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.

机器学习被用于理解 Q-Fano 变量的分类。我们开发了一个神经网络分类器，以 95% 的准确率预测拥有托里克对称性和 Picard 秩 2 的 8 维正曲率代数变量是否为 Q-Fano 变量。该研究为 Q-Fanos 在 8 维领域的分类提供了初步理解，并对机器学习如何能够如此准确地检测 Q-Fano 变量提出了深层次数学理论的线索。此外，通过使用量子周期进行可视化，我们观察到机器学习揭示的分类似乎落在一个有界区域内，并通过 Fano 指数进行分层。这暗示了以后可能能够对完备性提出并证明猜想。受机器学习分析的启发，我们提出并证明了一个新的正曲率托里克 Picard 秩 2 变量具有终端奇点的全局组合准则。与 Q-Fanos 在更高维度上的初步分类相结合，这为机器学习成为发展数学猜想和加速理论发现的基本工具提供了新的证据。