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 869316302666&_mathId=si1.gif&_user=111111111&_pii=S0021869316302666&_rdoc=1&_issn=00218693&md5=c2ce2e5aac6c1c13cb82e9a80c4939cd" title="Click to view the MathML source">C$\mathcal{C}$ parameterized by three homogeneous forms of the same degree in the polynomial ring 869316302666&_mathId=si2.gif&_user=111111111&_pii=S0021869316302666&_rdoc=1&_issn=00218693&md5=4807d8ccc05ee86b16972ca4bd4877b8" title="Click to view the MathML source">R=k[x,y]$R=k[x,y]$ over a field k. The ideal I   generated by these forms is presented by a homogeneous 869316302666&_mathId=si3.gif&_user=111111111&_pii=S0021869316302666&_rdoc=1&_issn=00218693&md5=a56cbaa67bf83633f45e2732f4fa44f4" title="Click to view the MathML source">3×2$3\times 2$ matrix φ   with column degrees 869316302666&_mathId=si4.gif&_user=111111111&_pii=S0021869316302666&_rdoc=1&_issn=00218693&md5=b0f308da4cc6749b5d5652874e5cce9e" title="Click to view the MathML source">d₁≤d₂$d_1\le d_2$. The Rees algebra 869316302666&_mathId=si5.gif&_user=111111111&_pii=S0021869316302666&_rdoc=1&_issn=00218693&md5=54f4250a0a8d0fba260674ad832cc556" title="Click to view the MathML source">R=R[It]$\mathcal{R}=R[It]$ of I   is the bi-homogeneous coordinate ring of the graph of the parameterization of 869316302666&_mathId=si1.gif&_user=111111111&_pii=S0021869316302666&_rdoc=1&_issn=00218693&md5=c2ce2e5aac6c1c13cb82e9a80c4939cd" title="Click to view the MathML source">C$\mathcal{C}$; and accordingly, there is a dictionary that translates between the singularities of 869316302666&_mathId=si1.gif&_user=111111111&_pii=S0021869316302666&_rdoc=1&_issn=00218693&md5=c2ce2e5aac6c1c13cb82e9a80c4939cd" title="Click to view the MathML source">C$\mathcal{C}$ and algebraic properties of the ring 869316302666&_mathId=si1108.gif&_user=111111111&_pii=S0021869316302666&_rdoc=1&_issn=00218693&md5=22e1e51bbbd8bb71e8c1523d1a18d8ab" title="Click to view the MathML source">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 869316302666&_mathId=si365.gif&_user=111111111&_pii=S0021869316302666&_rdoc=1&_issn=00218693&md5=b160292d7059363f59ccb0e9bfd237c2" title="Click to view the MathML source">A$\mathcal{A}$ of the natural map from the symmetric algebra 869316302666&_mathId=si324.gif&_user=111111111&_pii=S0021869316302666&_rdoc=1&_issn=00218693&md5=a384ed206583afe4cba7256b17666c60" title="Click to view the MathML source">Sym(I)$\mathrm{Sym}(I)$ onto 869316302666&_mathId=si1108.gif&_user=111111111&_pii=S0021869316302666&_rdoc=1&_issn=00218693&md5=22e1e51bbbd8bb71e8c1523d1a18d8ab" title="Click to view the MathML source">R$\mathcal{R}$. The ideal 869316302666&_mathId=si9.gif&_user=111111111&_pii=S0021869316302666&_rdoc=1&_issn=00218693&md5=02c991277f556e8f592fc7ec43a1f959" title="Click to view the MathML source">A_≥d2−1${\mathcal{A}}_{\ge d_2-1}$, which is an approximation of 869316302666&_mathId=si365.gif&_user=111111111&_pii=S0021869316302666&_rdoc=1&_issn=00218693&md5=b160292d7059363f59ccb0e9bfd237c2" title="Click to view the MathML source">A$\mathcal{A}$, can be obtained using linkage. We exploit the bi-graded structure of 869316302666&_mathId=si324.gif&_user=111111111&_pii=S0021869316302666&_rdoc=1&_issn=00218693&md5=a384ed206583afe4cba7256b17666c60" title="Click