The numerical dimension of a nef divisor 
on a projective variety 
of dimension 
is
 |
where 
is an ample Cartier divisor. This is independent
of the chosen ample divisor. It measures numerical intersection positivity, whereas
the Iitaka dimension measures actual sections.
The zero divisor has numerical dimension zero, and an ample divisor has numerical
dimension 
.
On a threefold, a nef divisor has numerical dimension one precisely when 
and 
. The numerical dimension can exceed the
Iitaka dimension, so the two hypotheses must
be distinguished in abundance statements.
For a Cartier divisor 
on a smooth projective 
-fold, Nakayama's section-growth convention (Nakayama 2004)
is the largest integer 
for which some fixed ample Cartier
divisor 
satisfies
 |
It is 
when no such integer exists. The ample twist is fixed
before taking the limit superior.
OpenAI (2026c) announced a finite sequence of canonical-negative divisorial contractions and flips from every smooth connected
complex projective variety of dimension
at least three with pseudo-effective 
and 
to a rationally factorial terminal model
with nef canonical divisor. On a common smooth
resolution, 
with 
effective and 
-exceptional,
containing the transforms of all contracted source divisors. The inverse birational
map contracts no prime divisor. The growth hypothesis
is neither Iitaka dimension one nor a claimed
linear upper bound for every ample-twisted section
count. The release proof has not been independently verified
here.
