Machine learning detects terminal singularities
Tom Coates Alexander M. Kasprzyk Sara Veneziale
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.