Consider a rational projective curve C of degree d over an algebraically closed field kk. There are n homogeneous forms g1,...,gn of degree d in B=kk[x,y] which parameterise C in a birational, base point free, manner. The authors study the singularities of C by studying a Hilbert-Burch matrix f for the row vector [g1,...,gn]. In the ""General Lemma"" the authors use the generalised row ideals of f to identify the singular points on C, their multiplicities, the number of branches at each singular point, and the multiplicity of each branch. Let p be a singular point on the parameterised planar...