On the Medvedev-Scanlon Conjecture for Minimal Threefolds of Non-Negative Kodaira Dimension

Abstract

Motivated by work of Zhang from the early `90s, Medvedev and Scanlon formulated the following conjecture. Let $K$ be an algebraically closed field of characteristic $0$ and let $X$ be a quasiprojective variety defined over $K$ endowed with a dominant rational self-map $\Phi$. Then there exists a point $\alpha \in X(K)$ with Zariski dense orbit under $\Phi$ if and only if $\Phi$ preserves no nontrivial rational fibration, i.e., there exists no non-constant rational function $f\in K(X)$ such that ${\Phi }^{\ast }(f)=f$. The Medvedev-Scanlon conjecture holds when $K$ is uncountable. The case where $K$ is countable (e.g., $K=\overline{\mathbb{Q}}$) is much more difficult; here the conjecture has only been proved in a small number of special cases. In this paper we show that the Medvedev-Scanlon conjecture holds for all varieties of positive Kodaira dimension, and explore the case of Kodaira dimension $0$. Our results are most complete in dimension $3$.

Minor changes limited to the Introduction concerning past work on the conjecture