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Sufficiently general arcs have minimal valuation above the Nagata threshold
Minimal valuation conjecture. If $C$ is sufficiently general and $s\geqslant 8+\frac{1}{36}$ , then
Conjectureopennagata-conjecture newton-okounkov-bodies valuative-treesarXiv posted 2 days ago

Sufficiently general arcs have minimal valuation above the Nagata threshold