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Reflection groups in algebraic geometry. (English) Zbl 1278.14001
This is a survey article, containing few full proofs but much enlightening detail. Algebraic geometry enters directly only half-way through. The first four sections are about the general theory of reflection groups, though the topics and more especially the numerous illustrative examples are chosen with applications in algebraic geometry in mind.
The first section is a largely historical introduction. The author draws attention in particular to the fact that reflection groups appeared in algebraic geometry at a very early stage in the modern development of both subjects, in Kantor’s 1895 monograph on subgroups of the plane Cremona group. Some special cases, such as \(W(E_6)\) as the symmetries of the 27 lines on a cubic surface, are of course older still.
Section 2, on real reflection groups, describes the basic types (spherical, Euclidean, hyperbolic) and the machinery used to work with them (Coxeter diagrams, Gram matrices), and states some classification theorems for spherical and Euclidean groups and for the Coxeter groups \(W(p,q,r)\). Section 3, on linear reflection groups, gives the classification by Shephard and Todd of finite complex linear reflection groups, and briefly discusses complex crystallographic and unitary reflection groups. Section 4 is concerned with quadratic lattices and their reflection groups, Dynkin diagrams, and reflective lattices. It gives some details of examples and partial classification of reflective lattices but mainly establishes terminology and presents basic facts.
These find an immediate use in Section 5, where the algebro-geometric fun starts with a section on automorphisms of algebraic surfaces. Letting \(X\) be a projective complex algebraic surface we consider the lattice \(H_X=H^2(X,{\mathbb Z})/\mathrm{Torsion}\) with the cup product, or more often the Picard lattice \(S_X\) or the sublattice \(S_X^0\subset S_X\) orthogonal to the canonical class (which are even lattices). Enriques’ classification of surfaces is expounded in terms of the properties of \(H_X\), \(S_X\) and \(S_X^0\), and then we go on to study \(\mathrm{Aut}(X)\) via its image in \(\roman O(S_X^0)\). The cases discussed here are rational surfaces, \(K3\) surfaces and (briefly) Enriques surfaces.
The discussion of rational surfaces deals first with the classical cases of \({\mathbb P}^2\), conic bundles and del Pezzo surfaces, and the images of their automorphism groups as subgroups of \(W(E_n)\) (\(n\leq 8\)). There is also a short but intriguing discussion of infinite automorphism groups of \({\mathbb P}^2\) blown up in \(n\geq 9\) points.
The \(K3\) case is given careful attention. In this case \(\mathrm{Aut}(X)\) is finite if and only if \(S_X(-1)\) is \(2\)-reflective, and all \(2\)-reflective lattices arise in this way. But not all even hyperbolic reflective lattices do: some of them have rank \(22\). These arise as Picard lattices of supersingular \(K3\) surfaces in characteristic \(p>0\), and the author conjectures that any even hyperbolic reflective lattice \(M\) can be transformed into \(S_X(-1)\) for some \(K3\) surface \(X\) in characteristic \(p>0\) by the operations of scaling and of replacing \(M\) by \(p^{-1}(M+p^2M^*)\) or by \((M^*+p^{-1}M)(p)\).
The section (Section 6) on Cremona groups, which follows, describes Coble’s action of the Coxeter groups \({W(2,n+1,m+1)}\) on \({\mathbb C}(z_1,\dots,z_{mn})\) and Mukai’s generalisation to \(W(p,q,r)\), with examples. The author points out that many interesting finite groups can be conveniently presented as quotients of \(W(p,q,r)\) (for instance the Monster is a quotient of \(W(4,5,5)\)) and it may be possible to give geometric descriptions of those groups by this route.
Section 7, on invariants of finite complex reflection groups, is limited to indicating a few classical and modern examples of complex hypersurfaces whose symmetries are (or are close to) such groups, including Klein’s quartic curve and the Burkhardt quartic, given in \({\mathbb P}^5\) by the vanishing of the first and fourth elementary symmetric polynomials in the coordinate functions.
Section 8, on monodromy groups, deals with Picard-Lefschetz transformations and the monodromy map \(\pi_1(S,s_0)\to \mathrm{Aut}(H^n_c(X_{s_0},{\mathbb Z}))\) associated with a family over a base \(S\ni s_0\). In particular, reflection groups arise when one considers the Milnor fibration associated with an isolated hypersurface singularity of even dimension. There is some discussion of the surface case, especially rational double points and simple elliptic singularities. This leads on to Section 9 on symmetries of singularities. In the presence of a group \(G\) of symmetries of a singularity, one has the notion of \(G\)-equivariant monodromy and this leads to the appearance of more lattices, arising for example as the \(G\)-invariant parts of Milnor lattices in work by Slodowy and others.
Finally, Section 10, on complex ball quotients, describes very quickly the work of Deligne and Mostow on hypergeometric integrals and then moves on to the more recent results of Allcock, Carlson and Toledo realising the moduli space of cubic surfaces as a ball quotient. The paper concludes with a mention of some further work in this direction, and some relations with the moduli of certain \(K3\) surfaces, due to Kondo, van Geemen, the author and others.

MSC:
14-02 Research exposition (monographs, survey articles) pertaining to algebraic geometry
14E07 Birational automorphisms, Cremona group and generalizations
14H20 Singularities of curves, local rings
11H55 Quadratic forms (reduction theory, extreme forms, etc.)
20F55 Reflection and Coxeter groups (group-theoretic aspects)
51F15 Reflection groups, reflection geometries
14J28 \(K3\) surfaces and Enriques surfaces
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