I have the following problem:

Let $\pi_1$, $\pi_2$, $\pi_3$ be three planes in a projective space such that $\pi_1 \cap \pi_2 \cap \pi_3$ = $\emptyset$, $\dim\,(\pi_1 \cap \pi_2) = 1$, $\dim\,(\pi_1 \cap \pi_3) = \dim (\pi_2 \cap \pi_3) =0.$ Find the dimension of $\pi_1 \lor \pi_2\lor \pi_3$.

I have almost solved it, but I'm struggling at the end. So far, I have already done the following:

Using Grassmann Formula, I get: $\dim\,(\pi_1\lor \pi_2\lor \pi_3)=5-\dim\,((\pi_1\lor \pi_2)\cap \pi_3)$. Since $\pi_1\cap \pi_3 \subseteq (\pi_1\lor \pi_2)\cap \pi_3 \subseteq \pi_3$, we know that $0 \le \dim\,((\pi_1 \lor \pi_2)\cap \pi_3)\le2$.

If $\dim\,((\pi_1\lor \pi_2)\cap \pi_3)=0$, then $(\pi_1\lor \pi_2)\cap \pi_3= \pi_1\cap\pi_3= p$, $(\pi_1\lor \pi_2)\cap \pi_3=\pi_2\cap\pi_3={p}$. Thus, $\pi_1\cap\pi_2\cap\pi_3={p}$, which leads to a contradiction.

Therefore, we know that $\dim\,((\pi_1\lor\pi_2)\cap\pi_3)$ can only be either 1 or 2.

I don't know how to solve this and I would appreciate a lot if you could help me out.


A projective space of dimension $n$ is the space of lines through the origin of a linear space of dimension $n+1$, and the projective planes correspond to 3 dimensional subspaces, and, generally, the dimension of the corresponding subspace $\hat H$ in the linear space is increased by one, compared to the original object projective subspace $H$, for any $H$.

So, $\dim(\pi_1\cap\pi_2)=1$ means that $\dim(\hat\pi_1\cap\hat\pi_2)=2$, thus we can choose a basis $u_1,u_2\in\hat\pi_1\cap\hat\pi_2$.
Now, $\dim(\pi_1\cap\pi_3)=0$ means $\dim(\hat\pi_1\cap\hat\pi_3)=1$, choose any nonzero element of it, $a$,
and similarly choose a nonzero $b$ in $\hat\pi_2\cap\hat\pi_3$.
Clearly, $a$ and $b$ are not scalar multiple of one another, so they are linearly independent. Extend them to a basis $a,b,c$ of $\hat\pi_3$.

Using $\pi_1\cap\pi_2\cap\pi_3=\emptyset\text{,}\ $ that is, $\ \hat\pi_1\cap\hat\pi_2\cap\hat\pi_3=\{0\}$, prove that $u_1,u_2,a,b,c$ are linearly independent, yielding in particular that $u_1,u_2,a$ is a basis for $\hat\pi_1$ and $u_1,u_2,b$ is a basis for $\hat\pi_2$.

Consequently, $u_1,u_2,a,b,c$ is a basis for $\hat\pi_1+\hat\pi_2+\hat\pi_3=(\pi_1\lor\pi_2\lor\pi_3)\hat{}$,
so its projective dimension is $4$.


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