Deep Reinforcement Learning for Fano Hypersurfaces

TL;DR

This paper introduces a deep reinforcement learning method to discover new Fano 4-fold hypersurfaces with terminal singularities, significantly expanding the known examples in algebraic geometry.

Contribution

It presents a novel RL-based search algorithm tailored for high-dimensional algebraic geometry problems, enabling the discovery of previously unknown Fano hypersurfaces.

Findings

Thousands of new Fano hypersurfaces identified

Hundreds of examples inaccessible to existing methods

Demonstrates effectiveness of RL in complex mathematical searches

Abstract

We design a deep reinforcement learning algorithm to explore a high-dimensional integer lattice with sparse rewards, training a feedforward neural network as a dynamic search heuristic to steer exploration toward reward dense regions. We apply this to the discovery of Fano 4-fold hypersurfaces with terminal singularities, objects of central importance in algebraic geometry. Fano varieties with terminal singularities are fundamental building blocks of algebraic varieties, and explicit examples serve as a vital testing ground for the development and generalisation of theory. Despite decades of effort, the combinatorial intractability of the underlying search space has left this classification severely incomplete. Our reinforcement learning approach yields thousands of previously unknown examples, hundreds of which we show are inaccessible to known search methods.