A study of singularities on rational curves via syzygies

Abstract

Consider a rational projective curve C of degree d over an algebraically closed field k. There are n homogeneous forms g1;:::;g n of degree d in B = kk(x;y) which parameterize C in a birational, base point free, manner. We study the singularities of C by studying a Hilbert-Burch matrix ' for the row vector (g1;:::;g n). In the we use the generalized 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 parameterized planar curve C which corresponds to a general- ized zero of '. In the we give a matrix ' 0 whose maximal minors parameterize the closure, in P 2 , of the blow-up at p of C in a neighborhood of p. We apply the General Lemma to ' 0 in order to learn about the singularities of C in the first neighborhood of p. If C has even degree d = 2c and the multiplicity of C at p is equal to c, then we apply the Triple Lemma again to learn about the singularities of C in the second neighborhood of p. Consider rational plane curves C of even degree d = 2c. We classify curves according to the configuration of multiplicity c singularities on or infinitely near C. There are 7 possible configurations of such singularities. We 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 generalized zeros of the fixed balanced Hilbert-Burch matrix ' for a parameterization of C. Let