# Kodaira-Iitaka Dimension on a Normal Prime Divisor

- Authors
- Type
- Preprint
- Publication Date
- Dec 18, 2008
- Submission Date
- Dec 18, 2008
- Identifiers
- arXiv ID: 0812.3454
- Source
- arXiv
- License
- Yellow
- External links

## Abstract

This paper was inspired by work by T. Peternell, M. Schneider and A.J. Sommese on the Kodaira dimension of subvarieties. In it I find a relation between the Kodaira-Iitaka dimension of a divisor on a normal variety and that of related divisors on an irreducible normal subvariety of codimension one. The main result may be stated in a simplified form as: For $X$ a complete normal variety, $Y \sub X$ an irreducible complete normal divisor and $\sL$ an invertible sheaf on $X$, there exist integers $n_1 > 0, n_2 \geq 0$ for which $\kappa(X,\sL) - 1 \leq \kappa(Y,\sL^{n_1}(-n_2Y)|_Y)$, where, if $Y$ is not a fixed component of large tensor powers of $\sL$, we may take $n_1 >> n_2$. This has implications for Kodaira-Iitaka dimension on a subvariety of any codimension.