On the class of an elliptic projective surface

It is known that properly elliptic surfaces S \subset P^n of degree d and class m satisfy the inequality m-3d \geq 2, equality implying that S is minimal, \chi(O_S)=0 and the base curve of the elliptic fibration is rational. Using the progress in understanding such surfaces made by Serrano, the result above is improved considerably. In fact it turns out that for S as above m-3d \geq 6, equality implying that S is an elliptic quasi-bundle over a smooth curve C of genus 0 or 1 and in both cases p_g, q and the multiplicities of the multiple fibers are determined. The result is effective and applies to describe smooth projective surfaces S \subset P^n satisfying the condition m \leq 3d+6.