Twenty-seven lines on a cubic surface

Remarks

One proof identifies $X$ with the blow-up of $\mathbb{P}^2$ at six points $p_1, \ldots, p_6$ in general position, so that $X$ is a del Pezzo surface of degree $3$. The lines on $X$ are exactly the curves $E$ with $E^2 = -1$ and $-K_X \cdot E = 1$. They are the $6$ exceptional curves over the $p_i$ and the strict transforms of the $15$ lines through two of the points. The other $6$ are the strict transforms of the conics through five of the points, and $6 + 15 + 6 = 27$. Each of the $27$ lines meets exactly $10$ of the others, and the symmetry group of the incidence configuration is the Weyl group of type $E_6$, of order $51840$. Smoothness is needed: the cone over a smooth plane cubic is a singular cubic surface. It contains infinitely many lines, one through the vertex and each point of the base curve. The result goes back to Cayley and Salmon in 1849.

Going deeper

For a student

The theorem is one of the first results of enumerative geometry and a model for counting curves on varieties. The lines are the $(-1)$-curves of the del Pezzo surface of degree $3$. Similar counts hold for the other degrees, for instance $16$ such curves on a del Pezzo surface of degree $4$ and $240$ on one of degree $1$. Over the real numbers a smooth cubic surface has $3$, $7$, $15$ or $27$ real lines.