Kodaira-Iitaka Dimension on a Normal Prime Divisor

TL;DR

This paper establishes a relationship between the Kodaira-Iitaka dimension of a divisor on a normal variety and that of related divisors on a subvariety, extending understanding of dimension behavior in algebraic geometry.

Contribution

It introduces a new inequality linking the Kodaira-Iitaka dimensions of divisors on a variety and its codimension-one subvariety, generalizing previous work.

Findings

Derived an inequality relating Kodaira-Iitaka dimensions of divisors on a variety and its subvariety.

Showed the inequality holds with specific integer parameters under certain conditions.

Extended the understanding of dimension behavior for subvarieties of any codimension.

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.