The qualitative Nagata-Biran-Szemberg conjecture asserts that multipoint Seshadri constants of a fixed polarized variety attain their volume upper
bound at sufficiently many very general points. For a smooth integral complex
projective variety 
of dimension 
with ample line bundle 
,
it asks for an integer 
such that
 |
at very general distinct tuples for every integer 
.
The surface form is also called the qualitative Nagata-Biran conjecture.
It asks for exact equality at each sufficiently large point count, rather than an
asymptotic ratio.
For the projective plane the Nagata conjecture makes the sharper prediction starting at ten points, including a strict degree-multiplicity inequality.
