Machine learning detects terminal singularities

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 eight-dimensional positively-curved algebraic varieties that have toric symmetry and Picard rank two, 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-Fano varieties in dimension eight.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 two to have terminal singularities. Together with the first sketch of the landscape of Q-Fano varieties in higher dimensions, this gives strong new evidence that machine learning can be an essential tool in developing mathematical conjectures and accelerating theoretical discovery.