# A study of singularities on rational curves via syzygies

### About this Title

**David Cox**, *Department of Mathematics, Amherst College, Amherst, Massachusetts 01002-5000*, **Andrew R. Kustin**, *Mathematics Department, University of South Carolina, Columbia, South Carolina 29208*, **Claudia Polini**, *Mathematics Department, University of Notre Dame, Notre Dame, Indiana 46556* and **Bernd Ulrich**, *Department of Mathematics, Purdue University, West Lafayette, Indiana 47907*

Publication: Memoirs of the American Mathematical Society

Publication Year
2012: Volume 222, Number 1045

ISBNs: 978-0-8218-8743-1 (print); 978-0-8218-9513-9 (online)

DOI: http://dx.doi.org/10.1090/S0065-9266-2012-00674-5

Published electronically: September 11, 2012

Keywords: Axial singularities, Balanced Hilbert-Burch matrix, Base point free locus,
birational locus, birational parameterizations, branches of a rational
plane
curve, conductor, configuration of singularities, generalized row ideal,
generalized zero of a matrix, generic Hilbert-Burch matrix, Hilbert-Burch
matrix, infinitely near singularities, Jacobian matrix, module of
Kähler
differentials, multiplicity, parameterization, parameterization of a
blow-up,
ramification locus, rational plane curve, rational plane quartics, rational
plane sextics, scheme of generalized zeros, singularities of multiplicity
equal to degree divided by two, strata of rational plane curves, Taylor
resultant,
universal projective resolution, Veronese subring

MSC (2010): Primary 14H20, 13H15, 13H10, 13A30, 14H50, 14H10, 14Q05, 65D17

### Table of Contents

**Chapters**

- Chapter 0. Introduction, terminology, and preliminary results
- Chapter 1. The general lemma
- Chapter 2. The triple lemma
- Chapter 3. The BiProj lemma
- Chapter 4. Singularities of multiplicity equal to degree divided by two
- Chapter 5. The space of true triples of forms of degree : the base point free locus, the birational locus, and the generic Hilbert-Burch matrix
- Chapter 6. Decomposition of the space of true triples
- Chapter 7. The Jacobian matrix and the ramification locus
- Chapter 8. The conductor and the branches of a rational plane curve
- Chapter 9. Rational plane quartics: A stratification and the correspondence between the Hilbert-Burch matrices and the configuration of singularities

### Abstract

Consider a rational projective curve of degree over an algebraically closed field . There are homogeneous forms of degree in which parameterize in a birational, base point free, manner. We study the singularities of by studying a Hilbert-Burch matrix for the row vector . In the “General Lemma” we use the generalized row ideals of to identify the singular points on , their multiplicities, the number of branches at each singular point, and the multiplicity of each branch.

Let be a singular point on the parameterized planar curve which corresponds to a generalized zero of . In the “Triple Lemma” we give a matrix whose maximal minors parameterize the closure, in , of the blow-up at of in a neighborhood of . We apply the General Lemma to in order to learn about the singularities of in the first neighborhood of . If has even degree and the multiplicity of at is equal to , then we apply the Triple Lemma again to learn about the singularities of in the second neighborhood of .

Consider rational plane curves of even degree . We classify curves according to the configuration of multiplicity singularities on or infinitely near . There are possible configurations of such singularities. We classify the Hilbert-Burch matrix which corresponds to each configuration. The study of multiplicity singularities on, or infinitely near, a fixed rational plane curve of degree is equivalent to the study of the scheme of generalized zeros of the fixed balanced Hilbert-Burch matrix for a parameterization of . Let

The group acts on by way of . We decompose into a disjoint union of 11 orbits. Each orbit has the form , where is the closed irreducible subspace of affine space defined by the maximal order minors of a generic matrix.

We introduce the parameter space . Each element of is an ordered triple of -forms from , and each induces a rational map . We define

In practice we are only interested in the subset of . Every element of which is not in corresponds to an unsuitable parameterization of a curve. We prove that if there is a multiplicity singularity on or infinitely near a point on a curve of degree , then the parameterization of is an element of . We prove that are open subsets of . We identify an open cover of and for each in this open cover, we identify a generic Hilbert-Burch matrix which specializes to give a Hilbert-Burch matrix for for each . As an application of this result, we identify a universal projective resolution for the graded Betti numbers

We decompose the space of balanced triples into strata. Each stratum consists of those triples in for which the corresponding curve exhibits one particular configuration of multiplicity singularities.

We use the Jacobian matrix associated to the parameterization to identify the non-smooth branches of the curve as well as the multiplicity of each branch. The starting point for this line of reasoning is the result that if is an algebra which is essentially of finite type over the ring , then the ramification locus of over is equal to the support of the module of Kähler differentials . The General Lemma is a local result. Once one knows the singularities on a parameterized curve , then the General Lemma shows how to read the multiplicity and number of branches at each point from the Hilbert-Burch matrix of the parameterization. The result about the Jacobian matrix is a global result. It describes, in terms of the parameterization, all of the points on and all of the branches of at for which the multiplicity is at least two. In contrast to the General Lemma one may apply the Jacobian matrix technique before one knows the singularities on .

We use conductor techniques to study the singularity degree . Let be an element of and be the corresponding parameterized plane curve. We produce a polynomial whose factorization into linear factors gives the value of the invariant at each singular point of . The polynomial is obtained in a polynomial manner from the coefficients of the entries of a Hilbert-Burch matrix for . We use these ideas to produce closed sets in which separate various configurations of singularities. To create , we start with the coordinate ring . If is the Veronese subring of and is the conductor , then generates the saturation of the extension of to .

In the final chapter of the paper, we apply our results to rational plane quartics. We exhibit a stratification of in which every curve associated to a given stratum has the same configuration of singularities and we compute the dimension of each stratum.

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