  In this note we study linear systems on the blow-up of $(\mathbb{P}^1)^n$ at
$r$ points in very general position. We prove that the fibers of the
projections $(\mathbb{P}^1)^n \rightarrow (\mathbb{P}^1)^s, 1\leq s \leq n-1$
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
$(\mathbb{P}^1)^n$ which we call fiber dimension. Finally, we state a
conjecture about linear systems on $(\mathbb{P}^1)^3$.
