Rational lines on smooth cubic surfaces

Abstract: We prove that the enumerative geometry of lines on smooth cubic surfaces is governed by the arithmetic of the base field. In 1949, Segre proved that the number of lines on a smooth cubic surface over any field is 0, 1, 2, 3, 5, 7, 9, 15, or 27. Over a given field, each of these line counts may or may not be realized by some cubic surface. We give a sufficient criterion for each of these line counts in terms of the Galois theory of the base field.