8 problems

Nagata's conjecture. If $n\geq 10$ and $t>\sqrt{n}$, then $A_t$ is ample. In particular,

Let $\xi$ be a sufficiently general valuation parameter and let $t\ge 8+1/36$. Let $\operatorname{ord}_{\xi,t}$ denote the valuation written in the source, and let maximality mean…

Let $X$ be the blow-up of ${\mathbb P}^2$ at a set of $n$ general points, let $H$ be the hyperplane class, and let $\mathbb E=E_1+\cdots+E_n$ be the exceptional divisor. Nagata's c…

Minimal valuation conjecture. If $C$ is sufficiently general and $s\geqslant 8+\frac{1}{36}$, then

Let $t$ be a real parameter, and let $\widehat\mu(t)$ denote the invariant for a very general quasimonomial valuation introduced in the paper. Nagata-type conjecture for quasimonom…

Let $X_n$ be the projective plane blown up at $n$ general points, and let $C$ be an irreducible curve of genus $g>0$ on $X_n$. Let $(3;1^9,0^{n-9})$ denote the indicated plane line…

Let $X_n$ be the projective plane blown up at $n$ general points, and let $C$ be a prime divisor on $X_n$ with self-intersection $C^2$ and arithmetic genus $g$. Negativity conjectu…

Nagata's numerical inequality. If