论文标题
代态的年度戈伦斯坦理想三
Equigenerated Gorenstein ideals of codimension three
论文作者
论文摘要
我们专注于标准多项式环中的均匀的戈伦斯坦理想$ i $ i $ $ i $ $ i $ r = \ kk [x_1，\ ldots，x_n] $在字段$ \ kk $上，假设$ i $是固定度$ d $ d $的$ i $。对于这样一个理想的$ i $，该学位与$ i $的发电机数量最少，以及简单公式中相关偏斜的矩阵的条目。我们给出了一个无基本特征的论点，其效果是，对于该公式链接的任何此类数据，都存在Gorenstein理想的$ i $ $ i $ codimension三填充它们。我们推测，对于任意$ n \ geq 2 $，理想的$ i \ subset \ kk [x_1，\ ldots，x_n] $由$ r \ geq n+2 $ $ d \ geq 2 $的一般集生成当$ n = 3 $时，我们仅在此猜想的含义时证明了“仅”。对于任意$ n \ geq 2 $，我们证明，如果$ d = 2 $和$ r \ geq（n+2）（n+1）/6 $，那么理想是Gorenstein，仅当且仅当$ r = {{n+1} \ select 2} -1 $，这会解决$ n \ n \ n \ leq的确定。最后，我们围绕Fröberg-Lundqvist的一个问题阐述。在不同的方向上，我们揭示了麦考拉（Macaulay）逆与所谓的牛顿二重奏之间的联系，到目前为止，这一问题尚未据我们所知。 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.
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.
