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## ag.algebraic geometry – Consequence of the failure of Nagata’s surmise

A contemporary model of the Nagata’s surmise says that

$$

L_{N,t}:=f_{N}^{*}(-K_{mathbb{P}^{2}})-tsum_{j=1}^{N}E_{j}

$$

is Kähler for any $t<frac{3}{sqrt{N}}$, the place $f_{N}:Y_{N}to mathbb{P}^{2}$ is the blow-up at $Ngeq 9$ factors in common positions and the place $E_{j}$ are the distinctive divisors.

Then if the Nagata’s surmise is disloyal for a inescapable $N$, there exists $t_{0}<frac{3}{sqrt{N}}, t_{0}in mathbb{Q}$ such that $L_{N,t_{0}}$ is Nef however not Kähler. In this illustration, I might love to know if there may be some progress within the following questions.

- Is $L_{N,t_{0}}$ semiample? Namely, is $mL_{N,t_{0}}$ basepoint-free for $min mathbb{N}$ divisible sufficient?
- Is $L_{N,t_{0}}$ semipositive? Namely, is there $minmathbb{N}$ divisible sufficient such that the road bundle related to $mL_{N,t_{0}}$ admits a non-negative flush hermitian metric?

Note that 1 implies 2.

Thank you!

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