When is an algebraic space a scheme? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T18:51:46Z http://mathoverflow.net/feeds/question/4573 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/4573/when-is-an-algebraic-space-a-scheme When is an algebraic space a scheme? mdeland 2009-11-08T01:19:00Z 2009-11-08T03:51:01Z <p>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? </p> http://mathoverflow.net/questions/4573/when-is-an-algebraic-space-a-scheme/4574#4574 Answer by Anton Geraschenko for When is an algebraic space a scheme? Anton Geraschenko 2009-11-08T01:23:15Z 2009-11-08T01:29:35Z <p>One result along those lines is that that any algebraic space which has a quasi-finite morphism to a scheme is itself a scheme.</p> <p>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 <a href="http://math.berkeley.edu/~anton/written/Stacks/Stacks.pdf" rel="nofollow">my notes from Martin Olsson's stacks course</a>.</p> http://mathoverflow.net/questions/4573/when-is-an-algebraic-space-a-scheme/4579#4579 Answer by David Zureick-Brown for When is an algebraic space a scheme? David Zureick-Brown 2009-11-08T02:26:58Z 2009-11-08T02:26:58Z <p>One example: an algebraic space $X$ is a scheme iff $X_{\text{red}}$ is a scheme. </p> <p>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.</p> http://mathoverflow.net/questions/4573/when-is-an-algebraic-space-a-scheme/4586#4586 Answer by David Zureick-Brown for When is an algebraic space a scheme? David Zureick-Brown 2009-11-08T03:51:01Z 2009-11-08T03:51:01Z <p>Another example: the <a href="http://en.wikipedia.org/wiki/Ample%5Fline%5Fbundle#Criteria%5Ffor%5Fampleness" rel="nofollow">Nakai-Moishezon</a> 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.</p> <p>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.</p>