The dimension of the space of Garnier equations with fixed locus of apparent singularities

TL;DR

This paper investigates the space of second order linear differential equations with fixed singularity data, showing it forms a maximal rank linear system and establishing a birational correspondence with a Hilbert scheme.

Contribution

It demonstrates that the conditions on differential equations with fixed and apparent singularities define a space of expected dimension and constructs a birational map to a Hilbert scheme.

Findings

The conditions impose a maximal rank linear system.

The family of such differential equations has the expected dimension.

A birational map to the Hilbert scheme is constructed.

Abstract

We show that the conditions imposed on a second order linear differential equation with rational coefficients on the complex line by requiring it to have regular singularities with fixed exponents at the points of a finite set $P$ and apparent singularities at a finite set $Q$ (disjoint from $P$) determine a linear system of maximal rank. In addition, we show that certain auxiliary parameters can also be fixed. This enables us to conclude that the family of such differential equations is of the expected dimension and to define a birational map between an open subset of the moduli space of logarithmic connections with fixed logarithmic points and regular semi-simple residues and the Hilbert scheme of points on a quasi-projective surface.