Counting lines on projective surfaces

In this note we prove a new bound on the number of lines on a smooth surface of degree d >= 3 in P-3. Building on work of Segre, we provide a rigorous justification of an idea of his while at the same time improving his bound. Our re-sult gives the lowest known bound ford >= 6 , it is valid both in characteristic 0 , in positive characteristic p > d.