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

JOURNAL OF ALGEBRA

卷:469

Article

Kustin, Andrew¹  Polini, Claudia²  Ulrich, Bernd³ 

[1] Univ South Carolina, Dept Math, Columbia, SC 29208 USA

[2] Univ Notre Dame, Dept Math, Notre Dame, IN 46556 USA

[3] Purdue Univ, Dept Math, W Lafayette, IN 47907 USA

关键词: Bi-graded structures;    Duality;    Elimination theory;    Generalized zero of a matrix;    Generator degrees;    Hilbert-Burch matrix;    Infinitely near singularities;    Koszul complex;    Local cohomology;    Linkage;    Matrices of linear forms;   

DOI  :  10.1016/j.jalgebra.2016.08.014

来源: Elsevier

【 摘 要 】

Consider a rational projective plane curve C parameterized by three homogeneous forms of the same degree in the polynomial ring R = k[x,y] over a field k. The ideal I generated by these forms is presented by a homogeneous 3 x 2 matrix phi with column degrees d(1) <= d(2). The Rees algebra R = R[It] of I is the bi-homogeneous coordinate ring of the graph of the parameterization of C; and accordingly, there is a dictionary that translates between the singularities of C and algebraic properties of the ring R and its defining ideal. Finding the defining equations of Rees rings is a classical problem in elimination theory that amounts to determining the kernel A of the natural map from the symmetric algebra Sym(I) onto R. The ideal A(>= d2-1), which is an approximation of A, can be obtained using linkage. We exploit the bi-graded structure of Sym(I) in order to describe the structure of an improved approximation A(>= d1-1) when d(1) < d(2) and phi has a generalized zero in its first column. (The latter condition is equivalent to assuming that C has a singularity of multiplicity d(2).) In particular, we give the bi-degrees of a minimal bi-homogeneous generating set for this ideal. When 2 = d(1) < d(2) 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 that are on or infinitely near C. We conclude with a table that translates between the bi-degrees of a bi-homogeneous minimal generating set for A and the configuration of singularities of C when the curve C has degree six. (C) 2016 Elsevier Inc. All rights reserved.

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