This is probably a very simple question, but I can't seem to see the answer. Given a (Zariski) closed subset $Z$ in $\mathbb P^n$ of codimension $d$, we can always find a linear subspace $L \cong \mathbb P^{d-1}$ such that $L \cap Z = \varnothing $. I'm wondering if a similar statement can be said about Grassmannians.

Specifically, let $G= \mathbb G(1,n)$ be the Grassmannian of lines in $\mathbb P^n$ with $n \geq 3$. If $Z$ is a closed subset in $G$ of codimension $2$, can we always find a $\Sigma_{x,P} \cong \mathbb P^1$ in $G$ with $\Sigma_{x,P} \cap Z = \varnothing $, where $\Sigma_{x,P}$ corresponds to the locus of lines through $x$ lying in a plane $P= \mathbb P^2 \subset \mathbb P^n$ containing $x$ ?

Similarly, if $Z$ is of codimension $3$ in $G$, can we always find a $\Sigma_P \cong \mathbb P^2$ with $\Sigma_P \cap Z = \varnothing $, where $\Sigma_P$ corresponds to the locus of lines lying on a plane $P= \mathbb P^2 \subset \mathbb P^n$ ?


Indeed much more is true.

Suppose $X$ is an irreducible algebraic manifold admitting a transitive action of a linear algebraic group $G$. If $Y$ and $Z$ are irreducible subvarieties of $X$ then for a general $g \in G$ the intersection of $gY$ (the translate of $Y$ by $g$) and $Z$ is empty or equidimensional of dimension $\mathrm{dim}(Y) + \mathrm{dim}(Z) -\mathrm{dim}(X)$.

See Kleiman's "The transversality of a general translate".

  • $\begingroup$ Note that some of Kleiman's results only hold in characteristic 0, but my guess (without having it at hand) would be that this one is general. $\endgroup$ – Allen Knutson Mar 27 '11 at 0:49
  • $\begingroup$ You are right the result above holds true in general, no need to restrict to char zero. If one further assume that $Y$ and $Z$ are smooth and want to conclude that the intersection is also smooth then one does need to restrict to char zero. $\endgroup$ – Jorge Vitório Pereira Mar 27 '11 at 2:42

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