# Consequence of the failure of Nagata's conjecture

A modern version of the Nagata's conjecture says that $$L_{N,t}:=f_{N}^{*}(-K_{\mathbb{P}^{2}})-t\sum_{j=1}^{N}E_{j}$$ is Ample for any $$t<\frac{3}{\sqrt{N}}$$, where $$f_{N}:Y_{N}\to \mathbb{P}^{2}$$ is the blow-up at $$N\geq 9$$ points in general positions and where $$E_{j}$$ are the exceptional divisors.

Then if the Nagata's conjecture is false for a certain $$N$$, there exists $$t_{0}<\frac{3}{\sqrt{N}}, t_{0}\in \mathbb{Q}$$ such that $$L_{N,t_{0}}$$ is Nef but not Ample. In this case, I would like to know if there is some progress in the following questions.

1. Is $$L_{N,t_{0}}$$ semiample? Namely, is $$mL_{N,t_{0}}$$ basepoint-free for $$m\in \mathbb{N}$$ divisible enough?
2. Is $$L_{N,t_{0}}$$ semipositive? Namely, is there $$m\in\mathbb{N}$$ divisible enough such that the line bundle associated to $$mL_{N,t_{0}}$$ admits a non-negative smooth hermitian metric?

Note that 1 implies 2.

Moreover, as $$L_{N,t_{0}}$$ is Big and Nef, 1 is equivalent to show that there exists $$m\in\mathbb{N}$$ such that $$R(Y_{n},mL_{N,t_{0}}):=\bigoplus_{k\geq 0}H^{0}(Y_{n},kmL_{N,t_{0}})$$ is finitely generated as $$\mathbb{C}$$-algebra.

Thank you!