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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?

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  • $\begingroup$ I believe you can prove this just by looking at the map $\widehat{S}\rightarrow C$ and observing that it's generally smooth. It follows that the exceptional curves must lie in finitely many fibers, but each of those fibers is dimension $1$ and finite type. $\endgroup$ – dhy Apr 19 '17 at 16:39
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    $\begingroup$ The rational curves cannot dominate $C$, so they are contained in fibres. It follows that you must have finitely many of them. $\endgroup$ – Francesco Polizzi Apr 19 '17 at 16:41
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    $\begingroup$ If $S$ has higher Kodaira dimension, there can only be finitely many exceptional curves (here I'm understanding this to mean "exceptional curves of the first kind", i.e. $-1$-curves). Pf: $|mK_S|$ has an effective representative $C$ for some $m$. A $-1$ curve has intersection $-1$ with $K_S$ by adjunction, so the only candidates are the (finitely many) components of $C$. $\endgroup$ – user47305 Apr 19 '17 at 17:41
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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 $|mK_X|$.

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