This preview shows page 1. Sign up to view the full content.
Unformatted text preview: surface , .
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 , . 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 ).
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 , ). 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
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
Next we consider the group L3 of order 648 (No. 25). The group Γ is of order 216
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.
2 It is known that any nonsingular plane cubic curve is projectively isomorphic to
View Full Document