Electronic ISBN:  9780821895139 
Product Code:  MEMO/222/1045.E 
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Book DetailsMemoirs of the American Mathematical SocietyVolume: 222; 2013; 116 ppMSC: Primary 14; 13; 65;
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 HilbertBurch 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 blowup 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 HilbertBurch 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 HilbertBurch matrix \(\varphi\) for a parameterization of \(\mathcal{C}\). 
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 HilbertBurch 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 HilbertBurch matrices and the configuration of singularities


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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 HilbertBurch 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 blowup 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 HilbertBurch 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 HilbertBurch matrix \(\varphi\) for a parameterization of \(\mathcal{C}\).

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 HilbertBurch 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 HilbertBurch matrices and the configuration of singularities