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

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• 作者：Andrew Kustin^a ; ^{kustin@math.sc.edu" class="auth_mail" title="E-mail the corresponding author} ; Claudia Polini^b ; ^{cpolini@nd.edu" class="auth_mail" title="E-mail the corresponding author} ; Bernd Ulrich^c ; ^{ulrich@math.purdue.edu" class="auth_mail" title="E-mail the corresponding author}
• 关键词：primary ; 13A30 ; 14H50 ; secondary ; 14H20 ; 13D02 ; 14E05
• 刊名：Journal of Algebra
• 出版年：2017
• 出版时间：1 January 2017
• 年：2017
• 卷：469
• 期：Complete
• 页码：188-250
• 全文大小：909 K

文摘

Consider a rational projective plane curve C$\mathcal{C}$ 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 3×2$3\times 2$ matrix φ   with column degrees d₁≤d₂$d_1\le d_2$. The Rees algebra R=R[It]$\mathcal{R}=R[It]$ of I   is the bi-homogeneous coordinate ring of the graph of the parameterization of C$\mathcal{C}$; and accordingly, there is a dictionary that translates between the singularities of C$\mathcal{C}$ and algebraic properties of the ring R$\mathcal{R}$ 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$\mathcal{A}$ of the natural map from the symmetric algebra Sym(I)$\mathrm{Sym}(I)$ onto R$\mathcal{R}$. The ideal A_≥d2−1${\mathcal{A}}_{\ge d_2-1}$, which is an approximation of A$\mathcal{A}$, can be obtained using linkage. We exploit the bi-graded structure of Sym(I)$\mathrm{Sym}(I)$ in order to describe the structure of an improved approximation A_≥d1−1${\mathcal{A}}_{\ge d_1-1}$ when d₁<d₂$d_1<d_2$ and φ   has a generalized zero in its first column. (The latter condition is equivalent to assuming that C$\mathcal{C}$ has a singularity of multiplicity d₂$d_2$.) In particular, we give the bi-degrees of a minimal bi-homogeneous generating set for this ideal. When 2=d₁<d₂$2=d_1<d_2$ and φ   has a generalized zero in its first column, then we record explicit generators for A$\mathcal{A}$. When d₁=d₂$d_1=d_2$, we provide a translation between the bi-degrees of a bi-homogeneous minimal generating set for A_d1−2${\mathcal{A}}_{d_1-2}$ and the number of singularities of multiplicity d₁$d_1$ that are on or infinitely near C$\mathcal{C}$. We conclude with a table that translates between the bi-degrees of a bi-homogeneous minimal generating set for A$\mathcal{A}$ and the configuration of singularities of C$\mathcal{C}$ when the curve C$\mathcal{C}$ has degree six.