The bi-graded structure of Symmetric Algebras with applications to Rees rings

Abstract

Consider a rational projective plane curve C parameterized by three homogeneous forms h1,h2,h3 of the same degree d in the polynomial ring R=k[x,y] over the field k. Extracting a common factor, we may harmlessly assume that the ideal I=(h1,h2,h3)R has height two. Let phi be a homogeneous minimal Hilbert-Burch matrix for the row vector [h1,h2,h3]. So, phi is a 3 by 2 matrix of homogeneous forms from R; the entries in column m have degree dm, with d1 \le d2 and d1+d2=d. The Rees algebra $cal R$ of I is the subring k[h1t,h2t,h3t] of the polynomial ring k[t]. The bi-projective spectrum of $cal R$ is the graph of the parameterization of C; and therefore, there is a dictionary which translates between the singularities of C and the algebra structure of $cal R$. The ring $cal R$ is the quotient of the symmetric algebra Sym(I) by the ideal, A, of local cohomology with support in the homogeneous maximal ideal of R. The ideal A_{\ge d2-1}, which is an approximation of A, can be calculated using linkage. We exploit the bi-graded structure of Sym(I) in order to describe the structure of an improved approximation A_{\ge d1-1} when $d1<d2$ and phi has a generalized zero in its first column. (The later condition is equivalent to assuming that C has a singularity of multiplicity d2.) In particular, we give the bi-degrees of a minimal bi-homogeneous generating set for this ideal. When 2=d1<d2 and phi has a generalized zero in its first column, then we record explicit generators for A. When d1=d2, we provide a translation between the bi-degrees of a bi-homogeneous minimal generating set for A_{d1-2} and the number of singularities of multiplicity d1 which are on or infinitely near C. We conclude with a table which translates between the bi-degrees of a bi-homogeneous minimal generating set for A and the configuration of singularities of C in the case that the curve C has degree six.