Let \(\mathcal {H}_{a,b}^n\) denote the component of the Hilbert scheme whose general point parameterizes an a-plane union a b-plane meeting transversely in \({\mathbf {P}}^n\). We show that \(\mathcal {H}_{a,b}^n\) is smooth and isomorphic to successive blow ups of \(\mathbf {Gr}(a,n) \times \mathbf {Gr}(b,n)\) or \(\text {Sym}^2 \mathbf {Gr}(a,n)\) along certain incidence correspondences. We classify the subschemes parameterized by \(\mathcal {H}_{a,b}^n\) and show that this component has a unique Borel fixed point. We also study the birational geometry of this component. In particular, we describe the effective and nef cones of \(\mathcal {H}_{a,b}^n\) and determine when the component is Fano. Moreover, we show that \(\mathcal {H}_{a,b}^n\) is a Mori dream space for all values of a, b, n.

让\(\mathcal {H}_{a,b}^n\)表示希尔伯特方案的组件，其一般点参数化a平面联合 a b平面在\({\mathbf {P}} ^n\)。我们证明\(\mathcal {H}_{a,b}^n\)是平滑的，并且与\(\mathbf {Gr}(a,n) \times \mathbf {Gr}(b ,n)\)或\(\text {Sym}^2 \mathbf {Gr}(a,n)\)沿着某些关联对应。我们对由\(\mathcal {H}_{a,b}^n\)参数化的子方案进行分类，并表明该组件具有唯一的 Borel 不动点。我们还研究了该组件的双有理几何。特别地，我们描述了有效和 nef 锥体\(\mathcal {H}_{a,b}^n\)并确定组件何时为 Fano。此外，我们证明\(\mathcal {H}_{a,b}^n\)是所有a ,  b ,  n值的 Mori 梦空间。