There is a famous fact that any smooth cubic surface has exactly 27 lines.
I am a undergraduate students who want to view some detailed proofs of this result. I know there are several different approaches to it. Can anyone recommend me a proof which is both beautiful and useful for my future math study?
There is a question on MathOverflow for interesting results in algebraic geometry that's both enticing and accessible to undergrads, and among the answers this outline of a proof of the 27 line theorem is given. In the comments, the author points to a more detailed presentation (page 222), which unfortunately is in Italian. (The link provided in the MO answer is dead, but I have linked to a more recent version of the same text here.) Another commenter points to a few other sources for some of the more technical results used in the proof.