1
$\begingroup$

When $X$ is a smooth projective variety and $D$ is a divisor on $X$, it is known that the nonnef locus $\mathbf{B}_-(D)$ is the union of the centers of the discrete valuations $\sigma$ such that $\overline\sigma(D):=\inf${$\sigma(E)|0\leq E\equiv D$}$>0$. But can we say that $\overline\sigma(D)>0$ if and only if Center$(\sigma)$ is contained in the nonnef locus of $D$? (In other words, can there be a discrete valuation $\sigma$ such that $\sigma(D)=0$ and the center of $\sigma$ is still in the nonnef locus of $D$?)

$\endgroup$

1 Answer 1

2
$\begingroup$

If $X$ is a smooth projective variety, then $\overline{\sigma}(D)>0$ if and only if $\mathrm{Center}(\sigma)\subseteq \mathbf{B}_-(D)$. A reference is "Asymptotic invariants of base loci" by Ein, Lazarsfeld, Mustata, Nakamaye, Popa, theorem 2.8

http://de.arxiv.org/PS_cache/math/pdf/0308/0308116v2.pdf

Actually they do this for a big divisor, but this should imply the same result for all pseudoeffective divisors, by using that $\mathbf{B}_-(D)$ is the union of the non-nef loci of $D+A_m$, where $A_m$ is a sequence of amples that goes to 0.

See also "Asymptotic base loci on singular varieties" by Cacciola and Di Biagio http://de.arxiv.org/PS_cache/arxiv/pdf/1105/1105.1253v2.pdf

for a generalization to the singular case.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.