The cone of curves and the Cox ring of rational surfaces over Hirzebruch surfaces

2024/01/14 by Carlos Galindo, Galindo, Carlos, Francisco Monserrat +3
Mathematics · #13A18 #14C20 #14C22 #14E15 #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Commutative Algebra and Its Applications #FOS: Mathematics

paper · pdf · doi:10.48550/arxiv.2401.07281

openalex publication_date 2024/01/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01

Abstract

Let X be a rational surface obtained by blowing up at a configuration C of infinitely near points over a Hirzebruch surface \mathbbF_δ. We prove that there exist two positive integers a ≤ b such that the cone of curves of X is finite polyhedral and minimally generated when δ≥ a, and the Cox ring of X is finitely generated whenever δ≥ b. The integers a and b depend only on a combinatorial object (a graph decorated with arrows) representing the strict transforms of the exceptional divisors, their intersections and those with the fibers and special section of \mathbbF_δ.

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