# We will be interested only in basic rational surfaces

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Unformatted text preview: itely many singular points, then this number is equal to 2g0 − 2 + 2δ , where g0 is the genus of a nonsingular model of C (e.g. the normalization) and δ depends on the nature of singular points of C (e.g. equal to their number if all singular points are ordinary nodes or cusps). This implies that pa (x) is always even. In particular, the sublattice (5.1) 0 ⊥ SX = kX = {x ∈ SX : x · kX = 0} REFLECTION GROUPS IN ALGEBRAIC GEOMETRY is always even. Its signature is equal to ⎧ ⎪(1, ρ − 1) ⎪ ⎪ ⎨(0, ρ − 2, 1) 0 sign(SX ) = ⎪(0, ρ − 1) ⎪ ⎪ ⎩ (1, ρ − 2) if if if if kX 2 kX 2 kX 2 kX 25 = 0, = 0, kX = 0, > 0, < 0. A (−1)-curve (resp. (−2)-curve ) on X is a nonsingular irreducible curve C of genus 0 (thus isomorphic to P1 (C)) with [C ]2 = −1 (resp. −2). By the adjunction formula this is equivalent to C ∼ P1 (C) and [C ] · KX = −1 (resp. 0). A (−1)-curve appears = as a ﬁbre of a blow-up map f : X → Y of a point on a nonsingular algebraic surface Y . A (−2)-curve appears as a ﬁbre of a resolution of an ordinary double point on a complex surface Y . A surface X which does not contain (−1)-curves is called minimal. The following results follow from the Enriques-Kodaira classiﬁcation of complex algebraic surfaces (see [9]). Theorem 5.1. Let X be a minimal complex algebraic surface. Then one of the following cases occurs. (1) X ∼ P2 or there exists a regular map f : X → B to some nonsingular curve = B whose ﬁbres are isomorphic to P1 . Moreover SX = HX , and we are in precisely one of the following cases: (i) X ∼ P2 , SX ∼ 1 , and kX = 3a, where a is a generator. = = (ii) SX ∼ U , and kX = 2a + 2b, where a, b are generators of SX with the = Gram matrix ( 0 1 ). 10 (iii) SX ∼ I1,1 = 1 ⊥ −1 and kX = 2a + 2b, where a, b are generators = 0 of SX with the Gram matrix 1 −1 . 0 (2) kX = 0 : (i) HX ∼ U 3 ⊥ E8 (−1)2 , SX is an even lattice of signature (1, ρ − 1), = where 1 ≤ ρ ≤ 20 ; (ii) HX = SX ∼ U ⊥ E8 (−1) ; = (iii) HX ∼ U 3 , SX is an even lattice of signature (1, ρ − 1), where 1 ≤ ρ ≤ = 4; (iv) HX = SX ∼ U . = 2 (3) kX = 0, kX = 0, S 0 /ZkX is a negative deﬁnite lattice. 2 0 (4) kX > 0, KX · [C ] ≥ 0 for any curve C on X , SX is a negative deﬁnite lattice. The four cases (1)-(4) correspond to the four possible values of the Kodaira dimension κ(X ) of X equal to −∞, 0, 1, 2, respectively. Recall that κ(X ) is equal to the maximal possible dimension of the image of X under a rational map given by some multiple of the canonical linear system on X . The four subcases (i)-(iv) in (2) correspond to K3-surfaces, Enriques surfaces, abelian surfaces (complex algebraic tori), and hyperelliptic surfaces, respectively. Let Aut(X ) denote the group of automorphisms of X (as an algebraic variety or as a complex manifold). Any g ∈ Aut(X ) acts naturally on HX via the pull-backs of cohomology classes. Since the latter is compatible with the cup-product, the action preserves the structure of a...
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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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