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.
For surfaces, Bruce (2026, Theorem B) reports an explicit threshold. Let 
be a smooth connected two-dimensional projective
variety over an uncountable algebraically
closed field of characteristic
zero, and let 
be an ample line bundle.
Choose an integer 
such that 
is very ample
and 
. At very general
distinct 
-tuples, the reported equality is
 |
for every 
, where 
is the self-intersection number. The argument is a tropical
reworking of earlier AI-generated methods. This surface result does not settle the
higher-dimensional assertion.
