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When it comes to defining a general plane with respect to a line in $\mathbb R^3$, I can think of this definition as: take any plane not containing the line.

Reading Fulton's "Young Tableau" I can't understand this situation in a proper way: we work in the Grassmanian $Gr(V,d)$ and have two schubert varieties $\Omega_{\lambda}(F_{\bullet})$ and $\Omega_{\lambda}(\tilde F_{\bullet})$, where $\tilde F_{\bullet}$ is the opposite flag to $F_{\bullet}$. We take a $\textbf{general linear subspace}$ $L \subset V$ to define another Schubert variety $\Omega(L)$.

What does he exactly mean with $\textbf{general}$ linear subspace? Is there any relation with the given flags or?

Thanks in advance!

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  • $\begingroup$ I think that here "general" means "any", "arbitrarily chosen" $\endgroup$
    – avs
    Feb 14, 2017 at 19:17

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Usually in algebraic geometry, one says that something is generic if it holds for every choice outside a well defined (but maybe not easy to determine) proper Zariski closed set depending on the problem.

For instance, the quadratic equation $ax^2+bx+c=0$ has two distinct solutions for generic choice of $a,b,c$ because the cases where it does not have two distinct solutions are the ones satisfying $b^2−4ac=0$ that is a Zariski closed subset of the space (say $\mathbb{C}^3$) where $(a,b,c)$ lives.

One can think of "general" as "randomly chosen", or "almost every", with respect to some honest probability (technically, any measure that is absolutely continuous with respect to the Lebesgue measure would work).

In this context, I would guess that there is some Zariski open condition that the linear subspace $L$ should satisfy in order for the construction to work. Maybe some transversal intersection property?

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