# Linear subspaces in quadric hypersurfaces

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.

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.
• So, if I got correctly you argument, $B$ is an $(m+1)\times (m+1)$ matrix. If $m$ is odd then $m+1$ is even, $\det(B)\neq 0$ and so $\det(A)\neq 0$ and the quadric is smooth. If $m$ is even then $m+1$ is odd, $\det(B)= 0$ and so $\det(A)= 0$ and the quadric is singular. Is this right?