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Unformatted text preview: ne of the curves in the pencil. The base points of the pencil (i.e. points common
to all curves from the pencil) are inﬂection points of each nonsingular member from
the pencil. The singular members of the pencil correspond to the values of the parameters (λ, µ) = (0, 1), (1, −3), (1, −3e2πi/3 ), (1, −3e−2πi/3 ). The corresponding
cubic curves are the unions of 3 lines; all together we get 12 lines which form the
12 reﬂection hyperplanes.
The smallest degree invariant of L3 in C4 is a polynomial of degree 6:
F6 = T0 + T1 + T2 − 10(T0 T1 + T0 T2 + T1 T2 ). The double cover of P2 branched along the curve F6 = 0 is a K3 surface. Its group of
automorphisms is an inﬁnite group containing a subgroup isomorphic to the Hesse
Next we turn our attention to complex reﬂection groups of types K5 , L4 and
E6 . Their orders are all divisible by 6! · 36 = 25, 920 equal to the order of the
simple group PSp(4, F3 ). The group of type E6 (the Weyl group of the lattice E6 )
contains this group as a subgroup of index 2, which consists of words in fundamental
reﬂections of even length. The group K5 (No. 35) is the direct product of PSp(4, F3 )
and a group of order 2. The group L4 (No. 32) is the direct product of a group of
order 3 and Sp(4, F3 ).
Let Γ be of type K5 . It acts in P4 with 45 reﬂection hyperplanes. The hypersurface deﬁned by its invariant of degree 4 is isomorphic to the Burkhardt quartic
in P4 . Its equation can be given in more symmetric form in P5 :
5 5 Ti4 = 0. Ti =
i=0 i=0 ¯
These equations exhibit the action of the symmetric group S6 contained in Γ. It is
easy to see that the hypersurface has 45 ordinary double points. This is a record
for hypersurfaces of degree 4 in P4 , and this property characterizes Burkhardt
quartics. The dual representation of Γ is a linear reﬂection representation too; it is
obtained from the original one by composing it with an exterior automorphism of
11 Not to be confused with the Hesse group related to 28 bitangents of a plane quartic. REFLECTION GROUPS IN ALGEBRAIC GEOMETRY 41 the group. The double points correspond to reﬂection hyperplanes in the dual space.
The reﬂection hyperplanes in the original space cut out the Burkhardt quartic in
special quartic surfaces (classically known as desmic quartic surfaces ). They are
birationally isomorphic to the Kummer surface of the product of an elliptic curve
Finally note that the Burkhardt quartic is a compactiﬁcation of the moduli space
of principally polarized abelian surfaces with level 3 structure. All of this and much
much more can be found in .
Let Γ be of type L4 . The number of reﬂection hyperplanes is 40. The stabilizer
subgroup of each hyperplane is the group L3 from above. The smallest invariant
is of degree 12. The corresponding hypersurface cuts out in each reﬂection hyperplane the 12 reﬂection lines of the Hessian group. Again for more of this beautiful
geometry we refer to Hunt’s book, in which also the geometry of the Weyl group
W (E6 ) in P5 is fully discussed. He c...
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