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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: 2013; Volume 222, Number 1045
ISBNs: 978-0-8218-8743-1 (print); 978-0-8218-9513-9 (online)
DOI: https://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: Primary 14H20, 13H15, 13H10, 13A30, 14H50, 14H10, 14Q05, 65D17

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Table of Contents

Chapters

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

Abstract

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. We 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” we 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” we 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 $p$. We 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 we apply 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 $d=2c$. We classify curves according to the configuration of multiplicity $c$ singularities on or infinitely near $\mathcal {C}$. There are $7$ possible configurations of such singularities. We 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 $\mathcal {C}$. Let \begin{equation*} \operatorname {BalH}_{d} = \left \{ \varphi \left \vert \begin {matrix} \text {$\varphi $ is a $3\times 2$ matrix; each entry in $\varphi $ is a homogeneous} \\ \text {form of degree $c$ from $B$; and $\operatorname {ht} I_{2}(\varphi )=2$} \end{matrix} \right . \right \}. \end{equation*} The group $G=\operatorname {GL}_{3}(\pmb k)\times \operatorname {GL}_{2}(\pmb k)$ acts on $\operatorname {BalH}_{d}$ by way of $(\chi ,\xi )\cdot \varphi =\chi \varphi \xi ^{-1}$. We decompose $\operatorname {BalH}_{d}$ into a disjoint union of 11 orbits. Each orbit has the form $G\cdot M$, where $M$ is the closed irreducible subspace of affine space defined by the maximal order minors of a generic matrix.

We introduce the parameter space $\mathbb {A}_{d}$. Each element $\pmb g$ of $\mathbb {A}_{d}$ is an ordered triple of $d$-forms from $B$, and each $\pmb g\in \mathbb {A}_{d}$ induces a rational map $\Psi _{\pmb g}\,:\, \xymatrix {\mathbb {P}^{1}\ar @{–>}[r]&\mathbb {P}^{2}}$. We define \begin{equation*} \begin {split} \mathbb {T}_{d}&{}=\left \{\pmb g\in \mathbb {A}_{d}\mid \text {$\Psi _{\pmb g}$ is birational onto its image without base points}\right \}\ \text {and}\\ \mathbb {B}_{d}&{}=\left \{\pmb g\in \mathbb {T}_{d}\mid \begin {matrix}\text {every entry in the corresponding homogeneous Hilbert-Burch}\\ \text {matrix has degree $d/2$}\end{matrix}\right \}. \end{split}\end{equation*} In practice we are only interested in the subset $\mathbb {T}_{d}$ of $\mathbb {A}_{d}$. Every element of $\mathbb {A}_{d}$ which is not in $\mathbb {T}_{d}$ corresponds to an unsuitable parameterization of a curve. We prove that if there is a multiplicity $c$ singularity on or infinitely near a point $p$ on a curve $\mathcal {C}$ of degree $d=2c$, then the parameterization of $\mathcal {C}$ is an element of $\mathbb {B}_{d}$. We prove that $\mathbb {B}_{d}\subseteq \mathbb {T}_{d}$ are open subsets of $\mathbb {A}_{d}$. We identify an open cover $\cup \mathbb {B}_{d}^{(i)}$ of $\mathbb {B}_{d}$ and for each $\mathbb {B}_{d}^{(i)}$ in this open cover, we identify a generic Hilbert-Burch matrix which specializes to give a Hilbert-Burch matrix for $\pmb g$ for each $\pmb g\in \mathbb {B}_{d}^{(i)}$. As an application of this result, we identify a universal projective resolution for the graded Betti numbers \begin{equation*}0\to B(-3c)^{2} \to B(-2c)^{3}\to B. \end{equation*}

We decompose the space $\mathbb {B}_{d}$ of balanced triples into strata. Each stratum consists of those triples $\pmb g$ in $\mathbb {B}_{d}$ for which the corresponding curve $\mathcal {C}_{\pmb g}$ exhibits one particular configuration of multiplicity $c$ 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 $D$ is an algebra which is essentially of finite type over the ring $C$, then the ramification locus of $D$ over $C$ is equal to the support of the module of Kähler differentials $\Omega _{D/C}$. The General Lemma is a local result. Once one knows the singularities $\{p_{i}\}$ on a parameterized curve $\mathcal {C}$, then the General Lemma shows how to read the multiplicity and number of branches at each point $p_{i}$ 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 $p$ on $\mathcal {C}$ and all of the branches of $\mathcal {C}$ at $p$ 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 $\mathcal {C}$.

We use conductor techniques to study the singularity degree $\delta$. Let $\pmb g=(g_{1},g_{2},g_{3})$ be an element of $\mathbb {T}_{d}$ and $\mathcal {C}_{\pmb g}$ be the corresponding parameterized plane curve. We produce a polynomial $c_{\pmb g}$ whose factorization into linear factors gives the value of the invariant $\delta$ at each singular point of $\mathcal {C}_{\pmb g}$. The polynomial $c_{\pmb g}$ is obtained in a polynomial manner from the coefficients of the entries of a Hilbert-Burch matrix for $\pmb g$. We use these ideas to produce closed sets in $\mathbb {B}_{d}$ which separate various configurations of singularities. To create $c_{\pmb g}$, we start with the coordinate ring $A_{\pmb g}=\pmb k[g_{1},g_{2},g_{3}]\subseteq B$. If $\pmb V$ is the $d^{\text {th}}$ Veronese subring of $B$ and $\mathfrak {c}_{\pmb g}$ is the conductor $A_{\pmb g}\,:\,\! \pmb V$, then $c_{\pmb g}$ generates the saturation of the extension of $\mathfrak {c}_{\pmb g}$ to $B$.

In the final chapter of the paper, we apply our results to rational plane quartics. We exhibit a stratification of $\mathbb {B}_{4}$ in which every curve associated to a given stratum has the same configuration of singularities and we compute the dimension of each stratum.