Why cubic surfaces contain straight lines? I often heard that each smooth cubic surface contains even $27$ straight lines. I cannot prove it today, but I'll do my best to do it soon. However how to prove that each cubic surface contains a straight line?
 A: Shafarevich Volume 1 has a pretty elementary argument: Chapter 1.6, Theorem 10. It's a standard incidence correspondence proof: one considers the variety $\mathscr I$ of pairs $(X,L)$ where $X$ is a cubic surface, and $L$ is a line in $\mathbf{P}^3$ such that $L \subset X$. One calculates the dimension of $\mathscr I$ by considering the projection to $G(2,4)$ (the parameter space of lines) and finds that it is 19, which is the same as the dimension of the space $\mathbf{P}^{19}$ of cubic surfaces. Finally, one writes down a single cubic that contains only finitely many lines, proving that the map $\mathscr I \rightarrow \mathbf{P}^{19}$ is generically finite onto its image, hence (since the dimensions are equal) surjective.
As Matt E comments, once you know there is one line, it's not that hard to find all 27: Shafarevich goes on to do this in Chapter 4.
A: You should look at Section 7 of Miles Reid's wonderful book.
If you're more advanced, you can without too much difficulty prove that if you blow up $6$ general points in $\Bbb P^2$, the surface embeds as a cubic surface in $\Bbb P^3$. You get $6$ lines from the $6$ exceptional divisors, $6$ more lines from the proper transforms of the $6$ conics through five of the six points, and $15$ more lines from the $\binom 62$ proper transforms of the lines through pairs of the points.
