aiqbench analysis
The main result is a large new catalogue of Fano 4-fold hypersurfaces, a pure algebraic geometry contribution. The machine learning part is the search method: a neural network trained online with deep reinforcement learning guides the exploration of high-degree cases where exhaustive search is infeasible. The ablation study, which attributes the gain to the network rather than to random exploration, is the most relevant piece for AI readers, though the evidence is limited to this one search problem.
Terminal Fano 4-fold Hypersurfaces via Deep Reinforcement Learning
arXiv:2603.15437v2 Announce Type: replace-cross 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.
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.
12 pages, 11 figures, 5 tables. v2: Certified the terminality of the discovered families, added an ablation study, and revised the experimental results
Algebraic Geometry (math.AG); Machine Learning (cs.LG)
arXiv:2603.15437 [math.AG]
(or arXiv:2603.15437v2 [math.AG] for this version)
Full text under the publisher's open licence · 214 words