Consider a rational projective curve \(\mathcal{C}\) of degree \(d\) over an algebraically closed field \(\pmb k\). There are \(n\) homogeneous forms \(g_{1},\dots ,g_{n}\) of degree \(d\) in \(B=\pmb k[x,y]\) which parameterize \(\mathcal{C}\) in a birational, base point free, manner. The authors study the singularities of \(\mathcal{C}\) by studying a Hilbert-Burch matrix \(\varphi\) for the row vector \([g_{1},\dots ,g_{n}]\). In the “General Lemma” the authors use the generalized row ideals of \(\varphi\) to identify the singular points on \(\mathcal{C}\), their multiplicities, the number of branches at each singular point, and the multiplicity of each branch.

Let \(p\) be a singular point on the parameterized planar curve \(\mathcal{C}\) which corresponds to a generalized zero of \(\varphi\). In the “Triple Lemma” the authors give a matrix \(\varphi'\) whose maximal minors parameterize the closure, in \(\mathbb{P}^{2}\), of the blow-up at \(p\) of \(\mathcal{C}\) in a neighborhood of \(p\). The authors apply the General Lemma to \(\varphi'\) in order to learn about the singularities of \(\mathcal{C}\) in the first neighborhood of \(p\). If \(\mathcal{C}\) has even degree \(d=2c\) and the multiplicity of \(\mathcal{C}\) at \(p\) is equal to \(c\), then he applies the Triple Lemma again to learn about the singularities of \(\mathcal{C}\) in the second neighborhood of \(p\).

Consider rational plane curves \(\mathcal{C}\) of even degree \(d=2c\). The authors classify curves according to the configuration of multiplicity \(c\) singularities on or infinitely near \(\mathcal{C}\). There are \(7\) possible configurations of such singularities. They classify the Hilbert-Burch matrix which corresponds to each configuration. The study of multiplicity \(c\) singularities on, or infinitely near, a fixed rational plane curve \(\mathcal{C}\) of degree \(2c\) is equivalent to the study of the scheme of generalized zeros of the fixed balanced Hilbert-Burch matrix \(\varphi\) for a parameterization of \(\mathcal{C}\).