A note on Iitaka's conjecture $C_{3,1}$ in positive characteristics

TL;DR

This paper proves a case of Iitaka's conjecture for 3-fold fibrations over curves in characteristic p > 5, establishing a lower bound for the Kodaira dimension of the total space.

Contribution

It extends Iitaka's conjecture to positive characteristic by proving the inequality for fibrations with big canonical fibers in characteristic p > 5.

Findings

Proves Iitaka's conjecture $C_{3,1}$ for certain 3-folds in characteristic p > 5.

Establishes the inequality $\, ext{κ}(X) \, ext{ge}\, ext{κ}(Y) + ext{κ}(X_η)$ under specified conditions.

Advances understanding of the behavior of Kodaira dimensions in positive characteristic settings.

Abstract

Let $f:X\to Y$ be a fibration from a smooth projective 3-fold to a smooth projective curve, over an algebraically closed field $k$ of characteristic $p >5$. We prove that if the generic fiber $X_{\eta}$ has big canonical divisor $K_{X_{\eta}}$, then $$\kappa(X)\ge\kappa(Y) + \kappa(X_{\eta}).$$