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Algebraic Surfaces with q = pg = 0

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Algebraic Surfaces

Part of the book series: C.I.M.E. Summer Schools ((CIME,volume 76))

Abstract

1. Notations. Let F be a complex algebraic surface. We will use the following standard notations:

  • O F : the structure sheaf of F.

  • O F(D) : the invertible sheaf associated with a divisor D on F.

  • KF, = − c1(F) : minus the first Chern class of F or a canonical divisor on F.

  • ωF = O F(KF) : the canonical sheaf of F.

  • hi (D) ; the dimension of the space Hi(F,O F(D)).

  • pg (F) = h0(KF) = h2(O F) ; the geometric genus of F.

  • q(F) = h1(KF) = h1(0 F) ; the irregularity of F.

  • \({\text{K}}_{\text{F}}^2 \) : the self-intersection index of KF.

  • \({\text{P}}^{(1)} \left( {\text{F}} \right) = {\text{K}}_{{\text{F'}}}^2 + 1\), where F is a minimal model of a non-rational surface F ; the linear genus of F.

  • c2(F) : the topological Euler-Poincare characteristic of F.

  • Pn(F) = h0(nKF : the n-genus of F.

  • NS(F) : the Neron-Severi group of F, the quotient of the Picard group Pic(F) by the subgroup of divisors algebraically equivalent to zero (= Pic(F) if q = 0).

  • Tors(F) = Tors(NS(F)) = Tors(H1(F,Z)).

If not stated otherwise F will be always assumed to be non-singular and projective.

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Authors
  1. Igor Dolgachev
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G. Tomassini

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Dolgachev, I. (2010). Algebraic Surfaces with q = pg = 0. In: Tomassini, G. (eds) Algebraic Surfaces. C.I.M.E. Summer Schools, vol 76. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-11087-0_3

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