bokomslag Study of Singularities on Rational Curves Via Syzygies
Vetenskap & teknik

Study of Singularities on Rational Curves Via Syzygies

David Cox Andrew R Kustin Claudia Polini Bernd Ulrich

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  • 2013
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 Hilbert-Burch matrix ? for the row vector [g1,?,gn]. In the ""General Lemma"" the authors use the generalised row ideals of ? 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 ?. In the ""Triple Lemma"" the authors give a matrix ?? whose maximal minors parameterise the closure, in P2, of the blow-up at p of C in a neighbourhood of p. The authors apply the General Lemma to ?? 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 Hilbert-Burch 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 Hilbert-Burch matrix ? for a parameterisation of C
  • Författare: David Cox, Andrew R Kustin, Claudia Polini, Bernd Ulrich
  • Format: Pocket/Paperback
  • ISBN: 9780821887431
  • Språk: Engelska
  • Utgivningsdatum: 2013-03-30
  • Förlag: American Mathematical Society