# A Study of Singularities on Rational Curves Via Syzygies

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*David Cox; Andrew R. Kustin; Claudia Polini; Bernd Ulrich*

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 \(\mathcal{C}\).

#### Table of Contents

# Table of Contents

## A Study of Singularities on Rational Curves Via Syzygies

- Chapter 0. Introduction, terminology, and preliminary results 112 free
- Chapter 1. The General Lemma 1122
- Chapter 2. The Triple Lemma 2334
- Chapter 3. The BiProj Lemma 2738
- Chapter 4. Singularities of multiplicity equal to degree divided by two 3748
- Chapter 5. The space of true triples of forms of degree 𝑑: the base point free locus, the birational locus, and the generic Hilbert-Burch matrix 5768
- Chapter 6. Decomposition of the space of true triples 7586
- Chapter 7. The Jacobian matrix and the ramification locus 8596
- Chapter 8. The conductor and the branches of a rational plane curve 91102
- Chapter 9. Rational plane quartics: a stratification and the correspondence between the Hilbert-Burch matrices and the configuration of singularities 101112
- Bibliography 115126