David A Cox 
Study of Singularities on Rational Curves Via Syzygies [PDF ebook] 

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 $/mathcal{C}$.