When is an algebraic space a scheme? Sometimes general theory is "good" at showing that a functor is representable by an algebraic spaces (e.g., Hilbert functors, Picard functors, coarse moduli spaces, etc).  What sort of general techniques are there to show that an algebraic space is a scheme?  Sometimes it's possible to identify your algebraic space with something "else" (e.g. it "comes" from GIT as is the case for the moduli space of curves), but are there other methods? 
 A: One example: an algebraic space $X$ is a scheme iff $X_{\text{red}}$ is a scheme. 
One application of this is that a quotient (I may be missing an adjective or two here) by a reductive group over an Artin ring is a scheme. Call the quotient $X$. Then it is easy to prove that $X$ is an algebraic space. On the other hand when your Artin ring is a field the classical theory of reductive groups tells you that the quotient is a scheme, i.e. $X_{\text{red}}$ is a scheme, and you can conclude that $X$ is in fact a scheme.
A: One result along those lines is that that any algebraic space which has a quasi-finite morphism to a scheme is itself a scheme.
More precisely, if $f\colon X\to Y$ is a separated, finite type, quasi-finite morphism of algebraic spaces, then the Stein factorization $X\to \mathrm{Spec}_Y(f_*\mathcal{O}_X)\to Y$ is an open immersion followed by an affine morphism, so $f$ is quasi-affine. In particular, if $Y$ is a scheme, so is $X$.
References
This is Proposition 3.1 of Quot Functors for Deligne-Mumford Stacks or Théorème A.2 of Champs algébriques.
I learned this from Martin Olsson; it's Corollary 17.8 in my notes from his stacks course. It's Theorem 7.2.10 in the book he's since written covering the material from that course. Jason Starr points out that there's a mistake in Lemma 6.2.9 which propagates to this theorem (I haven't checked this), and suggested the two alternative references above.
A: Another example: the Nakai-Moishezon theorem says that a divisor D on X is ample iff for every curve C on D, $D \cdot C > 0$ and $D^2 > 0$. This holds also for an algebraic space.
As an application, you can show for instance that the coarse space of $\bar{M_g}$, the Deligne-Mumford compactification of the moduli stack of smooth genus g curves, is represented by a projective variety. The point is that Artin's representibility theorem tells you that the coarse space exists as an algebraic space, and you can then use Nakai-Moishezon to show that it has an ample line bundle. This is cool because it avoids GIT.
