Intersection of 3 or 4 dimensional subspaces of $\mathbb{P}^5$

Question

What do we know about the intersection of 3 or 4 dimensional subspaces in $\mathbb{P}^5$? If we for example take two 4-dimensional subspaces of $\mathbb{P}^5$, what do we know about the dimension of their intersection? It seems logical that this has dimension 3, but I wouldn't know why exactly. And the same for two subspaces of dimension 3, does their intersection have dimension at least 2? And furthermore, do a subspace of dimension 4 en of dimensions 1 necessarily intersect in a point?

Answer 1

In the case of intersections, it is easier to work with codimensions rather than dimensions. (Definition: If $W \subset V$, then the codimension of $W$ inside $V$ is $\dim V - \dim W$.)

In general, if $W_1$ and $W_2$ are linear subspaces of a projective space $V$, then we have $$\mbox{codim} (W_1 \cap W_2) \le \mbox{codim} (W_1) + \mbox{codim} (W_2).$$

(The empty set is considered to have dimension $-1$ and codimension $n+1$, where $n = \dim V$.)

Also, equality holds except when $W_1$ and $W_2$ are "specially" situated inside $V$, or if the right side is too big (bigger than $\dim V$) in which case the intersection will "generically" be empty.

I won't attempt to define "specially" and "generically" here, but to illustrate this, consider that in 3 dimensional projective space, a line and a plane usually intersect in a point (except when the line is contained in the plane, which is a special situation), while two lines usually do not intersect at all (if two lines intersect in 3-dimensional space, a slight perturbation will make them not intersect).

Answer 2

Just move back to $\mathbb A ^{n+1}$ and answer the question there. Increase the dimensions of your spaces by $1$ while moving to $\mathbb A^{n+1}$. After you get the answer subtract $1$ from it. That would give you the answer in $\mathbb P^n$.

In $\mathbb A^{n+1}$, you can use the formula from linear algebra, $\dim (W_1 + W_2) = \dim W_1 + \dim W_2 - \dim (W_1 \cap W_2)$.

You will figure that if you intersect two projective subspaces of $\mathbb P^n$ dimension $p$ and $q$, then the dimension of intersection is at least $p+q - n$.

Also, prove that if $p+q \ge n$, then intersection must happen.