. Ann. 329 (2004), 535–552. MR 2127989. DOI 10.1007/s00208-004-0524-3.CrossRefGoogle Scholar
[P] Pardini, R., Abelian covers of algebraic varieties, J. Reine Angew. Math. 417 (1991), 191–213. MR 1103912. DOI 10.1515/crll.1991.417.191.Google Scholar
[Re] Reider, I., Vector bundles of rank 2 and linear systems on algebraic surfaces, Ann. of Math. (2) 127 (1988), 309–316. MR 0932299. DOI 10.2307/2007055.Google Scholar
[X] Xiao, G., Degree of the bicanonical map of a surface of general type, Amer. J. Math. 112 (1990), 713–736. MR 1073006. DOI 10.2307/2374804.Google Scholar