The Mordell–Weil rank bound via Mori fiber spaces

Let $X\to B$ be as in Theorem MMP, with $X$ a Calabi–Yau variety. Suppose that $B$ is birational to a Mori fiber space $B'\to U$ with fiber $F$, and let $\operatorname{rk}\mathrm{MW}(X/B)$ denote the Mordell–Weil rank of the elliptic fibration. Mordell–Weil rank conjecture. One has

$$\operatorname{rk}\mathrm{MW}(X/B)\leq 10\cdot(\dim F+1)-2\leq 10\cdot\dim X-2.$$

This conjecture extrapolates the rank bounds proved for elliptic Calabi–Yau fibrations in dimensions three and four and for fibrations admitting suitable rational curves on the base. It is proposed because the required curves with bounded anticanonical degree are not known to exist in higher dimensions.