The bi-graded structure of symmetric algebras with applications to Rees rings

Consider a rational projective plane curve CC parameterized by three homogeneous forms of the same degree in the polynomial ring R=k[x,y]R=k[x,y] over a field k. The ideal I   generated by these forms is presented by a homogeneous 3×23×2 matrix φ   with column degrees d1≤d2d1≤d2. The Rees algebra R=R[It]R=R[It] of I   is the bi-homogeneous coordinate ring of the graph of the parameterization of CC; and accordingly, there is a dictionary that translates between the singularities of CC and algebraic properties of the ring RR and its defining ideal. Finding the defining equations of Rees rings is a classical problem in elimination theory that amounts to determining the kernel AA of the natural map from the symmetric algebra Sym(I)Sym(I) onto RR. The ideal A≥d2−1A≥d2−1, which is an approximation of AA, can be obtained using linkage. We exploit the bi-graded structure of Sym(I)Sym(I) in order to describe the structure of an improved approximation A≥d1−1A≥d1−1 when d1