Infinitely many exceptional curves on ruled surfaces 
Take an algebraically closed field $k$. Let $C$ be a smooth projective variety of dimension $1$ over $k$(a curve). Consider a geometrically ruled surface $S :=C\times\mathbb{P}^1$. Does there exists a blow-up, say $\widehat{S}$ of $S$ at distinct points $p_1,\ldots,p_n$ on $S$, which has infinitely many exceptional curves?

This is in spirit of the analogous result for $C=\mathbb P^1$, where the answer is affirmative. The pith of the proof relies on the fact that blowing-up $\mathbb P^2$ at $k$ points in the general position might introduce large number of exceptional curve even for very small $k$. But for curves of higher genus it seems not to be the case. Is this observation correct?
Also could we say something for higher Kodaira dimension surfaces?
 A: Let me write a short answer summarizing the comments above, so that the question will not appear unanswered anymore.

Proposition. The following holds.
(1) If $C$, $D$ are smooth curves and  $g(C) \geq 1$, then any blow-up of $C \times D$ contains at most finitely many $(-1)$-curves.
(2) If $S$ is any smooth, complex, projective surface with non-negative Kodaira dimension, then $S$ contains at most finitely many $(-1)$-curves.
Thus, surfaces containing infinitely many $(-1)$-curves are necessarily rational.

Proof. (1) Let $\pi \colon S \to C \times D$ be a blow-up in a finite number of points, and $p \colon S \to C$ the composition of $\pi$ with the projection onto $C$. Since $g(C) \geq 1$, no rational curve in $S$ can dominate $C$. It follows in particular that the $(-1)$-curves of $S$ are contained in reducible fibres of $p$, so there are finitely many of them.
(2) If $E \subset S$ is a $(-1)$-curve, then by adjunction $K_XE = -1$.
If we take a positive integer $m$ such that $mK_X$ is effective, it follows that $E$ is necessarily one of the finitely many components of the base locus of the complete linear system $|mK_X|$.
