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!


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