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Unformatted text preview: surface [20], [24]. The inertia subgroups of finite index of W (p, q, r ) defined in these examples have the quotient groups isomorphic to simple groups O(8, F2 )+ , Sp(6, F2 ), O(10, F2 )+ and Sp(8, F2 ), respectively. Remark 6.4. It is popular in group theory to represent a sporadic simple group or a related group as a finite quotient of a Coxeter group W (p, q, r ). For example, the Monster group F1 is a quotient of W (4, 5, 5). The Bimonster group F1 2 is a quotient of W (6, 6, 6) by a single relation [61], [88]. Is there a geometric interpretation of these presentations in terms of the Cremona action of W (p, q, r ) on some special subset of points in Xp,q,r ? Mukai’s construction should relate the Monster group with some special configurations of 10 points in (P3 )4 or 9 points in (P4 )4 . The Bimonster group could be related to special configurations of 12 points in (P5 )5 (see related speculations in [3]). 7. Invariants of finite complex reflection groups Let Γ ⊂ GL(n + 1, C) be a finite linear complex reflection group in Cn+1 and ¯ let Γ be its image in PGL(n, C). The reflection hyperplanes of G define a set of hyperplanes in Pn and the zeroes of G-invariant polynomials define hypersurfaces in Pn . The geometry, algebra, combinatorics and topology of arrangements of reflection hyperplanes of finite complex reflection groups is a popular area in the theory of hyperplane arrangements (see [89], [90]). On the other hand, classical algebraic geometry is full of interesting examples of projective hypersurfaces whose symmetries are described in terms of a complex reflection group. We discuss only a few examples. We begin with the group J3 (4) of order 336 (No. 24 in the list). It has funda¯ mental invariants of degrees 4, 6 and 14. Its center is of order 2, and the group Γ 10 A pseudo-automorphism is a birational transformation which is an isomorphism outside a closed subset of codimension > 2. 40 IGOR V. DOLGACHEV is a simple group of order 168 isomorphic to PSL(2, F7 ). The invariant curve of degree 4 is of course the famous Klein quartic, which is projectively equivalent to the curve 3 3 3 F4 = T0 T1 + T1 T2 + T2 T0 = 0. There are 21 reflection hyperplanes in P2 . They intersect the curve at 84 points, forming an orbit with stabilizer subgroups of order 2. The invariant F6 of degree 6 defines a nonsingular curve of degree 6, the Hessian curve of the Klein quartic. Its equation is given by the Hesse determinant of second partial derivatives of F4 . The double cover of P2 branched along the curve F6 = 0 is a K3 surface X . The automorphism group of X is an infinite group which contains a subgroup isomorphic ¯ to Γ. ¯ Next we consider the group L3 of order 648 (No. 25). The group Γ is of order 216 11 and is known as the Hessian group. It is isomorphic to the group of projective transformations leaving invariant the Hesse pencil of plane cubic curves, (7.1) λ(t3 + t3 + t3 ) + µt0 t1 t2 = 0. 0 1 2 It is known that any nonsingular plane cubic curve is projectively isomorphic to o...
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