Normal forms for ordinary differential operators, I

TL;DR

This paper introduces a new normal form concept for ordinary differential operators, extending Schur theory, and uses it to explicitly parametrize certain sheaves on algebraic curves.

Contribution

It develops a generalized Schur theory for differential operators and introduces normal forms to explicitly parametrize torsion free rank one sheaves on curves.

Findings

Normal forms are effective for commuting differential operators.

Explicit parametrization of sheaves on curves is achieved.

Normal forms involve conjugation by invertible operators.

Abstract

In this paper we develop the generalised Schur theory offered in the recent paper by the second author in dimension one case, and apply it to obtain a new explicit parametrisation of torsion free rank one sheaves on projective irreducible curves with vanishing cohomology groups. This parametrisation is obtained with the help of normal forms - a notion we introduce in this paper. Namely, considering the ring of ordinary differential operators $D_1=K[[x]][\partial ]$ as a subring of a certain complete non-commutative ring $\hat{D}_1^{sym}$, the normal forms of differential operators mentioned here are obtained after conjugation by some invertible operator ("Schur operator"), calculated using one of the operators in a ring. Normal forms of commuting operators are polynomials with constant coefficients in the differentiation, integration and shift operators, which have a restricted finite order in each variable, and can be effectively calculated for any given commuting operators.