The qualitative Nagata-Biran-Szemberg conjecture: maximal multipoint Seshadri constants for many very general points

Surfaces (Theorem 1.1, principal): for every smooth integral complex projective surface $S$ and ample $L$ there is $r_0(S,L)$ such that for every $r\ge r_0$, at $r$ very general points, $L-\sqrt{L^2/r}\sum E_i$ is nef on the blowup, so $\varepsilon(S,L;p)=\sqrt{L^2/r}$; Corollary 5.5 extends this to uncountable algebraically closed fields of characteristic zero. Companion, $n\ge3$ over $\mathbb C$: $\varepsilon(X,L;p)=(L^n/r)^{1/n}$ at very general points for every $r\ge r_0(X,L)$. Companion, positive characteristic, $n\ge2$: the same equality at the geometric generic tuple. Not shown: an explicit or uniform threshold, the higher-cycle Seshadri analogues Roé-Ross distinguish, or very general points (rather than the geometric generic tuple) in positive characteristic. The sharp plane case $r\ge10$ is the separate Nagata entry.

Surfaces (Theorem 1.1, principal): for every smooth integral complex projective surface $S$ and ample $L$ there is $r_0(S,L)$ such that for every $r\ge r_0$, at $r$ very general points, $L-\sqrt{L^2/r}\sum E_i$ is nef on the blowup, so $\varepsilon(S,L;p)=\sqrt{L^2/r}$; Corollary 5.5 extends this to uncountable algebraically closed fields of characteristic zero. Companion, $n\ge3$ over $\mathbb C$: $\varepsilon(X,L;p)=(L^n/r)^{1/n}$ at very general points for every $r\ge r_0(X,L)$. Companion, positive characteristic, $n\ge2$: the same equality at the geometric generic tuple. Not shown: an explicit or uniform threshold, the higher-cycle Seshadri analogues Roé-Ross distinguish, or very general points (rather than the geometric generic tuple) in positive characteristic. The sharp plane case $r\ge10$ is the separate Nagata entry.