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A lower bound for $K_{X}L$ of quasi-polarized surfaces $(X,L)$ with non-negative Kodaira dimension
TLDR
In this paper, the Kodaira dimension of a smooth projective surface over the complex number field was shown to be 2q(X)-4, where q(X) is the irregularity of the surface.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 $\kappa(X)\geq 0$, then $K_{X}L\geq 2q(X)-4$, where $q(X)$ is the irregularity of $X$. In this paper, we prove that this conjecture is true if (1) the case in which $\kappa(X)=0$ or 1, (2) the case in which $\kappa(X)=2$ and $h^{0}(L)\geq 2$, or (3) the case in which $\kappa(X)=2$, $X$ is minimal, $h^{0}(L)=1$, and $L$ satisfies some conditions.read more
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A Lower Bound for the Sectional Genus of Quasi-Polarized Surfaces
TL;DR: In this article, a quasi-polarized version of the Kodaira dimension of a smooth projective variety over the complex numbers (X,L) is considered, and the conjecture that g(L) = q(X) is proved to be true.