Let $X$ be a non-singular projective variety, and $D$ a divisor on $X$.


Saying that $D$ has positive (meaning non-zero) Iitaka dimension is equivalent to the function $n \mapsto h^0(\cal{O}(D))$ being strictly increasing for sufficiently large $n$?


Does every effective divisor have positive Iitaka dimension? If not, what are counterexamples?



Divisors with positive Iitaka dimension
Theorem 1.2 in [Takayama 2002](http://www.ams.org/journals/tran/2003-355-01/S0002-9947-02-03068-4/S0002-9947-02-03068-4.pdf) should help compute some examples of effective divisors with 0 iitaka dimension. I think your first question is related to the "exponent" of the divisor (see Lazarsfeld Positivity I 2.1.1 and example 2.1.2).


edit: Note that a slight rewording of your first question might work by 2.1.10 in the same book.


