A first look at the twenty-seven lines on a cubic surface

These are working notes — corrections welcome.

Statement

A cubic surface is the zero locus in $\mathbb{P}^3_k$ of a homogeneous polynomial of degree three. The classical theorem of Cayley and Salmon (1849) asserts:

Every smooth cubic surface over an algebraically closed field of characteristic zero contains exactly twenty-seven lines.

Equivalently, denote by $X \subset \mathbb{P}^3$ a smooth cubic surface and by $F(X) \subset \mathrm{Gr}(2, 4)$ its Fano variety of lines. Then $F(X)$ is a reduced zero-dimensional scheme of length

Sketch

One pleasant way to see the number $27$ is via intersection theory on the Grassmannian. The space of lines in $\mathbb{P}^3$ has dimension four, and the condition that a line lie on a fixed cubic cuts out a degree-four subscheme. Computing the relevant Chern class of $\mathrm{Sym}^3 \mathcal{S}^\vee$ — where $\mathcal{S}$ is the tautological rank-two subbundle on $\mathrm{Gr}(2,4)$ — gives, after a small calculation,

I plan to flesh this out in a follow-up note with the actual Chern class calculation.

Why I find this interesting

Two reasons.

1. The argument is short once one accepts the formalism of Chern classes; the answer drops out of a finite computation in a polynomial ring. There is something pleasing about a number-theoretic answer — a small integer — emerging as the value of a topological invariant.

2. The configuration of the twenty-seven lines is an object of intrinsic interest. They form a graph (the Schläfli double-six) with rich automorphism group, isomorphic to the Weyl group $W(E_6)$.

[PLACEHOLDER: continue with worked example].

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