A Study of Singularities on Rational Curves Via Syzygies

ISBN-13: 9780821887431 / Angielski

Andrew R. Kustin;Claudia Polini;Bernd Ulrich

Opis

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

Spis treści

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