Linear systems on the blow-up of (P1)n(P1)n

In this note we study linear systems on the blow-up of (P1)n(P1)n at r   points in very general position. We prove that the fibers of the projections (P1)n→(P1)s(P1)n→(P1)s, 1≤s≤n−11≤s≤n−1 can give contribution to the speciality of the linear system. This allows us to give a new definition of expected dimension of a linear system in (P1)n(P1)n which we call fiber dimension. Finally, we state a conjecture about linear systems on (P1)3(P1)3.