Machine learning detects terminal singularities

Algebraic varieties are the geometric shapes defined by systems of polynomial equations; they are ubiquitous across mathematics and science. Amongst these algebraic varieties are Q-Fano varieties: positively curved shapes which have Q-factorial terminal singularities. Q-Fano varieties are of fundamental importance in geometry as they are "atomic pieces" of more complex shapes - the process of breaking a shape into simpler pieces in this sense is called the Minimal Model Programme. Despite their importance, the classification of Q-Fano varieties remains unknown. In this paper we demonstrate that machine learning can be used to understand this classification. We focus on 8-dimensional positively-curved algebraic varieties that have toric symmetry and Picard rank 2, and develop a neural network classifier that predicts with 95% accuracy whether or not such an algebraic variety is Q-Fano. We use this to give a first sketch of the landscape of Q-Fanos in dimension 8. How the neural network is able to detect Q-Fano varieties with such accuracy remains mysterious, and hints at some deep mathematical theory waiting to be uncovered. Furthermore, when visualised using the quantum period, an invariant that has played an important role in recent theoretical developments, we observe that the classification as revealed by ML appears to fall within a bounded region, and is stratified by the Fano index. This suggests that it may be possible to state and prove conjectures on completeness in the future. Inspired by the ML analysis, we formulate and prove a new global combinatorial criterion for a positively curved toric variety of Picard rank 2 to have terminal singularities. Together with the first sketch of the landscape of Q-Fanos in higher dimensions, this gives new evidence that machine learning can be an essential tool in developing mathematical conjectures and accelerating theoretical discovery.Comment: 20 pages, 11 figures, 3 table

Nuovi manoscritti latini e francesi prodotti a Genova a cavallo tra XIII e XIV secolo

Abstract Il saggio, partendo da una riflessione sulla produzione libraria genovese della fine del XIII secolo e dell’inizio del XIV, si pone l’obiettivo di presentare un codice attribuibile all’atelier domenicano attivo a Genova durante l’ultimo quarto del Duecento (una Bibbia latina oggi conservata frammentariamente in biblioteche statunitensi), e di ridiscutere il luogo di copia e l’origine del copista di un codice di prose religiose antico-francesi di inizio Trecento (Paris BNF fr. 187), la cui attribuzione oscilla negli studi tra Genova e Milano. Starting with some considerations about the Genovese book production at the end of the 13th century and at the beginning of the 14th, this paper aims to describe a codex whose production could be attributed to the Dominican atelier in activity in the last quarter of the Duecento (a Latin bible nowadays fragmentally preserved in several US libraries). This essay also debates the place of copying and the provenance of the scribe of an Old-French religious prose codex from the beginning of the 14th Century (Paris BNF fr. 187), whose attribution oscillates between Genova and Milan in previous studies. Keywords Manoscritti miniati a Genova – Produzione libraria – Bibbia – Manoscritti italiani di prose francesi – Biblioteca dei ViscontiManuscripts illuminated in Genoa – Book production – Bible – Italian manuscripts of French prose texts – Library of the Viscont