A lower bound for K X L of quasi-polarized surfaces (X,L) with non-negative Kodaira dimension
Abstract
Let
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
κ(X)≥0
, then
K
X
L≥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
κ(X)=0
or 1, (2) the case in which
κ(X)=2
and
h
0
(L)≥2
, or (3) the case in which
κ(X)=2
,
X
is minimal,
h
0
(L)=1
, and
L
satisfies some conditions.