Thanks for witting this. It is nice to see as many approaches as possible. I'll also need to sit down an understand all the details in this one!

Dear Francesco: Thanks! This seems much nicer, but it will take me a while to understand it completely! Would you mind explaining in more detail how the argument that $h^0(N_{L_2'/ X})=3$ implies the equality $N_{L_2' / X}=\mathcal{O}_{P^1}(1) \oplus \mathcal{O}_{P^1}$?

Thank you. I am taking a look at the paper. Do you think there is a good way in the general case to know if the resulting scheme will be normal? As you mention in a comment below, is the subscheme is not reduced then you might get something which is not normal.

This way to get the polytope is very nice! Is there a place where I can read about it? Thanks!

And thank you for the example on the lines too!