We will be interested only in basic rational surfaces

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: 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...
View Full Document

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.

Ask a homework question - tutors are online