Variations on Nagata's Conjecture

Here 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 blowup Xn of the plane at n ≥ 10 general points. Nagata’s original result was the existence of a good ray for Xn with n ≥ 16 a square number. Using degenerations, we give examples of good rays for Xn 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.