Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

The Zariski-Fujita theorem says that on a projective variety $X$, if a Cartier divisor $D$ is ample on its base locus, then some positive multiple $mD$ is base point free. I'm wondering if the following related statement is true.

In the same setting, suppose the base locus $Bs(|D|)= \bigcup_{i=1}^{N} C_i$ is a union of curves, and $D \cdot C_i >0$ for $i=1, \ldots , k$, i.e. $D$ is ample on the first $k$ curves. Then is it true that for some $m>0$, $C_i \nsubseteq Bs(|mD|)$ for $i=1, \ldots , k$? Specifically, that $Bs(|mD|) \subseteq \bigcup_{i=k+1}^N C_i$ ?

I'm interested in the case where $\dim X=3$ and $X$ is smooth, but I don't think that's particularly relevant. Any help is appreciated.

share|improve this question
add comment

1 Answer

up vote 3 down vote accepted

I don't think this is true. Let $X$ be obtained by blowing up $\mathbb P^2$ at one point, and then at a point on the exceptional curve. Call the exceptional divisors $E$ and $F$ respectively. Let $D = E+3F$. Then $D \cdot E = 1$, but $E$ is in the stable base locus.

share|improve this answer
add comment

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

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