It has been long conjectured that the crossing number of C m  C n is (m À 2)n, for all m; n such that n ! m ! 3. In this paper, it is shown that if n ! m(m þ 1) and m ! 3, then this conjecture holds. That is, the crossing number of C m  C n is as conjectured for all but finitely many n, for each m. The proof is largely based on techniques from the theory of arrangements, introduced by Adamsson and further developed by