Secant varieties of P2 × Pn embedded by O(1, 2)

We describe the defining ideal of the rth secant variety of P × P embedded by O(1, 2), for arbitrary n and r 5. We also present the Schur module decomposition of the space of generators of each such ideal. Our main results are based on a more general construction for producing explicit matrix equations that vanish on secant varieties of products of projective spaces. This extends previous work of Strassen and Ottaviani.

我们描述了由\(O(1, 2)\)嵌入的\(\mathbb{P}^n\times\mathbb{P}^n\)的第\(r\)个割线簇的定义理想，其中\(n\)是任意的且\(r\leqslant n\)。我们还给出了每个此类理想的生成元空间的舒尔模分解。我们的主要结果基于一种更一般的构造，用于生成在射影空间乘积的割线簇上消失的显式矩阵方程。这扩展了施特拉森（Strassen）和奥塔维亚尼（Ottaviani）之前的工作。