the twenty-seven lines on a cubic surface
Take a smooth surface in projective 3-space cut out by a single equation of degree three. A startling and exact fact, discovered by Cayley and Salmon in 1849, is that such a surface contains exactly twenty-seven straight lines — no more, no fewer — lying entirely on it. The number is not approximate and not 'generic': every smooth cubic surface over an algebraically closed field has precisely 27 lines. This is one of the most celebrated small miracles of enumerative geometry, a self-contained taste of how rigid and combinatorial algebraic geometry can be.
Concretely, a line in P^3 lying on the surface V(F) with F a cubic form imposes conditions: parametrize the line and require F to vanish identically along it, which forces the coefficients of the resulting cubic in the parameter to be zero. A dimension count in the Grassmannian of lines in P^3 (which is 4-dimensional) against these conditions shows the expected number is finite, and a careful analysis pins it at exactly 27 for a smooth cubic. The cleanest modern viewpoint realizes the smooth cubic surface as the projective plane blown up at six points in general position; the 27 lines are then beautifully accounted for as the 6 exceptional curves over the blown-up points, the 15 lines through pairs of those points, and the 6 conics through five of them — that is 6 + 15 + 6 = 27. Their incidence pattern carries a large symmetry group, the Weyl group of the root system E_6, of order 51840.
Beyond its charm, the 27 lines launched the systematic study of cubic surfaces, of del Pezzo surfaces, and of how exceptional Lie-theoretic symmetry (E_6) surfaces in geometry, and they are the historical seed of modern intersection and enumerative theory. Honest caveats. The count is exactly 27 only for a smooth cubic over an algebraically closed field: over the reals the lines can be 27, 15, 7, or 3 depending on the surface, and a singular cubic has fewer lines (a cone is an extreme degeneration). And 'in general position' for the six blown-up points is a genuine condition — no three collinear, not all six on a conic — without which the elegant 6 + 15 + 6 bookkeeping degenerates.
The Fermat cubic surface x^3 + y^3 + z^3 + w^3 = 0 in P^3 over C contains exactly 27 lines. One family comes from pairing the variables: for a primitive cube root of unity omega, the line where x + omega y = 0 and z + omega' w = 0 lies on the surface, and tallying all such pairings reproduces the full 27.
Every smooth cubic surface over an algebraically closed field has exactly 27 lines, with E_6 symmetry in their incidences.
The count is exactly 27 only for a smooth cubic over an algebraically closed field. Over R the number is 27, 15, 7, or 3, and singular cubics have fewer; the clean theorem is a closed-field statement.