**Memoirs of the American Mathematical Society**

2013;
116 pp;
Softcover

MSC: Primary 14; 13; 65;

Print ISBN: 978-0-8218-8743-1

Product Code: MEMO/222/1045

List Price: $72.00

Individual Member Price: $43.20

**Electronic ISBN: 978-0-8218-9513-9
Product Code: MEMO/222/1045.E**

List Price: $72.00

Individual Member Price: $43.20

# A Study of Singularities on Rational Curves Via Syzygies

Share this page
*David A Cox; Andrew R Kustin; Claudia B 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