Cubic surfaces whose points all lie on their 27 lines

Summary

Let F be a cubic surface with 27 lines in PG(3,q). Theorem 30.1 in Manin [2] states that, if q > 34, then there exists a point of F on none of its lines. There is, however, sufficient information in [1] to work out the precise list of cubic surfaces with no such point.

When the cubic curves of PG(2,q) are mapped by their coefficients to the points of PG(9,q), then the set of triple lines is mapped to the Del Vezzo surface v₂⁹. Successive projections, always from points of themselves, are also called Del Pezzo surfaces by Manin. A cubic surface with 27 lines in PG(3,q) is in this sense a Del Pezzo surface of order three, whose lines are its exceptional curves. In what follows, F is always such a surface. The 27 lines of F lie by threes in 45 tritangent planes, in e of which the three lines are concurrent at an Eckardt point. All the subsequent lemmas come from [1].

LEMMA 1: F exists over all fields except GF(q) with q = 2, 3 or 5.

LEMMA 2: The number of points on F is q² + 7q + 1.

LEMMA 3: The number of points on the 27 lines of F is 27(q - 4) + e.

LEMMA 4: For q odd, e ≥ 18; for q even, e ≥ 45.