  In this article formulas for the quantum product of a rational surface are
given, and used to give an algebro-geometric proof of the associativity of the
quantum product for strict Del Pezzo surfaces, those for which $-K$ is very
ample. An argument for the associativity in general is proposed, which also
avoids resorting to the symplectic category. The enumerative predictions of
Kontsevich and Manin concerning the degree of the rational curve locus in a
linear system are recovered. The associativity of the quantum product for the
cubic surface is shown to be essentially equivalent to the classical
enumerative facts concerning lines: there are $27$ of them, each meeting $10$
others.
