29 1910 98 118 44 r friedman j morgan and e witten

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Unformatted text preview: with quotient isomorphic to a partial compactification of the moduli space of Del Pezzo surfaces of degree 1. Independently such a construction was found in [56]. Finally, one can also re-prove the result of Allcock-Carlson-Toledo by using K3 surfaces instead of intermediate jacobians (see [32]). Remark 10.3. All reflection groups arising in these complex ball uniformization constructions are not contained in the Deligne-Mostow list (corrected in [110]). However, some of them are commensurable16 with some groups from the list. For example, the group associated to cubic surfaces is commensurable to the group Γ(µ), where d = 6, m1 = m2 = 1, m3 = . . . = m7 = 2. We refer to [25] for a construction of complex reflection subgroups of finite volume which are not commensurable to the groups from the Deligne-Mostow list. No algebraic-geometrical interpretation of these groups is known so far. Acknowledgements This paper is dedicated to Ernest Borisovich Vinberg, one of the heroes of the theory of reflection groups. His lectures for high school children in Moscow were influential (without his knowledge) in my decision to become a mathematician. The paper is an expanded version of my colloquium lecture at the University di Roma Terzo in May 2006. I am thankful to Alessandro Verra for giving me an opportunity to give this talk and hence to write the paper. I am very grateful to Daniel Allcock, Victor Goryunov and the referee for numerous critical comments on earlier versions of the paper. About the author Igor Dolgachev is a professor at the University of Michigan in Ann Arbor. He has held visiting positions at the University of Paris, MIT, and Harvard University; and at Research Institutes in Bonn, Kyoto, Seoul, and Warwick. References 1. D. Allcock, The Leech lattice and complex hyperbolic reflections, Invent. Math. J. 140 (2000), 283–31. MR1756997 (2002b:11091) 2. D. Allcock, J. Carlson, and D. Toledo, The complex hyperbolic geometry of the moduli space of cubic surfaces, J. Alg. Geom. 11 (2002), 659–724. MR1910264 (2003m:32011) 3. D. Allcock, A monstrous proposal, 9 pages, math.GR/0606043. To appear in Groups and symmetries. From the Nordic Scots to John McKay, April 27–29, 2007, CRM, Montreal. 4. D. Allcock, J. Carlson, and D. Toledo, The moduli space of cubic threefolds as a ball quotient, math.AG/0608287. 16 This means that the two groups share a common subgroup of finite index. REFLECTION GROUPS IN ALGEBRAIC GEOMETRY 55 5. E. Andreev, Convex polyhedra of finite volume in Lobachevskii space (Russian), Mat. Sb. (N.S.) 83 (125) (1970), 256–260. MR0273510 (42:8388) 6. V. Arnol’d, Critical points of functions on a manifold with boundary, the simple Lie groups Bk , Ck , F4 and sequences of evolutes, Uspkhi Mat. Nauk 33 (1978), 91–105. MR511883 (80j:58008) 7. V. Arnol’d, S. Gusein-Zade, and A. Varchenko, Singularities of differentiable maps. Vol. I. Translated from the Russian. Monographs in Mathematics, 82. Birkh¨user Boston, Inc., a...
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