0 a 1 curve appears as a bre of a blow up map f x y

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Unformatted text preview: ume. The root vector corresponding to the extreme right vertex is equal to the vector r + e, where r generates the kernel of the lattice E2,3,6 embedded naturally in E2,3,8 and the vector e is the root vector corresponding to the extreme right vector in the subdiagram deﬁning the sublattice isomorphic to E2,3,7 . 24 IGOR V. DOLGACHEV 5. Automorphisms of algebraic surfaces 5.1. Quadratic lattices associated to an algebraic surface. Let X be a complex projective algebraic surface. It has the underlying structure of a compact smooth oriented 4-manifold. Thus the cohomology group H 2 (X, Z) is a ﬁnitely generated abelian group equipped with a symmetric bilinear form H 2 (X, Z) × H 2 (X, Z) → Z deﬁned by the cup-product. When we divide H 2 (X, Z) by the torsion subgroup we obtain a unimodular quadratic lattice HX . We will denote the value of the bilinear form on HX induced by the cup product by x · y and write x2 if x = y . To compute its signature one uses the Hodge decomposition (depending on the complex structure of X ) H 2 (X, C) = H 2,0 (X ) ⊕ H 1,1 (X, C) ⊕ H 0,2 (X ), where dim H 2,0 = dim H 0,2 and is equal to the dimension pg (X ) of the space of holomorphic diﬀerential 2-forms on X . It is known that under the complex conjugation on H 2 (X, C) the space H 1,1 (X ) is invariant and the space H 2,0 (X ) is mapped to H 0,2 (X ), and vice versa. One can also compute the restriction of the cup-product on H 2 (X, C) to each H p,q to conclude that the signature of the cup-product on H 2 (X, R) is equal to (b+ , b− ), where b+ = 2pg + 1. 2 2 2 The parity of the lattice HX depends on the property of its ﬁrst Chern class c1 (X ) ∈ H 2 (X, Z). If is divisible by 2 in H 2 (X, Z), then HX is an even lattice. The lattice HX contains an important primitive sublattice which depends on the complex structure of X . For any complex irreducible curve C on X its fundamental class [C ] deﬁnes a cohomology class in H 2 (X, Z). The Z-span of these classes deﬁnes a subgroup of H 2 (X, Z), and its image SX in HX is called the Neron-Severi lattice (or the Picard lattice ) of X . It is a sublattice of HX of signature (1, ρ − 1), where ρ = rank SX . An example of an element of positive norm is the class of a hyperplane section of X in any projective embedding of X . Thus the lattice SX is hyperbolic in the sense of previous sections, and we can apply the theory of reﬂection groups to SX . It is known that the image in HX of the ﬁrst Chern class c1 (X ) ∈ H 2 (X, Z) belongs to the Picard lattice. The negative KX = −c1 (X ) is the canonical class of X . It is equal to the image in H 2 (X, Z) of a divisor of zeros and poles of a holomorphic diﬀerential 2-form on X . We denote the image of KX in HX by kX . For any x ∈ SX we deﬁne pa (x) = x2 + x · kX . It is always an even integer. If x is the image in HX of the fundamental class of a nonsingular complex curve C on X , then pa (x) is equal to 2g −2, where g is the genus of the Riemann surface C (the adjunction formula ). If C is an irreducible complex curve with ﬁn...
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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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