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Let $X$ be a smooth projective variety over $\mathbb{C}$. Let $D$ be a Cartier divisor on $X$. If $C$ is a curve on $X$ such that the intersection $C\cdot D <0$, we have $C \subseteq \mathbf{B}(D)$, where $$\mathbf{B}(D):= \bigcap_{m \in \mathbb{N}, F\in |mD|}F$$ is the stable base locus of $D$.

I want to know if there is any result related to the inverse direct. To be precise, I want to know if the followings are true or false:

(1)For a general point of $\mathbf{B}(D)$, is there a curve $C$ passing through that point with $C\cdot D<0$?

(2)The union of all curves with negative intersections with D has the same dimension as $\mathbf{B}(D)$, that is $$\dim \mathbf{B}(D) = \dim \bigcup_{C, C\cdot D < 0}{C}\qquad?$$

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Both are false. Let $X$ be the blow-up of $\mathbb P^3$ at $8$ very general points, and let $D = 2H - \sum E_i$ be the linear system of quadrics through the $8$ points. There is a pencil of such quadrics. The stable base locus of $D$ is a genus $1$, degree $4$ curve (obtained as the intersection of any two quadrics). But $D$ is nef.

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    $\begingroup$ You could a little closer if you instead considered the diminished base locus $\mathbf B_-(D) = \bigcup_{\text{$A$ ample}} \mathbf B(D+A)$, which is a subset of $\mathbf B(D)$. Both are probably still false, but it's at least true that if $\mathbf B_-(D)$ is non-empty, $D$ is not nef. $\endgroup$
    – user47305
    Commented Jan 4, 2015 at 16:20
  • $\begingroup$ Thank you for your answer! But I did not quite follow your example: In your example, $X$ is a toric variety, hence any nef divisor is base point free (in particular $D$ is basepoint free if it is nef). Where did I understand wrong? $\endgroup$
    – Li Yutong
    Commented Jan 5, 2015 at 6:12
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    $\begingroup$ $X$ is not toric! You can blow up $4$ general points in $\mathbb P^3$ and get a toric variety, but no more ([1,0,0,0] etc are torus-invariant points, but that's it). However, it's still a Mori dream space for up to $7$ points, and so nef divisors are semiample. One you hit $8$, everything goes bad; it's a lot like $\mathbb P^2$ blown up at $9$ points versus $8$. $\endgroup$
    – user47305
    Commented Jan 5, 2015 at 6:58

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