We prove that a nonzero effective plane curve of degree d at r ≥ 10 very general complex points has total multiplicity strictly less than $d\sqrt r$. The inequality holds simultaneously for all curves, including reducible and nonreduced curves, and establishes Nagata's conjecture in its strict, nonhomogeneous form.

We prove that, for every smooth integral complex projective surface S and every ample line bundle L, the multipoint Seshadri constant at r very general points equals $\sqrt{L^2/r}$ for every sufficiently large integer r. This resolves positively the qualitative Nagata–Biran conjecture for surfaces.

Let L be an ample line bundle on a smooth integral complex projective variety X of dimension n ≥ 3. We prove that there is a threshold $r_0=r_0(X,L)$ such that for every integer $r\ge r_0$, the ordinary multipoint Seshadri constant at r very general points equals the volume bound $(L^n/r)^{1/n}$. This establishes the qualitative Nagata–Biran–Szemberg assertion in these dimensions.

Let L be an ample line bundle on a smooth integral projective variety X over an algebraically closed field of positive characteristic, with $\dim X=n\ge3$. We prove that there is a threshold $r_0=r_0(X,L)$ such that for every integer $r\ge r_0$, the ordinary multipoint Seshadri constant at the geometric generic tuple of r points equals the volume bound $(L^n/r)^{1/n}$. The same conclusion holds in dimension two. This establishes the positive-characteristic form of the qualitative Nagata–Biran–Szemberg assertion at geometric generic tuples.