Equivalent conjectures on blowing-ups of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"> <mml:msup> <mml:mrow> <mml:mi mathvariant="double-struck">P</mml:mi> </mml:mrow> <mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup>

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 of the projective plane at points in very general position is equivalent to the fact that each nef class is non-special. Finally we prove that if r < 2 n then any nef divisor of the blowing-up of the n -dimensional projective space at r points in very general position is asymptotically non-special.