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arXiv · 2504.16187

The (noncommutative) geometry of difference equations

Eric M. Rains
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Abstract

The aim of this monograph is twofold: to explain various nonautonomous integrable systems (discrete Painlev\'e all the way up to the elliptic level, as well as generalizations \`a la Garnier) using an interpretation of difference and differential equations as sheaves on noncommutative projective surfaces, and to develop the theory of such surfaces enough to allow one to apply the usual GIT construction of moduli spaces of sheaves. This requires a fairly extensive development of the theory of birationally ruled noncommutative projective surfaces, both showing that the analogues of Cremona transformations work and understanding effective, nef, and ample divisor classes. This combines arXiv:1307.4032, arXiv:1307.4033, arXiv:1907.11301, as well as those portions of arXiv:1607.08876 needed to make things self-contained. Some additional results appear, most notably a proof that the resulting discrete actions on moduli spaces of equations are algebraically integrable.

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Eric M. Rains. 2025-04-22. The (noncommutative) geometry of difference equations. https://arxiv.org/abs/2504.16187

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