  We consider the vortex equations for a U(n) gauge field coupled to a Higgs
field with values on the n times n square matrices. It is known that when these
equations are defined on a compact Riemann surface, their moduli space of
solutions is closely related to a moduli space of tau-stable holomorphic
n-pairs on that surface. Using this fact and a local factorization result for
the Higgs matrix, we show that the vortex solutions are entirely characterized
by (1) the location in the surface of the zeros of the determinant of the Higgs
matrix and (2) by the choice of a vortex internal structure at each of these
zeros. We describe explicitly the vortex internal spaces and show that they are
compact and connected spaces.
