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Unformatted text preview: from a regular
action of the Coxeter group on any Zariski open subset of (Pn )m .
The following result of S. Mukai [82] extends the Cremona action to any group
W (p, q, r ).
Theorem 6.3. Let X = (Pq−1 )p−1 . Consider the natural diagonal action of the
group PGL(q, C)p−1 on X and extend it to the diagonal action on Xp,q,r := X q+r .
Let K (p, q, r ) be the ﬁeld of invariant rational functions on Xp,q,r isomorphic to
C(t1 , . . . , td ), d = (p − 1)(q − 1)(r − 1). Then there is a natural homomorphism,
crp,q,r : W (p, q, r ) → AutC (K (p, q, r )).
It coincides with the Coble action when p = 2.
It seems that the homomorphism crp,q,r is always injective when the group is
inﬁnite. The geometric meaning of the kernels in the case of ﬁnite groups W (2, q, r )
are discussed in [36] and [39].
The reﬂections corresponding to the vertices on the branches of the Tp,q,r diagram with q and r vertices act by permuting the factors of X . The reﬂections
corresponding to p − 2 last vertices of the pbranch permute the factors of X . The
second vertex on the pbranch acts via the Cremona transformation in (Pq−1 )p−1 ,
(1) (1) (p−1) (x0 , . . . , xq−1), . . . , (x0 (p−1) , . . . , x q −1 ) → ( 1
(1)
x0 ,..., 1
(1)
xq−1 (p−1) (p−1) ), . . . , ( x0 (1)
x0 ,..., xq−1
(1) xq−1 ). Let Y (p, q, r ) be a birational model of the ﬁeld K (p, q, r ) on which W (p, q, r ) acts
birationally via cpqr . For any g ∈ W (p, q, r ) let dom(g ) be the domain of deﬁnition
of g . Let Z be a closed irreducible subset of Y (p, q, r ) with generic point ηZ . Let
GZ = {g ∈ W (p, q, r ) : ηZ ∈ dom(g ) ∩ dom(g −1 ), g (ηZ ) = ηZ } REFLECTION GROUPS IN ALGEBRAIC GEOMETRY 39 be the decomposition subgroup of Z in W (p, q, r ) and Gi be the inertia subgroup
Z
of Z , the kernel of the natural map GZ → Aut(R(Z )), where R(Z ) is the ﬁeld of
rational functions of Z . These groups were introduced for any group of birational
transformations by M. Gizatullin [47]. Deﬁne a PGL(q, C)p−1 invariant closed irreducible subset S of Xp,q,r to be special if it deﬁnes a closed subset Z on some
birational model Y (p, q, r ) such that GZ = W (p, q, r ) and Gi is a subgroup of ﬁnite
Z
index of W (p, q, r ).
Consider a general point s ∈ S as a set of q + r distinct points in (Pq−1 )p−1 and
let V (s) → (Pq−1 )p−1 be the blowup of this set. One can show that the group GZ
is realized as a group of pseudoautomorphisms of V (s).10
I know only a few examples of special subsets when W (p, q, r ) is inﬁnite. Here
are some examples.
• (p, q, r ) = (2, 3, 6), S parametrizes ordered sets of base points of a pencil of
plane cubic curves in P2 [20], [36];
• (p, q, r ) = (2, 4, 4), S parametrizes ordered sets of base points of a net of
quadrics in P3 [20], [36];
• (p, q, r ) = (2, 3, 7), S parametrizes ordered sets of double points of a rational
plane sextic [20], [36];
• (p, q, r ) = (2, 4, 6), S parametrizes ordered sets of double points of a quartic
symmetroid...
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This note was uploaded on 02/24/2012 for the course MATH 285 taught by Professor Igordolgachev during the Fall '04 term at University of MichiganDearborn.
 Fall '04
 IgorDolgachev
 Algebra, Geometry, The Land

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