Equivalent conjectures on blowing-ups of \mathbb P²

2024/05/13 by Laface, Antonio, Ugaglia, Luca, Vilches, Macarena
#14C20 #Algebraic Geometry (math.AG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2405.07716

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^⊥. As a consequence we prove that the SHGH conjecture for linear systems on the blowing-up X_r² 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 X_rⁿ is asymptotically non-special.

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