Let $X$ be a smooth algebraic surface over $\mathbb{C}$, and $Y \xrightarrow{\phi} X$ the blowup at a (reduced) point with exceptional divisor $E$. Then, the we have the following universal property: Every morphism from $Y$ to an algebraic variety $Z$ that contracts $E$ to a point factors through $X$ (Beauville, Algebraic Surfaces p.17) I'm not sure to think of this (univ. prop. of blow. down) as a property of smooth surfaces over $\mathbb{C}$ or more generally. Here are some natural questions come to mind:
Is there a more general universal property of blowing down along these lines. I don't expect there to be an answer for arbitrary blowups (say of noeth. schemes), however I would like to replace the field $\mathbb{C}$ by other algebraically closed fields like $\overline{\mathbb{F}_p}$, and if the formalism allows it, even $\mathbb{Q}$. There are a couple of cases:
1.) I can't even figure out a universal property if we stay in the context schemes smooth over $\mathbb{C}$, and blow-up at smooth, irreducible subvarieties $Z$.
2.) Let $X, Y, \phi, E$ as above (in particular, surfaces). Assume that $X$ is reduced, but possibly singular, and allow $\phi$ to be a blow-up at an arbitrary (possibly non-reduced) point. Is there a universal property that blowing-down along the exceptional divisor satisfies in this case? (The way I see it, one issue is to replace "collapsing $E$ to a point" by something else.)
3.) I'm very much interested in the most general version of the universal property one can formulate. If there is a reference where this is covered (EGA?) I would love to see it. I also don't have "good" reasons to believe that a universal property doesn't hold in complete generality (say in the context of noetherian schemes). If you're convinced that this is the case, I'd love to hear your reasoning.