Title: Rationality proofs by curve counting,
Abstract: 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. Our experiments indicate that the
previously open cases $X_{4,6}$ and $X_{5,6}$ are both rational. However, this
result is not rigorously justified and depends on a heuristic argument and a
Monte Carlo type computer simulation. In an unexpected twist, existence of
lattices $D_6$, $E_8$ and $\Lambda_{10}$ turns out to be crucial.