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?