Variations on Nagata's Conjecture

2012/02/02 by Ciro Ciliberto, Brian Harbourne, Ciliberto, C. +5 · 3 citations
Mathematics · #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Commutative Algebra and Its Applications #FOS: Mathematics #Finite Group Theory Research

paper · pdf · doi:10.48550/arxiv.1202.0475

openalex publication_date 2012/02/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper we discuss some variations of Nagata's conjecture on linear systems of plane curves. The most relevant concerns non-effectivity (hence nefness) of certain rays, which we call good rays, in the Mori cone of the blow-up X_n of the plane at n≥ 10 general points. Nagata's original result was the existence of a good ray for X_n with n≥ 16 a square number. Using degenerations, we give examples of good rays for X_n for all n≥ 10. As with Nagata's original result, this implies the existence of counterexamples to Hilbert's XIV problem. Finally we show that Nagata's conjecture for n≤ 89 combined with a stronger conjecture for n=10 implies Nagata's conjecture for n≥ 90.

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Related

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