# What is a general linear subspace?

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?

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

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.
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?