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Moduli of surfaces with an anti-canonical cycle

Part of: Surfaces and higher-dimensional varieties

Published online by Cambridge University Press:  09 October 2014

Mark Gross ,
Paul Hacking  and
Sean Keel
Mark Gross
Affiliation:
DPMMS, Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WB, UK email mgross@dpmms.cam.ac.uk
Paul Hacking
Affiliation:
Department of Mathematics and Statistics, Lederle Graduate Research Tower, University of Massachusetts, Amherst, MA 01003-9305, USA email hacking@math.umass.edu
Sean Keel
Affiliation:
Department of Mathematics, 1 University Station C1200, Austin, TX 78712-0257, USA email keel@math.utexas.edu
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Abstract

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We prove a global Torelli theorem for pairs $(Y,D)$ where $Y$ is a smooth projective rational surface and $D\in |-K_{Y}|$ is a cycle of rational curves, as conjectured by Friedman in 1984. In addition, we construct natural universal families for such pairs.

Keywords

Looijenga pairsrational surfacesmoduli

MSC classification

Primary: 14J26: Rational and ruled surfaces 14J10: Families, moduli, classification
Type
Research Article
Information
Compositio Mathematica , Volume 151 , Issue 2 , February 2015 , pp. 265 - 291
Copyright
© The Author(s) 2014 

References

Abramovich, D., Corti, A. and Vistoli, A., Twisted bundles and admissible covers, Comm. Algebra 31 (2003), 3547–3618.CrossRefGoogle Scholar
Bauer, T., Küronya, A. and Szemberg, T., Zariski chambers, volumes, and stable base loci, J. Reine Angew. Math. 576 (2004), 209–233.Google Scholar
Dolgachev, I., Reflection groups in algebraic geometry, Bull. Amer. Math. Soc. (N.S.) 45 (2008), 1–60.Google Scholar
Fock, V. and Goncharov, A., Moduli spaces of local systems and higher Teichmüller theory, Publ. Math. Inst. Hautes Études Sci. 103 (2006), 1–211.Google Scholar
Fomin, S. and Zelevinsky, A., Cluster algebras I. Foundations, J. Amer. Math. Soc. 15 (2002), 497–529.Google Scholar
Friedman, R., The mixed Hodge structure of an open variety, Preprint (1984).Google Scholar
Friedman, R., On the ample cone of a rational surface with an anticanonical cycle, Algebra Number Theory 7 (2013), 1481–1504.CrossRefGoogle Scholar
Fulton, W., Introduction to toric varieties, Annals of Mathematics Studies, vol. 131 (Princeton University Press, Princeton, NJ, 1993).Google Scholar
Gross, M., Hacking, P. and Keel, S., Mirror symmetry for log CY surfaces I, Preprint (2011),arXiv:1106.4977v2.Google Scholar
Gross, M., Hacking, P. and Keel, S., Birational geometry of cluster algebras, Algebraic Geometry, to appear, arXiv:1309.2573v2.Google Scholar
Gross, M., Hacking, P. and Keel, S., Mirror symmetry for log CY surfaces II, in preparation.Google Scholar
Kawamata, Y., Matsuda, K. and Matsuki, K., Introduction to the minimal model problem, in Algebraic geometry, Sendai, 1985, Advanced Studies in Pure Mathematics, vol. 10 (North-Holland, Amsterdam, 1987), 283–360.Google Scholar
Looijenga, E., Rational surfaces with an anticanonical cycle, Ann. of Math. (2) 114 (1981), 267–322.CrossRefGoogle Scholar
Looijenga, E. and Peters, C., Torelli theorems for Kähler K3 surfaces, Compositio Math. 42 (1980), 145–186.Google Scholar
Miranda, R. and Persson, U., On extremal rational elliptic surfaces, Math. Z. 193 (1986), 537–558.CrossRefGoogle Scholar