Terminal Fano 4-fold Hypersurfaces via Deep Reinforcement Learning

Abstract

We construct $46{,}992$ new families of Fano $4$-fold hypersurfaces $X_d\subset\mathbb(a_1,\ldots,a_6)$ of Fano index $1$ with $d>200$, whose general members have terminal singularities. Unlike the $11{,}617$ quasismooth families classified by Brown and Kasprzyk, these are nonquasismooth, for which determining terminality can require the analysis of hyperquotient singularities. An exhaustive search has classified all nonquasismooth families with $d\leq200$, but extending this classification to higher degrees becomes computationally prohibitive. We therefore exploit the sparsity and spatial clustering of terminal examples using two best-first searches over $\mathbb{Z}^6$: one with a fixed heuristic, and the other with a neural network trained online by deep reinforcement learning. Together, the searches reach degrees up to $3{,}234$. The neural network additionally discovers an average of $79$ hard-to-reach families per run. For the most isolated examples, our conservative distance-based model gives an estimated lower bound of nearly $40\%$ of the fixed search's total step budget to discover a single such family. An ablation study demonstrates that this advantage arises from the neural network rather than stochastic exploration alone. The complete list of families is publicly available.