Maximal Multipoint Seshadri Constants in Higher Dimensions

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.