A Lower Bound For KxL Of Quasi-Polarized Surfaces (X, L) With Non-Negative Kodaira Dimension

Abstract

AbstractLet X be a smooth projective surface over the complex number field and let L be a nef-big divisor on X. Here we consider the following conjecture; If the Kodaira dimension ≥ 0, then KXL ≥ 2q(X) - 4, where q(X) is the irregularity of X. In this paper, we prove that this conjecture is true if (1) the case in which = 0 or 1, (2) the case in which = 2 and h0(L) ≥ 2, or (3) the case in which = 2, X is minimal, h0(L) = 1, and L satisfies some conditions.