2-步理想与交换矩阵
2-step ideals and commuting matrices
- University of Ljubljana, Faculty of mathematics and physics（卢布尔雅那大学数学物理学院）
- Institute of mathematics, physics and mechanics（数学、力学与物理研究所）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文利用 Hilbert 方案与交换矩阵的对应关系，定义与 2-步理想对应的矩阵轨迹，估计其维数，为 n=3 情形下 2-步理想轨迹维数提供新证明，并省略部分假设。
AI中文摘要：
2-步理想是多项式环中满足 $\mathfrak{m}^{k+2}\subsetneq I\subsetneq \mathfrak{m}^k$ 的理想 $I$，其中 $\mathfrak{m}$ 是由变量生成的极大理想。这类理想最近在 [F. Giovenzana, L. Giovenzana, M. Graffeo, P. Lella: New components of Hilbert schemes of points and 2-step ideals, 2025, arxiv: 2507.02789] 中被引入，目的是获得 $\mathrm{Hilb}^d(\mathbb{A}^n)$ 的新的不可约分支，特别是获得 $\mathrm{Hilb}^d(\mathbb{A}^3)$ 中维数较大的新轨迹。在本文中，我们利用 Hilbert 方案与交换矩阵簇之间的对应关系，定义与 2-步理想对应的交换矩阵轨迹。然后我们估计所得轨迹的维数，从而为 $n=3$ 情形下 2-步理想轨迹维数的估计提供新的证明。在这一情形下，我们还能够省略上述论文中维数估计中的某些假设。
英文摘要：
2-step ideals are ideals $I$ of the polynomial ring that satisfy $\mathfrak{m}^{k+2}\subsetneq I\subsetneq \mathfrak{m}^k$ where $\mathfrak{m}$ is the maximal ideal generated by variables. This class of ideals was recently introduced in [F. Giovenzana, L. Giovenzana, M. Graffeo, P. Lella: New components of Hilbert schemes of points and 2-step ideals, 2025, arxiv: 2507.02789] with the aim to obtain new irreducible components of $\mathrm{Hilb}^d(\mathbb{A}^n)$ and in particular to obtain new loci in $\mathrm{Hilb}^d(\mathbb{A}^3)$ of large dimension. In this paper we use the correspondence between Hilbert schemes and varieties of commuting matrices to define loci of commuting matrices that correspond to 2-step ideals. Then we estimate the dimensions of the obtained loci to get new proofs for the estimates of dimensions of loci of 2-step ideals in the case $n=3$. In this case we are also able to omit some assumptions in the dimension estimates in the above mentioned paper.
