Birational geometry of blow-ups of projective spaces along points and lines

2019/10/25 by Zhuang He, Lei Yang, He, Zhuang +1 · 1 citation
Mathematics · #14E07 #14E30 #14J28 #14J50 #Advanced Algebra and Geometry #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics

paper · pdf · doi:10.48550/arxiv.1910.11854

openalex publication_date 2019/10/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Consider the blow-up X of ℙ³ at 6 points in very general position and the 15 lines through the 6 points. We construct an infinite-order pseudo-automorphism ϕ_X on X, induced by the complete linear system of a divisor of degree 13. The effective cone of X has infinitely many extremal rays and hence, X is not a Mori Dream Space. The threefold X has a unique anticanonical section which is a Jacobian K3 Kummer surface S of Picard number 17. The restriction of ϕ_X on S realizes one of Keum's 192 infinite-order automorphisms of Jacobian K3 Kummer surfaces. In general, we show the blow-up of ℙⁿ (n≥ 3) at (n+3) very general points and certain 9 lines through them is not Mori Dream, with infinitely many extremal effective divisors. As an application, for n≥ 7, the blow-up of M_0,n at a very general point has infinitely many extremal effective divisors.

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• Isoresidual fibrations and the birational geometry of moduli of pointed stable curves

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