Removable base loci for non-projective varieties

Question

The theorem of Zariski-Fujita says the following: Given a line bundle $L$ with base locus $B$ on a projective variety $X$. If $L_{|B}$ is ample on $B$, then $L$ is semiample, i.e. $L^{\otimes m}$ is generated by global sections for some $m>0$.

Does this remain true if we only assume $X$ is complete?

Answer 1

Short answer: No.

In his original paper, Fujita posed the question whether we can weaken the assumptions in the theorem. (Fujita 1983, 1.16) There has been one paper written since then with an attempt at improvement.

Let $R$ be a commutative Noetherian ring, $X$ a scheme proper over $R$, and $\mathcal{L}$ a line bundle on $X$. Define

$H^p_\*(\mathcal{F},\mathcal{L}) = H^p(X, \mathcal{F}\otimes\text{Sym}\ \mathcal{L})=\bigoplus_{n\geq0} H^p(X,\mathcal{F}\otimes \mathcal{L}^{\otimes n})$, a module over the graded ring $\Gamma_\*(\mathcal{L}) = H^0(X,\text{Sym} \ \mathcal{L})$.

Then there is

Theorem. (Schröer, 2001) Let $B$ be the stable base locus of $\mathcal{L}$. The following are equivalent:

1. $\mathcal{L}$ is semi-ample,
2. For each coherent sheaf $\mathcal{F}$ and $p \geq 0$, $H^p_\*(\mathcal{F},\mathcal{L})$ is finitely generated over $\Gamma_\*(\mathcal{L})$.
3. For all coherent ideal sheaves $\mathcal{I} \subset \mathcal{O}(B)$, the module $H^1_*(\mathcal{I}, \mathcal{L})$ is finitely generated over $\Gamma_\*(\mathcal{L})$.

Schröer uses this generalization of Zariski-Fujita to characterize contractible curves in 1-dimensional families, where considers $X$ complete (i.e., proper over a base field) in a couple instances.

I think any other attempt will run into the same problem: the best you can do is a cohomological characterization.

Keeler (2003) worked in the setting of schemes with filters of line bundles, rather than a single individual line bundle. (This is again a cohomological condition.) He proved a generalization of Serre's vanishing theorem and some other results in Fujita's paper, but not a very promising lead on your question:

Proposition. (Keeler, 2003) Let $X$ be a complete variety, $\mathcal{L}$ a line bundle on $X$, and suppose the base locus is zero-dimensional or empty. Then $\mathcal{L}$ is nef.

There is also the paper of Ein (2000) where he finds a more elegant proof of Zariski-Fujita using Koszul complexes.

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• Ein, L., Linear systems with removable base loci, Comm. Algebra 28, n. 12 (2000), 5931–-5934
• Fujita, Takao, Semipositive line bundles, J. Fac. Sci. Univ. Tokyo 30 (1983), 353--378
• Keeler, Dennis S, Ample filters of invertible sheaves, J. Algebra 259 (2003), 243--283
• Schroer, Stefan, A characterization of semiampleness and contractions of relative curves, Kodai Math. J. Volume 24, Number 2 (2001), 207--213.