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

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

Ask a homework question - tutors are online