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Let $V$ be a vector space, and consider $G=Gr(V,2)\subset \mathbb{P}^N$ embedded via the Plucker embedding. Let $W\subset \mathbb{P}^N$ be a linear subspace. I want to find the class $[W\cap G]\in A(G)$.

I think that if $W$ is a hyperplane then we always have $[W\cap G]=\sigma_1$, but I'm not even sure about this?

Then it seems to me that we could conclude that $[W\cap G]=\sigma_1^{\text{codim}(W)}$, but this seems too easy...

I'd appreciate some pointers as to how to actually go about this correctly.

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    $\begingroup$ The formulas are correct if the intersection $W \cap G$ is dimensionally transverse, i.e., has codimension $codim(W)$ in $G$. $\endgroup$
    – Sasha
    Commented Apr 7, 2018 at 11:06

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Pull back the equation $[W] = [H]^{codim\ W} \in \mathbb P^N$. This cohomological pullback is computable set-theoretically, as $[W\cap G]$, if the intersection is transverse.

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