One of the jewels of 19th-century algebraic geometry is the Cayley–Salmon theorem (1849): every smooth cubic surface in CP³ contains exactly 27 lines. Not "at most" or "generically" — exactly 27, on every smooth cubic, over any algebraically closed field. These lines form a rigid combinatorial structure with deep connections to root systems, exceptional Lie algebras, and the topology of del Pezzo surfaces.

A smooth cubic surface S in CP³ is isomorphic to the blow-up of CP² at six points p₁, …, p₆ in general position. The 27 lines arise as: the 6 exceptional divisors E_i (one for each blown-up point), the 15 strict transforms of lines p_ip_jthrough pairs of points, and the 6 strict transforms of conics through five of the six points. That gives 6 + 15 + 6 = 27.

The count is invariant because the configuration of 27 lines is determined by the intersection form on the Picard group of S, the unimodular lattice I_1,6; inside it, the sublattice orthogonal to the canonical class is the E₆ root lattice. The Weyl group W(E₆), of order 51840, acts on the 27 lines and preserves the incidence structure. This is why the number 27 is rigid — it comes from a lattice, not a parameter count.

The 27 lines are not randomly arranged. Each line meets exactly 10 others, and the incidence structure is governed by the Weyl group of the exceptional root system E₆. The lines can be organized into Schläfli double-sixes— pairs of six mutually skew lines where each line of one set meets exactly five lines of the other. On the Clebsch diagonal surface, all 27 lines are real and can be seen simultaneously.

A Schläfli double-six is a pair of sets of 6 lines {a₁, …, a₆} and {b₁, …, b₆} where: each a_i is skew to all other a_j, each b_i is skew to all other b_j, and a_i meets b_j if and only if i ≠ j. There are 36 double-sixes on any smooth cubic surface, and each determines the remaining 15 lines uniquely.

A tritangent plane is a plane that meets S in three lines (since a plane section of a cubic is a cubic curve, which can split into three lines). There are exactly 45 tritangent planes, and the combinatorics of how they intersect encodes the full E₆ structure. The intersection graph of the 27 lines is a strongly regular graph with parameters (27, 10, 1, 5) — its complement is the Schläfli graph.