# Rationality proofs by curve counting

## TL;DR

The paper introduces a curve counting approach to determine the rationality of algebraic varieties, applying it heuristically to certain manifolds and suggesting new rationality results based on computational experiments.

## Contribution

It proposes a novel method using rational curve counts to assess rationality, applied heuristically to Ueno-Campana manifolds, with implications for previously unresolved cases.

## Key findings

- Heuristic evidence suggests $X_{4,6}$ and $X_{5,6}$ are rational.
- The approach links lattice existence to rationality conclusions.
- Computational experiments support the conjecture of rationality for certain varieties.

## 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.