Algebraic surface with infinitely many exceptional curves

Question

I am learning about the classification of Projective Algebraic Surfaces (in fact, Compact Complex Surfaces) and I am troubled with the following point.

If I understood correctly every surface $X$ admits a (not necessarily unique) minimal model $X_{min}$, which is a surface without exceptional curves (a rational curve with sel-intersection $-1$). Furhtermore $X$ is obtained from $X_{min}$ after a finite number of blow-ups.

On the other hand I read that there are examples of surfaces with infinitely many exceptional curves. My question is how can we obtain $X$ from $X_{min}$ after a finite number of blow-ups? Can a single blow-up (or a finite number of them) add an infinite number of exceptional curves?

Another way of phrasing this is the following: given $X$ with infinitely many exceptional curves how can we obtain $X_{min}$ performing only a finite number of contractions/blow-downs.

Thanks in advance for your answers!

Answer 1

The solution of the paradox is that you may simultaneously blow up $k$ points and obtain more than $k$ exceptional curve.

The simplest example is obtained by simply blowing up two points $P_1,P_2$ in the plane $\mathbb P^2$.
The blown up surface $X=\tilde {\mathbb P^2}$ contains as exceptional curves not only the inverse images $E_1,E_2$ of $P_1,P_2$ but also the strict transform $\tilde L$ of the line $L=\overline {P_1P_2}$ joining $P_1$ to $P_2$:
Indeed the self-intersection of $L$ is $+1$ and that self-intersection diminishes by $1$ at each $P_i$ after the blow up, so that $\tilde L$ has self-intersection $1-1-1=-1$.
And since $\tilde L$ is isomorphic to $\mathbb P^1$ it is an exceptional curve.
Conclusion:
$X$ has $3$ exceptional curves $E_1,E_2,\tilde L$, although it is obtained by blowing up only $2$ points in $\mathbb P^2$ (which $\mathbb P^2$ has no exceptional curve at all).

If now you simultaneously blow up at least $9$ points of $\mathbb P^2$ in suitably general position you will obtain, rather surprisingly I concede, a surface with infinitely many exceptional curves: cf. Hartshorne, Remark 5.8.1, page 418.