Negative intersection number between curve and effective divisor

Question

Let X be a complex projective variety and E be an irreducible effective divisor on it. Then, I want to know whether the following set is finite: {C | C be an irreducible curve and C.E<0}. I know that the above set is finite when X is a surface.

Answer 1

If $X$ is a smooth projective variety of dimension $d$ and $\tilde X \to X$ is the blowup of $X$ at a point $x$, then the exceptional divisor $E \subseteq \tilde X$ is isomorphic to $\mathbf P^{d-1}$ and $\mathcal O_{\tilde X}(E)|_E \cong \mathcal O_E(-1)$ (see e.g. Hartshorne, Thm. II.8.24). In particular, $\mathcal O_{\tilde X}(E)|_C$ is anti-ample for any curve $C \subseteq E$, so $C \cdot E < 0$. If $d \geq 3$, there are infinitely many curves in $E \cong \mathbf P^{d-1}$.

Answer 2

It is not. Let $X$ be a smooth surface and let $X^{[n]}$ be the Hilbert scheme of $n$ points on $X$, parameterizing zero-dimensional subschemes of $X$ of length $n$ . This is a smooth complex projective variety of dimension $2n$, and there is a Hilbert-Chow morphism $\pi : X^{[n]}\to X^{(n)}$ which is a resolution of singularities of the symmetric product $X^{(n)} := X^n / S_n$.

In $X^{[n]}$, the locus of nonreduced schemes is an irreducible effective divisor $B$. Let $C$ be a curve on $X^{[n]}$ given by fixing $n-2$ points in $X$ and "spinning" a length $2$ tangent vector at another point. In other words, these are curves which are contracted to a point by the Hilbert-Chow morphism. Then $B.C = -2$.

There are probably also simple examples involving blowups.