The crossing number of Cm × Cn is as conjectured for n ≥ m(m + 1)

It has been long conjectured that the crossing number of Cm × Cn is (m−2)n, for all m,n such that n ≥ m ≥ 3. In this paper it is shown that if n ≥ m(m+1) and m ≥ 3, then this conjecture holds. That is, the crossing number of Cm × Cn 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 Adamsson and Richter. 2000 Mathematics Subject Classification: 05C10, 05C62, 57M15