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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 reprove the result of AllcockCarlsonToledo 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 DeligneMostow 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 DeligneMostow list. No algebraicgeometrical
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
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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.
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