Memoirs of the American Mathematical Society 2013; 116 pp; softcover Volume: 222 ISBN10: 0821887432 ISBN13: 9780821887431 List Price: US$72 Individual Members: US$43.20 Institutional Members: US$57.60 Order Code: MEMO/222/1045
 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 HilbertBurch 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 blowup 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 HilbertBurch 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 HilbertBurch matrix \(\varphi\) for a parameterization of \(\mathcal{C}\). Table of Contents  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 HilbertBurch 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 plane quartics: A stratification and the correspondence between the HilbertBurch matrices and the configuration of singularities
 Bibliography
