9 problems

Let $X$ be a smooth algebraic surface, let $L$ be an ample line bundle on $X$ with degree $d$, and let $p_1,\ldots,p_r$ be sufficiently many general points. The Seshadri constant a…

Let $u$ be a divisorial or irrational valuation of the complex projective plane, with its dual graph fixed, and interpret “very general” in the corresponding parameter space of va…

Valuative Nagata conjecture. If

Let $\pi:{\rm Bl}_{r}(\mathbb{P}^{2})\rightarrow\mathbb{P}^{2}$ be the blow-up at $r\geq9$ general points, with exceptional divisors $E_{1},\dots,E_{r}$. Homogeneous Nagata conject…

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…

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

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