  In this paper, we study triples of the form (E, theta, phi) over a compact
Riemann Surface, where (E, theta) is a Higgs bundle and phi is a global
holomorphic section of the Higgs bundle. Our main result is a description of a
birational equivalence which relates geometrically the moduli space of Higgs
bundles of rank r and degree d to the moduli space of Higgs bundles of rank
(r-1) and degree d.
