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Mat. Sb., 2018, Volume 209, Number 8, Pages 29–55 (Mi msb9031)  

This article is cited in 1 scientific paper (total in 1 paper)

Surprising examples of nonrational smooth spectral surfaces

A. B. Zheglov

Lomonosov Moscow State University, Faculty of Mechanics and Mathematics

Abstract: In this paper we study necessary and sufficient algebro-geometric conditions for the existence of a nontrivial commutative subalgebra of rank $1$ in $\widehat{D}$, a completion of the algebra of partial differential operators in two variables, which can be thought of as a simple algebraic analogue of the algebra of analytic pseudodifferential operators on a manifold.
These are conditions on a projective (spectral) surface; they are encoded in a new notion of pre-spectral data. For smooth surfaces the sufficient conditions look especially simple. On a smooth projective surface there should exist 1) an ample integral curve $C$ with $C^2=1$ and $h^0(X,\mathscr{O}_X(C))=1$; 2) a divisor $D$ with $(D, C)_X=g(C)-1$, $h^i(X,\mathscr{O}_X(D))=0$, $i=0,1,2$, and $h^0(X,\mathscr{O}_X(D+C))=1$. Amazingly, there are examples of such surfaces for which the corresponding commutative subalgebras do not admit isospectral deformations.
Bibliography: 45 titles.

Keywords: commuting differential operators, commuting difference operators, quantum integrable systems, algebraic KP theory, algebraic surfaces, Godeaux surfaces.

Funding Agency Grant Number
Russian Science Foundation 16-11-10069
This work was supported by the Russian Science Foundation under grant no. 16-11-10069.


DOI: https://doi.org/10.4213/sm9031

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English version:
Sbornik: Mathematics, 2018, 209:8, 1131–1154

Bibliographic databases:

Document Type: Article
UDC: 517.957+512.72+512.71
MSC: Primary 13N15, 14H81; Secondary 37K20
Received: 31.10.2017 and 06.02.2018

Citation: A. B. Zheglov, “Surprising examples of nonrational smooth spectral surfaces”, Mat. Sb., 209:8 (2018), 29–55; Sb. Math., 209:8 (2018), 1131–1154

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. Vik. S. Kulikov, “On divisors of small canonical degree on Godeaux surfaces”, Sb. Math., 209:8 (2018), 1155–1163  mathnet  crossref  crossref  adsnasa  isi  elib
  • Математический сборник Sbornik: Mathematics (from 1967)
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