# Negative intersection number between curve and effective divisor

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.

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 $$E \cong \mathbf P^{d-1}$$.
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 $$B.C = -2$$.