Identifying lines in $\mathbb P^2$

Question

Let $L$ and $M$ be two lines in $\mathbb P^2$. Does there exist a map $f : \mathbb P^2 \to X$ that "identifies" $L$ and $M$, in the sense that $f\vert \mathbb P^2 \setminus (L \cup M)$ is an isomorphism, and $f(L) = f(M)$, with $f\vert_L$ and $f\vert_M$ isomorphisms as well?

Answer 1

This kind of construction is usually called "pinching" or "glueing along closed subschemes".

I'm really not an expert on these things, but as far as I know, the definitive treatment is the article Conducteur, Descente, et Pincement by Daniel Ferrand.

If I understand correctly, your question is answered by Théorème 5.4 in that article: paraphrasing, this says

If $X$ is a scheme, $Y'$ a closed subscheme of $X$ and $\phi: Y' \rightarrow Y$ a finite morphism, we can "pinch" $X$ along $\phi$, changing $Y'$ into $Y$ and leaving the rest of $X$ unchanged.

Applying this with $X=\mathbf P^2$, $Y'=L \cup M$, $Y=\mathbf P^1$, and $Y' \rightarrow Y$ the obvious finite map should give what you want.