Every smooth cubic surface contains exactly 27 lines — find them all
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.
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.
The Clebsch diagonal surface — the unique smooth cubic on which all 27 lines are real — rendered via marching cubes. The lines are colored by group and can be filtered. Toggle the surface transparency to see the lines clearly, or hide the lines to appreciate the surface shape.
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 Ei (one for each blown-up point), the 15 strict transforms of lines pipjthrough 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, which is the lattice E6⊥inside the unimodular lattice I1,6. 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.
A Schläfli double-six is a pair of sets of 6 lines {a1, …, a6} and {b1, …, b6} where: each ai is skew to all other aj, each bi is skew to all other bj, and ai meets bj 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 dual graph of the 27 lines is the Schläfli graph, a strongly regular graph with parameters (27, 10, 1, 5).