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Machine learning, toric Fano varieties, and terminal singularities

Sara Veneziale (Imperial College London)
Tuesday, January 16, 2024 - 11:00am to 12:00pm

Abstract: In this talk, I will describe recent work in the application of machine learning to explore questions in algebraic geometry, specifically in the context of the study of Q-Fano varieties. Q-Fano varieties are Fano varieties with mild singularities (called terminal singularities) and they are the key players in the Minimal Model Program. We ask if machine learning can determine if a toric Fano variety has terminal singularities, since there is no global efficient algorithm that does this (except for the weighted projective spaces case). A simple feedforward neural network is able to detect terminal singularities with 95% accuracy for varieties in dimension eight and Picard rank two. Building this high-accuracy model has two consequences. Firstly, it inspires the formulation of a new global, combinatorial criterion to determine if a toric variety of Picard rank two has terminal singularities. Secondly, the machine learning model is used directly to give the first sketch of the landscape of Q-Fano varieties in dimension eight. This is joint work with Tom Coates and Al Kasprzyk.