to view the MathML source">Sym(I)$\mathrm{Sym}(I)$ in order to describe the structure of an improved approximation 869316302666&_mathId=si10.gif&_user=111111111&_pii=S0021869316302666&_rdoc=1&_issn=00218693&md5=ff4b180559cee40eda23ddd0bc50c8e3" title="Click to view the MathML source">A_≥d1−1${\mathcal{A}}_{\ge d_1-1}$ when 869316302666&_mathId=si11.gif&_user=111111111&_pii=S0021869316302666&_rdoc=1&_issn=00218693&md5=3e302e1f1be388b955231dcb38c41e53" title="Click to view the MathML source">d₁<d₂$d_1<d_2$ and φ   has a generalized zero in its first column. (The latter condition is equivalent to assuming that 869316302666&_mathId=si1.gif&_user=111111111&_pii=S0021869316302666&_rdoc=1&_issn=00218693&md5=c2ce2e5aac6c1c13cb82e9a80c4939cd" title="Click to view the MathML source">C$\mathcal{C}$ has a singularity of multiplicity 869316302666&_mathId=si1004.gif&_user=111111111&_pii=S0021869316302666&_rdoc=1&_issn=00218693&md5=b6c268e68b2ac694e85584325b7b0daa" title="Click to view the MathML source">d₂$d_2$.) In particular, we give the bi-degrees of a minimal bi-homogeneous generating set for this ideal. When 869316302666&_mathId=si13.gif&_user=111111111&_pii=S0021869316302666&_rdoc=1&_issn=00218693&md5=7b671947a62da0f7e7b4a27001d43d8f" title="Click to view the MathML source">2=d₁<d₂$2=d_1<d_2$ and φ   has a generalized zero in its first column, then we record explicit generators for 869316302666&_mathId=si365.gif&_user=111111111&_pii=S0021869316302666&_rdoc=1&_issn=00218693&md5=b160292d7059363f59ccb0e9bfd237c2" title="Click to view the MathML source">A$\mathcal{A}$. When 869316302666&_mathId=si14.gif&_user=111111111&_pii=S0021869316302666&_rdoc=1&_issn=00218693&md5=16e2e472ada19bf0456d71fdd8a845e6" title="Click to view the MathML source">d₁=d₂$d_1=d_2$, we provide a translation between the bi-degrees of a bi-homogeneous minimal generating set for 869316302666&_mathId=si1067.gif&_user=111111111&_pii=S0021869316302666&_rdoc=1&_issn=00218693&md5=4416406c4cd9d58d9299669e23f54f22" title="Click to view the MathML source">A_d1−2${\mathcal{A}}_{d_1-2}$ and the number of singularities of multiplicity 869316302666&_mathId=si16.gif&_user=111111111&_pii=S0021869316302666&_rdoc=1&_issn=00218693&md5=d3f7693bb2c3c8477bde3abcba4a25e9" title="Click to view the MathML source">d₁$d_1$ that are on or infinitely near 869316302666&_mathId=si1.gif&_user=111111111&_pii=S0021869316302666&_rdoc=1&_issn=00218693&md5=c2ce2e5aac6c1c13cb82e9a80c4939cd" title="Click to view the MathML source">C$\mathcal{C}$. We conclude with a table that translates between the bi-degrees of a bi-homogeneous minimal generating set for 869316302666&_mathId=si365.gif&_user=111111111&_pii=S0021869316302666&_rdoc=1&_issn=00218693&md5=b160292d7059363f59ccb0e9bfd237c2" title="Click to view the MathML source">A$\mathcal{A}$ and the configuration of singularities of 869316302666&_mathId=si1.gif&_user=111111111&_pii=S0021869316302666&_rdoc=1&_issn=00218693&md5=c2ce2e5aac6c1c13cb82e9a80c4939cd" title="Click to view the MathML source">C$\mathcal{C}$ when the curve 869316302666&_mathId=si1.gif&_user=111111111&_pii=S0021869316302666&_rdoc=1&_issn=00218693&md5=c2ce2e5aac6c1c13cb82e9a80c4939cd" title="Click to view the MathML source">C$\mathcal{C}$ has degree six.