Consider $H\_1,H\_2,H\_3\subset\mathbb{P}^{2m+1}$ three general linear subspaces of projective dimension $m$.


Then there exists a quadric hypersurface $Q^{2m}\subset\mathbb{P}^{2m+1}$ containing $H\_1,H\_2,H\_3$. This is just a dimension count.


*Does there exist a **smooth** quadric hypersurface $Q^{2m}\subset\mathbb{P}^{2m+1}$ containing $H\_1,H\_2,H\_3$?*


The answer is positive when $m = 1$, since if $Q^2$ is a cone then the three lines must intersect, and if $Q^2$ is the union of two planes or a double plane then the three lines must intersect as well. 



Linear subspaces in quadric hypersurfaces
This is true when $m$ is odd and false otherwise. 


Indeed, let $\mathbb{P}^{2m+1} = \mathbb{P}(V)$. Assuming $H\_1 \cap H\_2 = \emptyset$ (by genericity), we have $H\_1 = \mathbb{P}(V\_1)$, $H\_2 = \mathbb{P}(V\_2)$ and
$$
V = V\_1 \oplus V\_2.
$$
Furthermore, assuming $H\_3 \cap H\_1 = H\_3 \cap H\_2 = \emptyset$, we see that $H\_3 = \mathbb{P}(V\_3)$, where $V\_3 \subset V\_1 \oplus V\_2$ is the graph of an isomorphism $V\_1 \to V\_2$. Thus, choosing bases appropriately we may assume
$$
V\_1 = \langle e\_1,\dots,e\_{m+1} \rangle,\quad
V\_2 = \langle e\_{m+2},\dots,e\_{2m+2} \rangle,\quad 
V\_3 = \langle e\_1 + e\_{m+2}, \dots, e\_{m+1} + e\_{2m+2} \rangle.
$$
Let $A = (a\_{ij})$ be the matrix of the quadric $Q$. It follows that
$a\_{ij} = 0$ when either $1 \le i,j \le m+1$ or $m+2 \le i,j \le 2m+2$, and
$$
b\_{ij} = a\_{i,m+1+j}
$$
is a skew-symmetric matrix. It remains to note that
$$
\det(A) = \pm \det(B)^2,
$$
and that a skew-symmetric matrix of odd size is always degenerate.


