Unformatted text preview: quadratic lattice on HX and also leaves invariant the
sublattice SX . This deﬁnes a homomorphism
(5.2) Aut(X ) → O(SX ), g → (g ∗ )−1 . 26 IGOR V. DOLGACHEV Since g ∗ (KX ) = KX , we see that the image of this homomorphism is contained
in the stabilizer subgroup O(SX )kX of the vector kX . In particular, it induces a
homomorphism
0
a : Aut(X ) → O(SX ). (5.3) The group Aut(X ) is a topological group whose connected component of the identity
Aut(X )0 is a complex Lie group. One can show that Aut(X )0 acts identically on
SX and the kernel of the induced map of the quotient group Aut(X )/Aut(X )0 is
ﬁnite (see [31]). The group Aut(X )0 can be nontrivial only for surfaces of Kodaira
dimension −∞, or abelian surfaces, or surfaces of Kodaira dimension 1 isomorphic
to some ﬁnite quotients of the products of two curves, one of which is of genus 1.
It follows from Theorem 5.1 that Aut(X ) is always ﬁnite for surfaces of Kodaira
dimension 2.
5.2. Rational surfaces. A rational surface X is a nonsingular projective algebraic
surface birationally isomorphic to P2 . We will be interested only in basic rational
surfaces, i.e. algebraic surfaces admitting a regular birational map π : X → P2 .7
It is known that any birational regular map of algebraic surfaces is equal to the
composition of blowups of points ([55]). Applying this to the map π , we obtain a
factorization
(5.4) π π N −1 π π N
2
1
π : X = XN −→ XN −1 −→ . . . −→ X1 −→ X0 = P2 , where πi : Xi → Xi−1 is the blowup of a point xi ∈ Xi−1 . Let
−
E i = π i 1 ( xi ) , (5.5) Ei = (πi+1 ◦ . . . πN )−1 (Ei ). Let ei denote the cohomology class [Ei ] of the (possibly reducible) curve Ei . It
satisﬁes e2 = ei · kX = −1. One easily checks that ei · ej = 0 if i = j . Let
i
e0 = π ∗ ([ ]), where is a line in P2 . We have e0 · ei = 0 for all i. The classes
e0 , e1 , . . . , eN form a basis in SX = HX which we call a geometric basis. The Gram
matrix of a geometric basis is the diagonal matrix diag[1, −1, . . . , −1]. Thus the
factorization (5.4) deﬁnes an isomorphism of quadratic lattices
φπ : I1,N → SX , ei → ei ,
where e0 , . . . , eN is the standard basis of I1,N . It follows from the formula for the
behavior of the canonical class under a blowup that kX is equal to the image of
the vector
kN = −3e0 + e1 + . . . + eN .
0
⊥
This implies that the lattice SX is isomorphic to the orthogonal complement kN in
I1,N .
For N ≥ 3, the vectors (5.6) a1 = e 0 − e 1 − e 2 − e 3 , a2 = e 1 − e 2 , ... , aN = e N −1 − e N ⊥
kN form a basis of
with Gram matrix equal to −2C , where C is the Gram matrix
of the Coxeter group W (EN ) := W (2, 3, N − 3) if N ≥ 4 and W (E3 ) = W (A2 × A1 )
7 For experts: Nonbasic rational surfaces are easy to describe: they are either minimal rational
surfaces diﬀerent from P2 or surfaces obtained from minimal ruled surfaces Fn , n ≥ 2, by blowing
up points on the exceptional section and their inﬁnitely near points. The automorphism groups
o...
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 Fall '04
 IgorDolgachev
 Algebra, Geometry, Vector Space, The Land, Igor V. Dolgachev, Reflection group

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