New directions in the Minimal Model Program

Cascini, Paolo

doi:10.1007/s40574-020-00250-9

Cited by 1 publication

(2 citation statements)

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“…Understanding this classification is a fundamental problem in algebraic geometry, and is the motivation behind a huge amount of research; see e.g. [9,11,33,35] and the references therein. Q-Fano varieties also play an important role elsewhere in mathematics, for example in the study of K-stability and the existence of Kähler-Einstein metrics [5].…”

Section: Supplementary Materials a Mathematical Backgroundmentioning

confidence: 99%

“…A key class of algebraic varieties are Fano varieties: positively curved shapes that are basic building blocks in algebraic geometry. Fano varieties are 'atomic pieces' of more complex shapes, in the sense of the Minimal Model Programme [11,33,35]. Running the Minimal Model Programme -that is, breaking an algebraic variety 𝑋 into atomic pieces -involves making birational transformations of 𝑋.…”

Section: Introductionmentioning

confidence: 99%

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Machine learning the dimension of a Fano variety

Coates,

Kasprzyk,

Veneziale

2023

Nat Commun

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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.

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