A study of singularities on rational curves via Syzygies
 Author/Creator
 Cox, David A.
 Language
 English.
 Publication
 Providence, Rhode Island : American Mathematical Society, 2013.
 Physical description
 ix, 116 pages ; 25 cm.
 Series
 Memoirs of the American Mathematical Society ; no. 1045.
Access
Available online

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QA3 .A57 NO.1045

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QA3 .A57 NO.1045
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Contributors
Contents/Summary
 Bibliography
 Includes bibliographical references (pages 115116) and index.
 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 HilbertBurch 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 HilbertBurch matrices and the configuration of singularities Bibliography.
 (source: Nielsen Book Data)
 Publisher's Summary
 Consider a rational projective curve C of degree d over an algebraically closed field kk. There are n homogeneous forms g1, ..., gn of degree d in B=kk[x, y] which parameterise C in a birational, base point free, manner. The authors study the singularities of C by studying a HilbertBurch matrix f for the row vector [g1, ..., gn]. In the ""General Lemma"" the authors use the generalised row ideals of f to identify the singular points on 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 parameterised planar curve C which corresponds to a generalised zero of f. In the ""Triple Lemma"" the authors give a matrix f' whose maximal minors parameterise the closure, in P2, of the blowup at p of C in a neighbourhood of p. The authors apply the General Lemma to f' in order to learn about the singularities of C in the first neighbourhood of p. If C has even degree d=2c and the multiplicity of C at p is equal to c, then he applies the Triple Lemma again to learn about the singularities of C in the second neighbourhood of p. Consider rational plane curves C of even degree d=2c. The authors classify curves according to the configuration of multiplicity c singularities on or infinitely near 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 C of degree 2c is equivalent to the study of the scheme of generalised zeros of the fixed balanced HilbertBurch matrix f for a parameterisation of C.
(source: Nielsen Book Data)
Subjects
Bibliographic information
 Publication date
 2013
 Responsibility
 David Cox, Andrew R. Kustin, Claudia Polini, Bernd Ulrich.
 Series
 Memoirs of the American Mathematical Society, 00659266 ; no. 1045
 Note
 "March 2013, Volume 222, Number 1045 (fourth of 5 numbers)."
 ISBN
 9780821887431 (alk. paper)
 0821887432 (alk. paper)