27 lines on a Cubic Surface

Theorem: On every smooth cubic surface in $\mathbb{P}^3 (\mathbb C)$ , there are exactly 27 lines.

How do we prove such a result?

Proof Sketch/Ideas:

Find a nice cubic surface where you can exactly compute all the lines on the surface.
Show that this count doesn’t change under deformations.
And that the space of smooth cubic surface is connected.

Compute the lines on the Fermat’s cubic $X^3+Y^3+Z^3+ T^3=0.$ Computing the gradient we can see the surface is smooth and plugging in $X= aZ+bT, Y=cZ+dT$ (equation of a line in $\mathbb{P}^3$ ) we can solve to get 27 solutions. The solutions looks like $a=0, d=0, b=-\omega^s, c=-\omega^t$ for $\omega$ a primitive cube root of unity. (This gives you 9 solutions, and 27 solutions if you permute $a,b,c,d$
Consider the space $\mathcal I =\{(l, S) l \subset S|\}$ incidences of lines and smooth cubic surfaces. It’s easy to show that this space is defined by polynomial equations inside the space of $L \times \mathcal S$ , where $L$ is the space (Grassmanian) of lines in $\mathbb{P}^3$ and $\mathcal S$ is the set of smooth cubic surfaces. (This is an open set defined as the complement of singular surfaces, the conditions of singularity is defined by polynomial equations). By choosing local coordinates for these spaces and computing the Jacobian, one can show (by implicit function theorem) that the projection from space of incidences $\mathcal I =\{(l, S) l \subset S|\}$ to $\mathcal S$ is a covering- that is for every $S$ and $l$ , if $l \subset S$ then every surface in the neighbourhood of $S$ contains a unique line in the neighbourhood of $l.$ If $l$ is not contained in $S$ , there is a neigbourhood of $S$ which doesnt contain any line from a neighbourhood of $l$ .
This above property of the incidence space (along with compactness of the projective varieties) show that the number of lines contained on a surface is locally a constant function.
Now because the space of smooth surfaces is defined by complement of singular discriminant locus, we can see that it is connected.
Thus we see that if we know a single smooth surface with exactly 27 lines, all the smooth cubic surfaces have 27 lines.

Moreover we can prove that the arrangement graph of the 27 lines on the surface is the same for all the smooth surfaces. For instance,
1. Every line on the surface intersects exactly 10 other lines on the surface
2. Every two disjoint lines meets exactly 5 other lines.

Complement of the graph (a graph with edges between disjoint lines) is called The Schläfli graph

As you move in closed loops in the space of smooth cubic surfaces, there will be a permutation of these 27 lines. The permutations induced are related to the exceptional group $E_6$ (A group of reflections on the rootsystem (Weyl group) of $E_6$ )

Every smooth cubic surface can constructed by the construction of “Blowing up” the projective plane $\mathbb P^2$ at 6 generic points. (And the 27 lines would corresponds to exceptional curves above these 6 points, 15 lines through pairs of the points and 6 conics passing through any 5 of the points. Thus a cubic surface is rational.

There are many other questions above lines on cubic surfaces. What about the real cubic surfaces? Many of the above arguments fail. For instance, the space of real cubic surfaces is not connected and has many connected components. (And the number of lines depend on the connected component you lie in). A real cubic surface need not be rational.

Clebsch Cubic Surface:

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