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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)
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


  • 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 $p$: 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

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