发表机构
Università di Trento; Max Planck Institute for Mathematics in the Sciences（特伦托大学; 马克斯·普朗克科学促进研究所数学系）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文发展 Schur 极性理论以研究旗簇割线簇，引入几何 Schur 平方和 Schur 对偶 Terracini 引理，并确定 $\operatorname{Fl}(1,2;V_n)$ 在 $\mathcal{O}(1,1)$ 嵌入下所有割线簇的维数，仅两个缺陷情形。
AI 中文摘要
我们发展了一种一般的 Schur 极性一阶理论，用于研究任意齐次嵌入中旗簇的割线簇。将经典的 Veronese 簇的极性与肥点对应关系进行推广，我们证明在 Schur 设定下，极性理想的代数平方不必与几何双点条件一致。我们引入一个几何 Schur 平方，其相关分量是余法空间，从而得到 Schur 对偶 Terracini 引理。我们的构造在对称情形下恢复了经典极性理论。一个逐槽一致性定理将这些内在条件实现为多分次双点。作为应用，我们确定了由 $\mathcal{O}(1,1)$ 嵌入的 $\operatorname{Fl}(1,2;V_n)$ 的所有割线簇的维数：唯一的缺陷情形是 $\sigma_2(\operatorname{Fl}(1,2;V_3))$ 和 $\sigma_3(\operatorname{Fl}(1,2;V_4))$，两者缺陷均为 1。
英文摘要
We develop a general first-order theory of Schur apolarity for the study of secant varieties of flag varieties in arbitrary homogeneous embeddings. Extending the classical apolarity--fat-point correspondence for Veronese varieties, we show that in the Schur setting the algebraic square of the apolar ideal need not coincide with the geometric double-point conditions. We introduce a geometric Schur square whose relevant component is the conormal space, yielding a Schur Dual Terracini Lemma. Our construction recovers classical apolarity in the symmetric case. A slot-by-slot Consistency Theorem realizes these intrinsic conditions as multigraded double points. As an application, we determine the dimensions of all secant varieties of $\operatorname{Fl}(1,2;V_n)$ embedded by $\mathcal{O}(1,1)$: the only defective cases are $σ_2(\operatorname{Fl}(1,2;V_3))$ and $σ_3(\operatorname{Fl}(1,2;V_4))$, both of defect one.
Comments37 pages. Comments are welcome!
