Enumerative projective geometry

Question

I am wondering whether for any two lines $\mathfrak{L}, \mathfrak{L'}$ and any point $\mathfrak{P}$ in $\mathbf{P}^3$ there is a line having nonempty intersection with all of $\mathfrak{L}, \mathfrak{L'}$, $\mathfrak{P}$. I don't really know how to approach this, because I was never taught thinking about such a problem, not even related ones. I think the answer should be no, but have no means to justify it. Perhaps use the Klein representation of lines in 3 space? Could someone also recommend a book where similar problems are solved or at least posed as exercises? Also feel free to give an algebraic geometry perspective, but I don't really know how to approach this with algebraic geometry.

Answer 1

HINT: Consider the plane spanned by $\mathfrak L$ and $\mathfrak P$.

A lovely elementary book is Pedoe's Geometry: A Comprehensive Course.

Answer 2

Yes, this is true.

One way to see it is projection away from $P$: this is a rational map $ \mathbf P^3 \dashrightarrow \mathbf P^2$. Points in the target $\mathbf P^2$ correspond to lines in $\mathbf P^3$ passing through $P$. Now under this map, the two lines $L$ and $L'$ map to lines $\Lambda$ and $\Lambda'$ in $\mathbf P^2$. Since they are lines in the plane, they intersect in a point, and this point corresponds to the line in $\mathbf P^3$ you're looking for.

This is an elementary example of Schubert calculus on the Grassmannian $\mathbb G(1,3)$ of lines in $\mathbf P^3$. You can look at the book Principles of Algebraic Geometry by Griffiths--Harris for a thorough treatment, explaining how questions of this kind can be solved systematically.