Variations on Nagata's Conjecture

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 \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 $n\ge 90$.

13 pages

References in corpus (3)

• Limit linear systems and applications
• On the Mori cone of blow-ups of the plane
• Negative curves on very general blow-ups of P^2

Cited by in corpus (6)

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• Vanishing theorems for linearly obstructed divisors
• Tropical approach to Nagata's conjecture in positive characteristic
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• Linear systems on the blow-up of (P^1)^n