Equivalent conjectures on blowing-ups of ${\mathbb{P}}^2$

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 $D^{\perp }$. As a consequence we prove that the SHGH conjecture for linear systems on the blowing-up $X_r^2$ 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 $r<2^n$ then any nef divisor of $X_r^n$ is asymptotically non-special.