论文标题
二次呈现的戈伦斯坦理想
Quadratically presented Gorenstein ideals
论文作者
论文摘要
让$ j $是标准分级多项式环$ r = k [x，y，z] $的四级戈伦斯坦理想的四级gorenstein理想，其中$ k $是一个字段。假设$ r/j $满足弱的lefschetz物业。我们将$ j $的演示文稿矩阵按照$ j $的Macaulay反向系统的系数。 （此演示文稿矩阵是一个交替的矩阵，$ j $由演示文稿矩阵的最大订单Pfaffians生成。）我们的公式是计算机友好的；它们仅涉及矩阵乘法；它们不涉及多线性代数或复杂的总结。作为一个应用程序，我们给出了$ j_1 =（x^{n+1}的演示矩阵，y^{n+1}，z^{n+1}）:( x+y+y+z）^{n+1} $，当$ n $均匀，$ k $的特征为零。以前已经确定了$ J_1 $的发电机；但是以前尚不知道$ J_1 $的演示文稿矩阵。我们证明的第一步是，在麦考雷逆系统的系数上以$ i $ $ $ $ $ $ $而言，为线性呈现的三级Gorenstein级别的介绍矩阵提供了改进的公式。
Let $J$ be a quadratically presented grade three Gorenstein ideal in the standard graded polynomial ring $R= k[x,y,z]$, where $k$ is a field. Assume that $R/J$ satisfies the weak Lefschetz property. We give the presentation matrix for $J$ in terms of the coefficients of a Macaulay inverse system for $J$. (This presentation matrix is an alternating matrix and $J$ is generated by the maximal order Pfaffians of the presentation matrix.) Our formulas are computer friendly; they involve only matrix multiplication; they do not involve multilinear algebra or complicated summations. As an application, we give the presentation matrix for $J_1=(x^{n+1},y^{n+1},z^{n+1}):(x+y+z)^{n+1}$, when $n$ is even and the characteristic of $k$ is zero. Generators for $J_1$ had been identified previously; but the presentation matrix for $J_1$ had not previously been known. The first step in our proof is to give improved formulas for the presentation matrix of a linearly presented grade three Gorenstein ideal $I$ in terms of the coefficients of the Macaulay inverse system for $I$.
