Let $C$ and $T$ be compact connected Riemann surfaces (or: smooth projective connected curves over $\mathbb{C}$) of genus at least two and let $X:=C\times T$. Let $(c,t)$ be a point of $X$, and let $X'\to X$ be the blow-up of $X$ in $(c,t)$. By Grauert's contraction theorem, we may contract the strict transform of $\{c\}\times T$ on $X'$ and obtain a normal complex-analytic surface $X'\to S$.

Under what conditions (if any) is $S$ projective?

Note that $S$ contains a unique rational curve (given by the image of the exceptional curve $E$ of $X'\to X$), and that $S$ has a unique singular point $\sigma$ in $S$.

My interest in this surface is related to Lang's conjectures, and I first learned about this surface from Frederic Campana. Indeed, the surface $S$ has the peculiar property that, for any point $s$ which does *not* lie on the rational curve and any pointed curve $(D,d)$, the set of pointed maps $(D,d)\to (S,s)$ is finite. However, for the pointed curve $(C,c)$ and the singular point $\sigma$, the space of pointed maps $(C,c)\to (S,\sigma)$ covers $S$.

I was not able to prove projectivity of $S$, not even whilst assuming it is proper (so that one might appeal to https://arxiv.org/abs/1112.0975 )