Rationality proofs by curve counting


We propose an approach for showing rationality of an algebraic variety $X$. We try to cover $X$ by rational curves of certain type and count how many curves pass through a generic point. If the answer is $1$, then we can sometimes reduce the question of rationality of $X$ to the question of rationality of a closed subvariety of $X$. This approach is applied to the case of the so-called Ueno-Campana manifolds. Assuming certain conjectures on curve counting, we show that the previously open cases $X_{4,6}$ and $X_{5,6}$ are both rational. Our conjectures are evidenced by computer experiments. In an unexpected twist, existence of lattices $D_6$, $E_8$, and $\Lambda_{10}$ turns out to be crucial.

Experimental Mathematics DOI: 10.1080/10586458.2019.1691088