Rational normal scrolls and the defining equations of Rees algebras
Rational normal scrolls and the defining equations of Rees algebras
复制标题
有理常态卷轴和里斯代数的定义方程
DOI：
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发表时间：
2008
期刊：
影响因子：
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通讯作者：
B. Ulrich
中科院分区：
文献类型：
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作者：
A. Kustin;C. Polini;B. Ulrich
考虑多项式环R = k[x，y]中由m个d次齐次形式极小生成的高度为2的理想I.假设I的齐次表示矩阵φ中的一列有n次元素，而φ的所有其他元素都是线性的。我们确定了一个明确的生成集的理想，它定义了里斯代数S = R[它]，所以对于多项式环S = R[T 1，. . .，Tm ]。对于所有的幂s，我们将ε分解为S-模，将Is分解为R-模。证明使用齐次坐标环，A = S/H，一个合理的正常涡卷，与。理想同构于A的高1素理想K的n次符号幂。理想K（n）由单项式生成。只要有可能，我们就研究A/K（n）来代替，因为K（n）的生成元比的生成元简单得多。我们得到了K（n）的一个滤子，其中因子是多项式环，超曲面环，或由广义Dupon-Northcott复形分解的模. I参数的生成元表示射影m - 1空间中的一条代数曲线.特殊纤维环的定义方程给出了隐式化问题的解.
Abstract Consider a height two ideal, I, which is minimally generated by m homogeneous forms of degree d in the polynomial ring R = k[x, y]. Suppose that one column in the homogeneous presenting matrix φ of I has entries of degree n and all of the other entries of φ are linear. We identify an explicit generating set for the ideal which defines the Rees algebra ℛ = R[It]; so for the polynomial ring S = R[T 1, . . . , Tm ]. We resolve ℛ as an S-module and Is as an R-module, for all powers s. The proof uses the homogeneous coordinate ring, A = S/H, of a rational normal scroll, with . The ideal is isomorphic to the n th symbolic power of a height one prime ideal K of A. The ideal K (n) is generated by monomials. Whenever possible, we study A/K (n) in place of because the generators of K (n) are much less complicated then the generators of . We obtain a filtration of K (n) in which the factors are polynomial rings, hypersurface rings, or modules resolved by generalized Eagon–Northcott complexes. The generators of I parameterize an algebraic curve in projective m – 1 space. The defining equations of the special fiber ring ℛ/(x, y)ℛ yield a solution of the implicitization problem for .