Where can I find a proof of the following fact:


If $X\_1$ and $X\_2$ are subvarieties of $\mathbb{G}(k,n)$ of codimension $c\_1$ and $c\_2$ satisfying $c\_1+c\_2<n+1-2k$, then the intersection $X\_1\cap X\_2$ is nonempty.


(This is Lemma 3.13 in the paper *Hypersurfaces of Low Degree*, and it was said that it is a standard fact and stated without proof or reference.)



Nonempty intersection in Grassmannian
By recursion on $k$.


If $k=1$, this is Bezout's theorem in projective space. Now let $k>1$. We have the Pl\"ucker embedding $\mathbb{G}(k,n) \subset \mathbb{P}(\bigwedge^k \mathbb{C}^n)$. Let $p$ be a general point in $\mathbb{C}^n$. Let $Y\_i = \mathbb{P}(\bigwedge^{k-1} \mathbb{C}^n/\langle p\rangle) \cap X\_i$.


The variety $Y\_i$ is the variety of subspaces parametrized by $X\_i$ which contains the point $p$. The codimension $Y\_i$ in $\mathbb{G}(k-1,n-1) = \mathbb{P}(\bigwedge^{k-1} \mathbb{C}^n/\langle p\rangle) \cap \mathbb{G}(k,n)$ is $c\_i$ (because $p$ is generic).


By hypothesis, we have $c\_1 + c\_2 < n+1 - 2k$, so that $c\_1 + c\_2 < (n-1)+1 - 2(k-1)$. Hence by hyptohesis, we have $Y\_1 \cap Y\_2 \neq \emptyset$. 


Let $L \subset \mathbb{C}^{n}/\langle p \rangle$ be a linear space of dimension $k-1$ which is in $Y\_1$ and $Y\_2$. Then, by construction, the inverse image of $L$ in $\mathbb{C}^n$ is in $X\_1$ and $X\_2$. 


This concludes the recursion.


**EDIT** : correction of a misleading typo about codimensions


