Let k be a field, R=k[x,y] a polynomial ring in 2 variables over k, and I a height 2$ideal of R minimally generated by 3 forms, g_1,g_2,g_3 of the same positive degree d. The Hilbert-Burch Theorem guarantees that there is a 3x2 matrix \varphi, with homogeneous entries from R, so that the signed 2x2 minors of \varphi are equal to g_1, g_2, and g_3. We arrange \varphi so that each entry in column i of \varphi has degree d_i, with d_1< d_2.
Let \mathcal R be the Rees algebra of I, that is
\mathcal R=R\oplus I\oplus I^2\oplus I^3\oplus \dots=R[It],
\mathcal A be the kernel of the natural surjection
Sym(I)\twoheadrightarrow \mathcal R
from the symmetric algebra of I to the Rees algebra of I and let S and B be the polynomial rings S=k[T_1,T_2,T_3] and B=R\otimes_kS=k[x,y,T_1,T_2,T_3]. View B as a bi-graded k-algebra, where x and y have bi-degree (1,0) and each T_i has bi-degree $(0,1).
We describe the S-module structure of \mathcal A_{(*,\ge d_1-1)} under the hypothesis that \varphi has a generalized zero in its first column. This module is free and we identify the bi-degrees of its basis. We also identify the bi-degrees of a minimal generating set of \mathcal A_{(*,\ge d_1-1)} as an ideal of Sym(I)$. Our proof is motivated by a Theorem of Weierstrass and Kronecker which classifies matrices with homogeneous linear entries in two variables. When one views this result in a geometric context, that is,
[g_1,g_2,g_3]:\mathbb P^1\to \mathcal C\subseteq \mathbb P^2
is a birational parameterization of a plane curve, then \operatorname{Bi-Proj} \mathcal R is the graph, \Gamma, of the parameterization of \mathcal C, the hypothesis concerning the existence of a generalized zero is equivalent to assuming that \mathcal C has a singularity of multiplicity d_1, and the ideal \mathcal A_{(*,\ge d_1-1)} is an approximation of the ideal \mathcal A which defines \Gamma. |