Introduction, terminology, and preliminary results -- The general lemma -- The triple lemma -- The BiProj lemma -- Singularities of multiplicity equal to degree divided by two -- The space of true triples of forms of degree d : the base point free locus, the birational locus, and the generic Hilbert-Burch matrix -- Decomposition of the space of true triples -- The Jacobian matrix and the ramification locus -- The conductor and the branches of a rational plane curve -- Rational place quartics : a stratification and the correspondence between the Hilbert-Burch matrices and the configuation of singularities.