Deep Reinforcement Learning for Fano Hypersurfaces

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.

Figures (10)

• Figure 1: The $6$-dimensional dynamic heuristic (deep reinforcement learning) search for terminal Fano $4$-fold hypersurfaces projected onto $2$-dimensions. While the quasismooth terminal points are fully classified, the search discovers previously unknown nonquasismooth terminal ones. See Figure \ref{['fig:search_dynamic']} for full details.
• Figure 2: Flowchart of the fixed heuristic search algorithm.
• Figure 3: Flowchart of the dynamic heuristic search algorithm.
• Figure 4: Exhaustive search of Fano 4-fold hypersurfaces with terminal singularities. In total $84,733$ terminal examples were found, $7,346$ quasismooth, and $77,387$ nonquasismooth. Each frame shows points in $\mathbb{Z}^2$, obtained by projecting the original $\mathbb{Z}^6$ search space onto consecutive coordinate pairs via $(a_1,\ldots,a_6)\mapsto(a_i,a_{i+1})$.
• Figure 5: Classification of $11,617$ quasismooth Fano $4$-fold hypersurfaces with terminal singularities. Each frame shows points in $\mathbb{Z}^2$, obtained by projecting the original $\mathbb{Z}^6$ search space onto consecutive coordinate pairs via $(a_1,\ldots,a_6)\mapsto(a_i,a_{i+1})$.