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arXiv 2005.12953math.AC
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余维3的等生成Gorenstein理想

Equigenerated Gorenstein ideals of codimension three

Dayane Lira, Zaqueu Ramos, Aron Simis

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摘要：

我们专注于结构的齐次Gorenstein理想$I$的余维3在一个标准的多项式环$R=\kk[x_1,\ldots,x_n]$在一个字段$\kk$，假设$I$是产生在一个固定的程度$d$。对于这样一个理想的$I$，这个度伴随着$I$的生成元的最小数量和相关反对称矩阵的条目的度在一个简单的公式中。我们给出了一个基本的无特征参数的效果，任何这样的数据由这个公式，存在一个Gorenstein理想$I$的余维3填充它们。我们猜想，对于任意的$n\geq 2$，由一般的$r\geq n+2$形式的次数$d\geq 2$生成的理想$I\subset \kk[x_1,\ldots,x_n]$是Gorenstein当且仅当$d=2$和$r= {{n+1}\choose 2}-1$ .我们证明了这个猜想的“只有当”的含义时，$n=3$。对于任意的$n\geq 2$，我们证明了如果$d=2$和$r\geq (n+2)(n+1)/6$，则理想是Gorenstein当且仅当$r={{n+1}\choose 2}-1$，从而解决了$n\leq 5$猜想的“如果”断言.最后，我们围绕弗勒贝格-伦德奎斯特的问题之一进行阐述。在另一个方向，我们揭示了麦考利逆和所谓的牛顿对偶之间的联系，这是一个迄今为止我们所知的问题。最后，我们考虑了链$(\ell_1^m,\ldots,\ell_n^m):\mathfrak{f}$何时等生成的问题，其中$\ell_1,\ldots,\ell_n$是独立线性形式，$\mathfrak{f}$是一个形式，在某些重要情况下给出了一个解.

英文摘要：

We focus on the structure of a homogeneous Gorenstein ideal $I$ of codimension three in a standard polynomial ring $R=\kk[x_1,\ldots,x_n]$ over a field $\kk$, assuming that $I$ is generated in a fixed degree $d$. For such an ideal $I$ this degree comes along with the minimal number of generators of $I$ and the degree of the entries of the associated skew-symmetric matrix in a simple formula. We give an elementary characteristic-free argument to the effect that, for any such data linked by this formula, there exists a Gorenstein ideal $I$ of codimension three filling them. We conjecture that, for arbitrary $n\geq 2$, an ideal $I\subset \kk[x_1,\ldots,x_n]$ generated by a general set of $r\geq n+2$ forms of degree $d\geq 2$ is Gorenstein if and only if $d=2$ and $r= {{n+1}\choose 2}-1$. We prove the `only if' implication of this conjecture when $n=3$. For arbitrary $n\geq 2$, we prove that if $d=2$ and $r\geq (n+2)(n+1)/6$ then the ideal is Gorenstein if and only if $r={{n+1}\choose 2}-1$, which settles the `if' assertion of the conjecture for $n\leq 5$. Finally, we elaborate around one of the questions of Fröberg--Lundqvist. In a different direction, we reveal a connection between the Macaulay inverse and the so-called Newton dual, a matter so far not brought out to our knowledge. Finally, we consider the question as to when the link $(\ell_1^m,\ldots,\ell_n^m):\mathfrak{f}$ is equigenerated, where $\ell_1,\ldots,\ell_n$ are independent linear forms and $\mathfrak{f}$ is a form, is given a solution in some important cases.

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