Algebraic varieties are the geometric shapes defined by systems of polynomial equations; they are ubiquitous across mathematics and science. Amongst these algebraic varieties are Q-Fano varieties: positively curved shapes which have Q-factorial terminal singularities. Q-Fano varieties are of fundamental importance in geometry as they are "atomic pieces" of more complex shapes - the process of breaking a shape into simpler pieces in this sense is called the Minimal Model Programme. Despite their importance, the classification of Q-Fano varieties remains unknown. In this paper we demonstrate that machine learning can be used to understand this classification. We focus on 8-dimensional positively-curved algebraic varieties that have toric symmetry and Picard rank 2, and develop a neural network classifier that predicts with 95% accuracy whether or not such an algebraic variety is Q-Fano. We use this to give a first sketch of the landscape of Q-Fanos in dimension 8. How the neural network is able to detect Q-Fano varieties with such accuracy remains mysterious, and hints at some deep mathematical theory waiting to be uncovered. Furthermore, when visualised using the quantum period, an invariant that has played an important role in recent theoretical developments, we observe that the classification as revealed by ML appears to fall within a bounded region, and is stratified by the Fano index. This suggests that it may be possible to state and prove conjectures on completeness in the future. Inspired by the ML analysis, we formulate and prove a new global combinatorial criterion for a positively curved toric variety of Picard rank 2 to have terminal singularities. Together with the first sketch of the landscape of Q-Fanos in higher dimensions, this gives new evidence that machine learning can be an essential tool in developing mathematical conjectures and accelerating theoretical discovery.

翻译：代数簇是由多项式方程组定义的几何形状，广泛存在于数学与科学领域。其中包含Q-法诺簇：具有正曲率且拥有Q-因子末端奇点的几何结构。Q-法诺簇在几何学中具有基础重要性，它们是复杂形状的"原子构件"——将形状拆解为更简单组件的这一过程被称为极小模型纲领。尽管其重要性不言而喻，Q-法诺簇的分类问题至今尚未解决。本文证明机器学习可用于理解这一分类问题。我们聚焦于具有环面对称性且皮卡秩为2的8维正曲率代数簇，开发了一个神经网络分类器，能以95%的准确率预测该代数簇是否为Q-法诺簇。基于此，我们首次勾勒出8维Q-法诺簇的全局图景。神经网络如何实现如此高精度的Q-法诺簇检测仍是一个谜题，暗示着有待揭示的深层数学理论。此外，当使用量子周期（这一不变量在近期理论发展中扮演重要角色）进行可视化时，我们观察到机器学习揭示的分类结果似乎落于有界区域内，并按法诺指数分层。这表明未来或可提出并证明关于完备性的猜想。受机器学习分析启发，我们提出并证明了一个新的全局组合判别准则，用于判定具有皮卡秩2的正曲率环面带状簇是否具有末端奇点。结合高维Q-法诺簇的初步图景，这一成果为"机器学习可成为发展数学猜想、加速理论发现的关键工具"提供了新证据。