# 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?

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 $f:X\to Y$ is a separated, locally quasi-finite, locally finite type morphism from an algebraic space to a scheme, then by the Stein factorization theorem, $f$ is quasi-affine, so $X$ must be a scheme. If you want more details, this is Corollary 17.8 in my notes from Martin Olsson's stacks course.

• Cool! Then one might wonder how to produce "enough line bundles" on X to start trying to make these maps to something projective, for example. Or simply how to make them in general. – mdeland Nov 8 '09 at 1:49

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.

• Ah, cool! Doing a little research I see that Koll\'ar has done exactly this in his paper "Projectivity of Complete Moduli". You let $D$ to be a power of the relative dualizing sheaf. Then for curves $C$ not contained in the boundary it's more or less clear the Nakai-Moishezon condition is satisfied and for curves contained in the boundary you work a little harder but it's still true. He also applies this to compactifications of surfaces of general type. Are there other known applications of this method? – mdeland Nov 8 '09 at 15:36
• One is David Smyth's thesis. Its similar though, he gives alternative compactificaitons of $\bar{M_{g,n}}$ and shows that they have projetive coarse moduli spaces (here GIT doesn't work). – David Zureick-Brown Nov 8 '09 at 15:59

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.

• How do you show that X is a scheme if X_{red} is? I remember we worked this out before, but I don't remember how. I feel like the first thing to do (assuming X is noetherian) is to reduce to showing that a square-zero thickening of a scheme is a scheme. – Anton Geraschenko Nov 8 '09 at 3:57
• I believe this is in Knutson's book. – David Zureick-Brown Nov 8 '09 at 8:20
• If $X$ is noetherian and locally separated, this is Corollary III.3.6 of Knutson's book Algebraic Spaces. – Anton Geraschenko Feb 3 '15 at 3:43
• Apparently the noetherian and locally separated hypotheses are not needed: mathoverflow.net/q/195528. – Anton Geraschenko Feb 3 '15 at 5:57