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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 $det(T_{\Sigma})$?

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    $\begingroup$ 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 $\endgroup$ – Libli Jul 15 '20 at 14:29

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