Деформационное квантование и квантовые группы, Семинар 9, Г.Б.Шабат

Title: On the 3-generated commutative rings of differential operators Abstract: The general theory of commutative rings of differential operators (containing the operators of almost all orders) was basically completed in 1970-s; the complete classification in algebro-geometric terms can be found, e.g., in [Drinfeld1977] and in [Krichever1978]. This theory established the bijection of the isomorphic classes of such rings and certain linear flows on the Jacobians of projective models of their spectra. A special attention was paid to the rings, generated by two operators of the coprime order, usually of order 2 and of some odd order; the theory of such rings turned out to be equivalent to the theory of KdV hierarchy. However, the corresponding algebraic curves were always hyperelliptic. In order to handle the general (canonical curves), one should consider the rings, generated by more than two operators. In the paper [Shabat1980] the author considered the simplest possible case of this kind – that of generators of orders 3,4,5. Some details of the 1980-calculations were enigmatic. The talk will be devoted to their algebro-geometric interpretation, based on the smooth models of genus-two curves as quintics in the projective 3-space. References [Drinfeld1977] V.G. Drinfeld. Commutative subrings of certain noncommutative rings. Functional Analysis and Its Applications, 1977, 11:1, 9-12. [Krichever1978] I. M. Krichever. Commutative rings of ordinary linear differential operators. Functional Analysis and Its Applications, 1978, 12:3, 175–185. [Shabat1980] G. B. Shabat. A system of equations of S. P. Novikov. Functional Analysis and Its Applications, 1980, 14:2, 158–160.