A study of singularities on rational curves via Syzygies
QA3 .A57 NO.1045
- Unknown QA3 .A57 NO.1045
- Includes bibliographical references (pages 115-116) and index.
- 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 Hilbert-Burch 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 Hilbert-Burch 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 Hilbert-Burch 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 blow-up 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 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 f for a parameterisation of C.
(source: Nielsen Book Data)
- Publication date
- David Cox, Andrew R. Kustin, Claudia Polini, Bernd Ulrich.
- Memoirs of the American Mathematical Society, 0065-9266 ; no. 1045
- "March 2013, Volume 222, Number 1045 (fourth of 5 numbers)."