ACTA issues

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

Szilárd Szabó

Acta Sci. Math. (Szeged) 79:1-2(2013), 107-128

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.

AMS Subject Classification (1991): 34M03, 34M35

Keyword(s): second-order linear ordinary differential equation, regular singularity, apparent singularity

Received November 4, 2011, and in final form December 20, 2012. (Registered under 59/2011.)