-dimensions of algebraic surfaces and numerically effective divisors
Let an elliptic fibration with general fibre . Let be the minima of the non-zero intersection numbers where runs successively through the following sets: effective divisors on , invertible sheaves spanned by global sections, ample divisors and very ample divisors. Let be the maximum of the multiplicities of the fibres of . We prove that if and only if and that if and only if .