# Spaces intersecting a plane non-trivially in $G(3,6)$

I want to understand the Schubert variety $$\Sigma\subseteq G(3,6)$$ representing 3-dim subspaces intersecting a given 2-dim subspace non-trivially. Is it smooth? How to describe $$det(T_{\Sigma})$$?

• if you denote by $A$ the given two dimensional subspace, then the locus you are looking for is the degenracy locus of $\mathcal{R} \longrightarrow \mathbb{C}^6/A \otimes \mathcal{O}_{G(3,6)}$, where $\mathcal{R}$ is the tautological bundle on $G(3,6)$. It should be of codimension 2 in $G(3,6)$. is not smooth and its singular locus is the strata $\{L \in G(3,6), \ L \supset A \}$. There is a huge litterature on degeneracy loci Jul 15, 2020 at 14:29