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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 blowup 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 EnriquesKodaira 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 K3surfaces, 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 pullbacks of
cohomology classes. Since the latter is compatible with the cupproduct, 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 MichiganDearborn.
 Fall '04
 IgorDolgachev
 Algebra, Geometry, The Land

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