Suppose $X$ is a surface, can it have infinitely many $(-1)$ curves?(If they are disjoint, we can see this since Neron Severi group has finite rank, but how to deal with the case when they are not disjoint?) 


Suppose $X$ is a projective variety, is it true that the codimension $1$ subvarieties with negative top self-intersection (degree of the zero cycle) is finite?



Negative self-intersection and finiteness
Let $C\_1$ and $C\_2$ be two general cubic plane curves. Let $S$ be the surface obtained by blwoing-up the nine base points. Then $S$ is a rational elliptic surface. Every section of the elliptic fibration $S\to \mathbb{P}^1$ is a $-1$ curve. After choosing one section as the zero-section you find that the sections form an abelian group. If $C\_1$ and $C\_2$ are sufficiently general then this group is isomorphic with $\mathbb{Z}^8$, in particular you have infinitely many $-1$ curves. 


