Let $S$ be a minimal, non-singular complex projective surface. $\widehat S$ is the surface obtained by $r$ blow-ups of $S$ at the points $x_1,\ldots,x_r\in S$. Clearly $\widehat S$ contains exactly $r$ (-1)-curves and now if we contract all these curves we obtain another minimal surface $X$ such that $K^2_X=K^2_S$. Under this hypothesis can we conclude that $S$ and $X$ are isomorphic?
I suppose that the answer is no, because in this way I simply find two minimal models of $\widehat S$.
What do you think?
The statement
is false. For example, blow up $\mathbf P^2$ in 2 points. Then the exceptional curves are (-1)-curves, but there is another (-1)-curve on this surface: namely the proper transform of the line that went through the two points. If we contract just that last (-1)-curve, we get the minimal surface $\mathbf P^1 \times \mathbf P^1 \neq \mathbf P^2.$ This is the point about non-uniqueness of minimal models for Kodaira dimension $-\infty$: blowing up can create more (-1)-curves than just the exceptional curves.
On the other hand, if the (-1)-curves you contract are exactly the exceptional curves of the blowups, then yes, certainly $S$ and $X$ will be isomorphic: contracting the curves is just undoing what you did when you blew up the points.