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 $$\dim \mathbf{B}(D) = \dim \bigcup\_{C, C\cdot D < 0}{C}\qquad?$$



Stable base locus of a divisor and negative intersection with curves
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.


