Variations on Nagata's Conjecture
By C. Ciliberto, Brian Harbourne, R. Miranda, Joaquim Roé. 2012
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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 \emph{good rays}, in the Mori cone of the blow-up $X_n$ of the plane at $n\ge 10$ general points. Nagata's original result was the existence of a good ray for $X_n$ with $n\ge 16$ a square number. Using degenerations, we give examples of good rays for $X_n$ for all $n\ge 10$. 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 $n\le 89$ combined with a stronger conjecture for $n=10$ implies Nagata's conjecture for $n\ge 90$.
Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics, Geometry and Topology
View Paper: https://www.semanticscholar.org/paper/39c5688db3fb52b85029b4636096e1f94652c7ce
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Citations in the corpus
- The Hitchhiker Guide to: Secant Varieties and Tensor Decomposition (2018, background): “Our brief presentation is taken from [63,64], which we suggest as excellent and very instructive deepening on this topic.”