论文标题
高度分级的Artinian代数的约旦单元中的理想发电机数量
Number of generators of ideals in Jordan cells of the family of graded Artinian algebras of height two
论文作者
论文摘要
我们让$ a = r/i $成为标准分级的Artinian代数商$ r = {\ sf k} [x，y] $，由理想$ i $ $ $ $的字段$ {\ sf k} $上的两个变量中的多项式戒指，让$ i $，让$ n $是其矢量空间空间。 A_1 $中的线性表单$ \ ell \的Jordan类型$ p_ \ ell $是$ n $的分区，以确定$ a $ a $ a $ \ ell $ $ \ ell $上的Jordan Block块分解 - 这是Nilpotent。前三位作者先前确定了$ n = \ dim _ {\ sf k}的哪些分区可能会作为Jordan类型作为Jordan类型的某种线性形式的$ \ Ell $，在分级的完整交点$ \ ell $上，$ a = r/（f，g）$ r $的$ a $ $ $ $ $，并且它们计算了每个完整互动$ t $ pare的数量。 我们在这里考虑了分级Artinian标准的namily $ \ mathrm {g} _t $ $ a = r/i $ of $ r = {\ sf k} [x，y] $，具有任意的希尔伯特函数$ h（a）= t $。 Jordan Cell $ \ MATHBB V（E_P）$对应于具有对角线长度$ t $的分区$ P $的所有理想$ i $ in $ r $中的所有理想是由$ p $确定的所有理想$ e_p $。这些单元格将品种$ \ mathrm {g} _t $分解为仿射空间。我们确定了每个单元格$ \ Mathbb V（e_p）$中的理想的通用数字$κ（p）$，从而概括了Arxiv的结果：1810.00716。特别是，我们确定了$κ（p）=κ（t）$的分区，这是定义代数$ a $ in $ \ mathrm {g} _t $的理想的通用发电机数。我们还计算具有给定$κ（p）$的对角线长度的分区数量。一个主要工具是组合和几何结果，使我们可以分配$ t $，任何分区$ p $对角度长度$ t $ s $ t $ simple $ t_i $和分区$ p_i $，因此$ \ mathbb v（e_p）$是单元格的产物，是单元格的产物$ \ mathrm {g} _ {t_i} $是Grassmannian。
We let $A=R/I$ be a standard graded Artinian algebra quotient of $R={\sf k}[x,y]$, the polynomial ring in two variables over a field ${\sf k}$ by an ideal $I$, and let $n$ be its vector space dimension. The Jordan type $P_\ell$ of a linear form $\ell\in A_1$ is the partition of $n$ determining the Jordan block decomposition of the multiplication on $A$ by $\ell$ -- which is nilpotent. The first three authors previously determined which partitions of $n=\dim_{\sf k}A$ may occur as the Jordan type for some linear form $\ell$ on a graded complete intersection Artinian quotient $A=R/(f,g)$ of $R$, and they counted the number of such partitions for each complete intersection Hilbert function $T$ arXiv:1810.00716.\par We here consider the family $\mathrm{G}_T$ of graded Artinian quotients $A=R/I$ of $R={\sf k}[x,y]$, having arbitrary Hilbert function $H(A)=T$. The Jordan cell $\mathbb V(E_P)$ corresponding to a partition $P$ having diagonal lengths $T$ is comprised of all ideals $I$ in $R$ whose initial ideal is the monomial ideal $E_P$ determined by $P$. These cells give a decomposition of the variety $\mathrm{G}_T$ into affine spaces. We determine the generic number $κ(P)$ of generators for the ideals in each cell $\mathbb V(E_P)$, generalizing a result of arXiv:1810.00716. In particular, we determine those partitions for which $κ(P)=κ(T)$, the generic number of generators for an ideal defining an algebra $A$ in $\mathrm{G}_T$. We also count the number of partitions $P$ of diagonal lengths $T$ having a given $κ(P)$. A main tool is a combinatorial and geometric result allowing us to split $T$ and any partition $P$ of diagonal lengths $T$ into simpler $T_i$ and partitions $P_i$, such that $\mathbb V(E_P)$ is the product of the cells $\mathbb V(E_{P_i})$, and $T_i$ is single-block: $\mathrm{G}_{T_i}$ is a Grassmannian.
