Linear systems on the blow-up of (P^1)^n

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 $(\mathbb{P}^1)^3$.

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