Gfrktyl

Consider $H_1,H_2,H_3subsetmathbb{P}^{2m+1}$ three general linear subspaces of projective dimension $m$.

Then there exists a quadric hypersurface $Q^{2m}subsetmathbb{P}^{2m+1}$ containing $H_1,H_2,H_3$. This is just a dimension count.

Does there exist a smooth quadric hypersurface $Q^{2m}subsetmathbb{P}^{2m+1}$ 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.

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.