Divisors on surfaces isogenous to a product of mixed type with $p_g=0$

In this paper, we study effective, nef and semiample cones of surfaces isogenous to a product of mixed type with $p_g=0$. In particular, we prove that all reducible fake quadrics are Mori dream surfaces.Comment: 10 pages; v2: Minor changes, final version to appear on Pacific Journal of Mathematic