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On some questions related to the Krichever correspondence. (English. Russian original) Zbl 1134.14023
Math. Notes 81, No. 4, 467-476 (2007); translation from Mat. Zametki 81, No. 4, 528-539 (2007).
The paper is concerned with the Krichever correspondence for surfaces; namely, on the association of a subspace of a two-dimensional skew field to each set of data \((X,C,p,F,e_p,t,u)\) where \(X\) is a projective irreducible normal algebraic surface, \(C\) is an ample Cartier divisor on \(X\), \(p\) is a smooth point on \(X\) and \(C\), \(F\) is a torsion free sheaf on \(X\), \(e_p\) is a local formal trivialization of \(F\) at \(p\), and \(u,t\) are local parameters at \(p\) such that \(u=0\) is a local equation of \(C\) on \(X\) near \(p\). This construction, which generalizes the classical Krichever construction [I. M. Krichever, Russ. Math. Surv. 32, No. 6, 185–213 (1977; Zbl 0386.35002)], was introduced by A. N. Parshin [Funct. Anal. Appl. 35, No. 1, 74–76 (2001); translation from Funkts. Anal. Prilozh. 35, No. 1, 88–90 (2001; Zbl 1078.14525); Commun. Algebra 29, No. 9, 4157–4181 (2001; Zbl 1014.14015)].
In the paper under review, some conditions on a subring of a two dimensional skew field are given for it being finitely generated over the ground field and being of transcendence degree two. On the other hand, the authors show that the subspace associated to the geometrical data above fulfill some conditions in terms of finite dimension of vector spaces that resembles M. Mulase’s conditions for the case of algebraic curves [J. Differ. Geom. 19, 403–430 (1984; Zbl 0559.35076)]. However, a full characterization of subspaces arising from geometrical data is still open. The paper is well written and contains very detailed examples and proofs that are very illustrating for the reader.

MSC:
14H70 Relationships between algebraic curves and integrable systems
12E15 Skew fields, division rings
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