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.
