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

TL;DR

This paper characterizes the asymptotic speciality of nef and big divisors on algebraic surfaces, relates the SHGH conjecture to nef classes, and proves non-speciality for certain nef divisors on blown-up projective spaces.

Contribution

It establishes a new characterization of asymptotic speciality, links the SHGH conjecture to nef classes, and proves non-speciality for nef divisors when the number of points is below a power of two.

Findings

Characterization of asymptotic speciality via arithmetic genus.

Equivalence of SHGH conjecture to nef class non-speciality.

Proof that nef divisors on blown-up spaces are asymptotically non-special when r<2^n.

Abstract

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.