Equivalent conjectures on blowing-ups of P²

We provide a characterization of asymptotical speciality of a nef and big divisor D on an algebraic surface in terms of the arithmetic genus of curves in D⊥. As a consequence we prove that the SHGH conjecture for linear systems on the blowing-up Xᵣ² of the projective plane at points in very general position is equivalent to the fact that each nef class of is non-special. Finally we prove that if r < 2ⁿ then any nef divisor of Xᵣⁿ is asymptotically non-special.