New symplectic 4--manifolds with $b_+{=}1$

Symplectic 4-manifolds $(X,\omega)$ with $b_+{=}1$ are roughly classified by the canonical class $K$ and the symplectic form $\omega$ depending upon the sign of $K^2$ and $K\cdot \omega$. Examples are known for each category except for the case when the manifold satisfies $K^2=0$, $K\cdot \omega >0$, $b_1=2$, and fails to be of Lefschetz type. The purpose of this paper is to construct an infinite number of examples of such manifolds. Furthermore, we will show that these manifolds have very special properties -- they are not complex manifolds, their Seiberg-Witten invariants are independent of the chamber structure, and they do not have metrics of positive scalar curvature.