Non finitely generated graded ring of a divisor in dimension >2

Question

I spent some time looking for an example of non finitely generate graded ring $R=\oplus H^0(X,mD)$ where $D$ is a divisor on a variety $X$ of dimensison $>2$. I know there are several such examples (e.g Zariski's), but they are all on surfaces. I believe there must exist many examples. Do you know any?

Answer 1

Here is a way of generating lots of examples:

Start with a variety $X$ with an effective cone which is not rational polyhedral (i.e., not finitely generated) and let $L_1,\ldots,L_r$ be a collection of line bundles on $X$ such that their span $\{L_1^{a_1}\otimes \cdots\otimes L_r^{a_r} | a_1,\ldots,a_r \ge 0\}\,\,$ includes the effective cone. For example, one could take $X$ to be the blow-up of projective space at sufficiently many points, or a K3 surface of maximal Picard number.

Now consider the variety $Y=\mathbb{P}(E)$ where $E=L_1\oplus \cdots \oplus L_r$ and the line bundle $O(1)$ on $Y$. We have for $n\geq 0$, $$ H^0(Y,O(n))\cong H^0(X,Sym^n(E))=\bigoplus_{a_1+\ldots +a_r=n}H^0(X,L_1^{a_1}\otimes \cdots\otimes L_r^{a_r}) $$so that $R(Y,O(1))$ is isomorphic to the sum of all sections of all effective line bundles on $X$: this is usually called the Cox ring of $X$. When the effective cone of $X$ is non-rational polyhedral it is clear that this ring is infinitely generated, since it requires sections from all effective divisor classes.

It is usually a difficult problem in birational geometry to decide when a nef and big divisor is semiample, i.e., some multiple of $D$ is base-point free. A well-known theorem of Zariski says that if $X$ is normal and projective, then $D$ is semiample if and only if the section ring $R(X,D)$ is finitely generated. So this theorem gives a way of producing nef and big divisors which are not semiample. In particular, choosing $X$ such that the nef-big cone is not rational polyhedral, $L=O(1)$ is nef and big, but not semiample, since $R(Y,L)$ is not finitely generated.

Answer 2

I give Siu's analytical method to test that the ring $$R(X,L)=\bigoplus_{m=1}^\infty H^0(X,mL)$$ is not finitely generated

Let $X$ be a compact complex manifold and the ring $R(X,L)=\bigoplus_{m=1}^\infty H^0(X,mL)$ is finitely generated and let $s^{(m)}_1, … , s^{(m)}_{q_m}\in H^0 (X,mL) $

be a basis over $\mathbb C$. Let

$$ \Phi =\sum_{m=1}^\infty \epsilon_m( ∑_{j=1}^{q_m} |s^{(m)}_j|^2)^{1/m} $$

where $\epsilon_m$ is some sequence of positive numbers decreasing fast enough to guarantee convergence of the series. Then all the Lelong numbers of the closed positive $(1,1)$-current $$T=\frac{\frak{\sqrt{-1}}}{2\pi}\partial\bar\partial \log Φ$$ are rational numbers.

So one of ways to find some examples is to show that the Lelong number of $T$ is not rational at some point, hence the ring $R(X,L)=\bigoplus_{m=1}^\infty H^0(X,mL)$ is not finitely generated

Definition: Let $W\subset \mathbb C^n$ be a domain, and $\Theta$ a positive current of degree $(q,q)$ on $W$. For a point $p\in W$ one defines $$\mathfrak v(\Theta,p,r)=\frac{1}{r^{2(n-q)}}\int_{|z-p|<r}\Theta(z)\wedge (dd^c|z|^2)^{n-q}$$ The Lelong number of $\Theta$ at $p$ is defined as

$$\mathfrak v(\Theta,p)=\lim_{r \to 0}\mathfrak v(\Theta,p,r)$$

Let $\Theta$ be the curvature of singular hermitian metric $h=e^{-u}$, one has

$$\mathfrak v(\Theta,p)=\sup\{\lambda\geq 0: u\leq \lambda\log(|z-p|^2)+O(1)\}$$

Answer 3

In Section 7 of this paper the authors gave an example of a smooth 3-fold $X$ with a divisor $D$ such that:

$\lim_{n \rightarrow \infty} \frac{h^0(\mathcal O_X(nD))}{n^3} = 6+ \frac{2\sqrt 3}{9}$

I would love to here about more natural examples!

Answer 4

In their paper "Monge-Ampère equations in big cohomology classes", Boucksom, Eyssidieux, Guedj and Zeriahi give an example (Ex 5.4 page 46 here : http://arxiv.org/abs/0812.3674) of a nef and big line bundle over a smooth projective 3-fold which is not semi-ample. More precisely, every positive current in its cohomology class has poles along some subvariety.

Furthermore, it is well-known (Lazarsfeld, PAG e.g) that a nef and big line bundle has a finitely generated sections ring iff it is semi-ample.

In one word, their construction consists in using the famous example of Serre (and studied by Demailly-Peternell-Schneider) of a flat rank 2 vector bundle $E$ on some elliptic curve $C$, and considering on $V:=\mathbb P(E\oplus A)$ (for $A$ ample on $C$) the tautological line bundle $\mathcal O_{\mathbb P(V)}(1)$.