In Hartshorne's "Algebraic Geometry", exercise I-3.7(a) asks us to show that any two curves in $\mathbb{P}^2$ have non-empty intersection. This lead me to think about whether this was generalizable to higher dimensional projective space, $\mathbb{P}^n$ for $n \geq 3$.
In particular, I read in an article that one can already find skew lines in $\mathbb{P}^3$ so this non-empty intersection principle does not hold for higher dimensions. I was wondering whether there were any examples of this?
Hartshorne later suggests in I-7 Theorem 7.2 a criterion for this not to be the case.
However, assuming we took two lines (1-dimensional varieties) in $\mathbb{P}^3$, of the general form:
$L_1=V(a_1X+b_1Y+c_1Z+d_1W)$
$L_2=V(a_2X+b_2Y+c_2Z+d_2W)$
wouldn't it be true that the point $[0:0:0:0] \in V(L_1,L_2)$, so the intersection would be non-empty? Is there an example where this would not be the case?
Thanks in advance and sorry!
Remember, $\Bbb P^n$ is $\Bbb A^{n+1}\setminus\{(0,\dots,0)\}$ up to scaling, so $[0:\dots:0]$ is not an element of $\Bbb P^n$. One easy example of two skew lines in $\Bbb P^3$ are $\{[a:b:0:0]\mid [a:b]\in\Bbb P^1\}$ and $\{[0:0:c:d]\mid [c:d]\in\Bbb P^1\}$.