# How close is $h^0(mD)$ to be a polynomial?

Let $X$ be a normal (or smooth if it helps) projective variety over an algebraically closed field $k$. Fix a Cartier divisor $D$: I am interested in knowing how $h^0(mD)$ behaves as $m$ varies.

At page 171 in Lazarsfeld's book "Positivity in Algebraic Geometry, I" there are some results in this direction. Above all, there are a result of Zariski about surfaces (see below), and the famous theorem by Kollàr and Matsusaka. Is there anything else worth to be mentioned? Or more recent?

In particular, I would like to know whether there are analogues to Zariski's result (then refined by Cutkosky and Srninivas) saying that on a smooth (I think the generalization includes the normal case) surface, given $D \geq 0$, then $h^0(mD)$ is a polynomial up to a bounded error, which is periodic in $m$. Is there anything like this in higher dimension? Are there examples that show this is not the case?

• For Toric variety it is e Ehrhart polynomial
– user21574
Feb 4, 2017 at 7:35
• – user21574
Feb 4, 2017 at 9:07
• sebastien.boucksom.perso.math.cnrs.fr/publis/…
– user21574
Feb 4, 2017 at 9:11
• The Annals paper of Cutkosky-Srinivas contains a higher-dimensional counterexample. Feb 7, 2017 at 15:06