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