A Study of Singularities on Rational Curves Via Syzygies

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 $mathcal{C}$.