Symplectic Classes on Elliptic Surfaces with positive Euler Number

A key question for $4$-manifolds $M$ admitting symplectic structures is to determine which cohomology classes $\alpha \in H^2(M,\mathbb{R})$ admit a symplectic representative. The collection of all such classes, the symplectic cone ${\mathcal{C}}_M$, is a basic smooth invariant of $M$. This paper describes the symplectic cone for elliptic surfaces with positive Euler number.