Intersecting e0 with ei we obtain that m0 0 mi 0 i 0

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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 blow-ups 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 blow-up 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 blow-up 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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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 Michigan-Dearborn.

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