Normal forms for ordinary differential operators, III

TL;DR

This paper extends the parametrization of torsion free sheaves on projective curves from rank one to arbitrary rank, providing explicit examples and broadening the understanding of sheaf structures on algebraic curves.

Contribution

It introduces a generalized parametrization method for torsion free sheaves of any rank on projective irreducible curves with vanishing cohomology, building on previous work for rank one.

Findings

Explicit parametrization for rank two sheaves on a Weierstrass cubic curve

Generalization of sheaf parametrization to arbitrary rank

Illustrative example demonstrating the new parametrization

Abstract

In this paper, which is a follow-up of our first paper "Normal forms for ordinary differential operators, I", we extend the explicit parametrisation of torsion free rank one sheaves on projective irreducible curves with vanishing cohomology groups obtained earlier to analogous parametrisation of torsion free sheaves of arbitrary rank with vanishing cohomology groups on projective irreducible curves. As an illustration of our theorem we calculate one explicit example of such parametrisation, namely for rank two sheaves on a Weierstrass cubic curve.