Linear span of two non-intersecting lines in projective space

Question

If i have 2 lines, say $L$ and $M$ in the projective space $\Bbb P^4$. It is given that they don't intersect. Is $\left<L,M\right>$ the smallest subspace of $\Bbb P^4$ to contain them both? Just like in affine geometry? So in this case $\left<L,M\right>$ would be a plane of $2$ dimensions?

Answer 1

The span of $L, M$ will be a hyperplane (dimension $3$) in $\mathbb{P}^4$. This can be seen by looking at the corresponding vector space $V$ of $\mathbb{P}^4$ of dimension $5$. The lines $L, M$ correspond to $2$-dimensional subspaces of $V$ which intersect trivially. The span of the subspaces induced by $L, M$ then is a $4$-dimensional subspace of $V$ since $L \cap M = \{0\}$ by assumption. Therefore, the span of $L, M$ is a subspace of $\mathbb{P}^4$ of dimension $3$.

Note: I have ignored the assumption that $L, M$ are not contained in the same hyperplane since this is the case as showed.