Birational geometry and positivity of blow-ups of products of projective spaces

We prove that certain blow-ups of P¹ × P² and P¹ × P³ are log Fano and develop a method which we use to compute their cones of effective divisors. We also compute the decomposition of the movable cone of P¹ × P² blown up at six points in very general position into the union of nef cones of its small Q-factorial modifications, which we show are sequences of flops of certain curves.