2025/08/02 by Ciro Ciliberto, Ciliberto, Ciro, Rick Miranda +3
Computer Science · Mathematics · #14H45 #14H50 #14J26 #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Polynomial and algebraic computation
paper · pdf · doi:10.48550/arxiv.2508.01462
openalex publication_date 2025/08/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let Xn be the projective plane blown up at n ≥ 10 general points. In this paper we give several consequences of the Segre-Harbourne-Gimigliano-Hirschowitz Conjecture, that pertain to complete linear systems on Xn. We begin by classifying such systems |C| with general irreducible member of genus g ≥ 2 (up to Cremona equivalence), in terms of invariants of the adjoint systems |C+mK|. We then use this to prove that, for fixed n ≥ 10 and g≥ 2, up to the action of the Cremona group, there exist finitely many complete linear systems on Xn whose general member is irreducible of genus g. Further, there is a function g↦ n(g) such that every such (effective) system is Cremona equivalent to a system in Xn(g). The latter result is based on the explicit computation of the minimum possible self-intersection of an irreducible linear system with given n and dim(|C|). We classify those systems which achieve the minimal self-intersection. We also classify the systems with C2 ≤ 5, whether or not they have minimal C2 for the given n and dimension. We finish by proving several statements concerning systems that are base-point-free, and systems that give birational maps to their image.