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A Study of Singularities on Rational Curves Via Syzygies
David Cox, Amherst College, MA, Andrew R. Kustin, University of South Carolina, Columbia, SC, Claudia Polini, University of Notre Dame, IN, and Bernd Ulrich, Purdue University, West Lafayette, IN
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Memoirs of the American Mathematical Society
2013; 116 pp; softcover
Volume: 222
ISBN-10: 0-8218-8743-2
ISBN-13: 978-0-8218-8743-1
List Price: US$72
Individual Members: US$43.20
Institutional Members: US$57.60
Order Code: MEMO/222/1045
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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

  • 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 Hilbert-Burch 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 Hilbert-Burch matrices and the configuration of singularities
  • Bibliography
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