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

Abstract

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 $({\mathbb{P}}^1{)}^n\to ({\mathbb{P}}^1{)}^s,1\le s\le n-1$ 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 $({\mathbb{P}}^1{)}^3$.

12 pages

References in corpus (1)

• Variations on Nagata's Conjecture