Abstract:
Several algebro-geometric properties of commutative rings of partial differential operators (PDOs) as well as several geometric constructions are investigated. In particular, we show how to associate a geometric data by a commutative ring of PDOs, and we investigate the properties of these geometric data. This construction is in some sense similar to the construction of a formal module of Baker–Akhieser functions. On the other hand, there is a recent generalization of Sato’s theory, which belongs to the third author of this paper. We compare both approaches to the commutative rings of PDOs in two variables. As a by-product, we get several necessary conditions on geometric data describing commutative rings of PDOs.
The second author was partially supported by Russian Foundation for Basic Research (Grant Nos. 14-01-00178-a and 12-01-33024 mol_a_ved) and by the Programme for the Support of Leading Scientific Schools of the Russian Federation (Grant No. NSh-2998.2014.1). The third author was partially supported by the RFBR Grant Nos. 14-01-00178-a, 13-01-00664 and by Grant NSh No. 581.2014.1.
This publication is cited in the following 7 articles:
Alexander B. Zheglov, “The Schur–Sato Theory for Quasi-elliptic Rings”, Proc. Steklov Inst. Math., 320 (2023), 115–160
Igor Burban, Alexander Zheglov, “Cohen–Macaulay modules over the algebra of planar quasi–invariants and Calogero–Moser systems”, Proc. London Math. Soc., 121:4 (2020), 1033
A. B. Zheglov, “Surprising examples of nonrational smooth spectral surfaces”, Sb. Math., 209:8 (2018), 1131–1154
Vik. S. Kulikov, “On divisors of small canonical degree on Godeaux surfaces”, Sb. Math., 209:8 (2018), 1155–1163
Igor Burban, Alexander Zheglov, “Fourier–Mukai transform on Weierstrass cubics and commuting differential operators”, Int. J. Math., 29:10 (2018), 1850064
D. A. Pogorelov, A. B. Zheglov, “An algorithm for construction of commuting ordinary differential operators by geometric data”, Lobachevskii J Math, 38:6 (2017), 1075
A. B. Zheglov, H. Kurke, “Geometric properties of commutative subalgebras of partial differential operators”, Sb. Math., 206:5 (2015), 676–717
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