Study of Singularities on Rational Curves Via Syzygies

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 φ for the row vector [g1,…,gn]. In the ""General Lemma"" the authors use the generalised row ideals of φ 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 φ. In the ""Triple Lemma"" the authors give a matrix φ′ 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 φ′ 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 φ for a parameterisation of C