Algebraic Surfaces with q = pg = 0

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)).