# Machine learning the dimension of a Fano variety

arXiv: https://arxiv.org/abs/2309.05473 · HTML: https://mlsanity.com/paper/2309.05473/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- Builds on 7 of its 64 references that are in the map.
- First posted on arXiv 11 Sep 2023.

## Builds on (most cited first, 7 of 7)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [A Unified Approach to Interpreting Model Predictions](https://mlsanity.com/paper/1705.07874/) (2017)
- [Constructions in combinatorics via neural networks](https://mlsanity.com/paper/2104.14516/) (2021)
- [Machine-Learning Mathematical Structures](https://mlsanity.com/paper/2101.06317/) (2021)
- [Inception Neural Network for Complete Intersection Calabi-Yau 3-folds](https://mlsanity.com/paper/2007.13379/) (2020)
- [Big Data Approaches to Knot Theory: Understanding the Structure of the Jones Polynomial](https://mlsanity.com/paper/1912.10086/) (2019)
- [Turning Mathematics Problems into Games: Reinforcement Learning and Gr"obner bases together solve Integer Feasibility Problems](https://mlsanity.com/paper/2208.12191/) (2022)

## Built on by (most cited first, 7 of 7)

- [Machine learning detects terminal singularities](https://mlsanity.com/paper/2310.20458/) (2023)
- [Can Transformers Do Enumerative Geometry?](https://mlsanity.com/paper/2408.14915/) (2024)
- [A Triumvirate of AI Driven Theoretical Discovery](https://mlsanity.com/paper/2405.19973/) (2024)
- [Learning BPS Spectra and the Gap Conjecture](https://mlsanity.com/paper/2405.09993/) (2024)
- [Calabi-Yau Four/Five/Six-folds as mathbbPnw Hypersurfaces: Machine Learning, Approximation, and Generation](https://mlsanity.com/paper/2311.17146/) (2023)
- [Murmurations: a case study in AI-assisted mathematics](https://mlsanity.com/paper/2603.09680/) (2026)
- [Deep Reinforcement Learning for Fano Hypersurfaces](https://mlsanity.com/paper/2603.15437/) (2026)

## Abstract (arXiv)

Fano varieties are basic building blocks in geometry - they are `atomic pieces' of mathematical shapes. Recent progress in the classification of Fano varieties involves analysing an invariant called the quantum period. This is a sequence of integers which gives a numerical fingerprint for a Fano variety. It is conjectured that a Fano variety is uniquely determined by its quantum period. If this is true, one should be able to recover geometric properties of a Fano variety directly from its quantum period. We apply machine learning to the question: does the quantum period of X know the dimension of X? Note that there is as yet no theoretical understanding of this. We show that a simple feed-forward neural network can determine the dimension of X with 98% accuracy. Building on this, we establish rigorous asymptotics for the quantum periods of a class of Fano varieties. These asymptotics determine the dimension of X from its quantum period. Our results demonstrate that machine learning can pick out structure from complex mathematical data in situations where we lack theoretical understanding. They also give positive evidence for the conjecture that the quantum period of a Fano variety determines that variety.