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Let $S$ be a scheme, and let $f:X\to S$ be a morphism of algebraic spaces.

If $f$ is smooth proper curve of genus at least two, then $X$ is a scheme. (Here I mean that $f$ is a smooth proper morphism whose geometric fibers are actual curves of genus at least two.)

Are there any other types of properties which one can impose on $f$ so that $X$ becomes a scheme?

My question is to see how far one can get without algebraic spaces. My guess is "not so far".

For instance, what if $f$ is a smooth proper morphism whose geometric fibres have ample or anti-ample canonical bundle?

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For a scheme $S$, every proper finitely presented map $f:X \rightarrow S$ from an algebraic space $X$ admitting a line bundle $L$ that is ample on each geometric fiber (which of course forces such fibers to be projective schemes, essentially by the very definition of ampleness for algebraic spaces) is automatically a scheme.

Indeed, by standard limit methods we may assume $S= {\rm{Spec}}(R)$ is local and then we assume only that $L$ is ample on the special fiber. That case can be descended to $S= {\rm{Spec}}(R)$ local noetherian, and the aim is to prove that for some large $n$ the power $L^n$ is generated by global sections with the resulting map $f:X \rightarrow {\mathbf{P}}(\Gamma(X,L^n))$ quasi-finite (so $X$ is a scheme, being quasi-finite and separated over a scheme). These are properties which are sufficient to check after faithfully flat base change to the completion of $R$, so we can assume $R$ is complete. It suffices to prove here that $X$ is now a scheme, as then the cohomological theory of ample line bundles would then apply to provide the desired $n$, the crux being the remarkable EGA IV$_3$, 9.6.4 (which has no flatness hypotheses).

Every infinitesimal fiber is a scheme, so we can make sense of the formal completion along the special fiber as a proper formal scheme equipped with a residually ample line bundle. By formal GAGA with the help of $L$, that formal scheme algebraizes to an actual proper scheme. But we have formal GAGA for algebraic spaces too, so the algebraic space algebraization and the scheme algebraization coincide.

QED

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  • $\begingroup$ Thank you for your quick answer. If the fibres of $f:X\to S$ have ample canonical bundle, then do I understand correctly that the mere existence of a line bundle $L$ which restricts to the canonical bundle on each fibre implies that $X$ is a scheme? Is it clear that there exists a relative dualizing sheaf $\omega_{X/S}$ in the setting you consider (i.e., $f$ proper finitely presented)? $\endgroup$
    – user235
    Commented Jun 9, 2015 at 16:35
  • $\begingroup$ In the smooth proper case (what you asked about) the canonical bundle $\Omega^d_{X/S}$ is such an $L$! (Construction same as for schemes.) $\endgroup$
    – grghxy
    Commented Jun 9, 2015 at 16:37
  • $\begingroup$ Dear @grghxy , thank you again for this answer. I was merely curious if $\omega_{X/S}$ exists in your more general setting. But you are right, in the smooth proper case this settles it. $\endgroup$
    – user235
    Commented Jun 9, 2015 at 16:57
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    $\begingroup$ @user235: It exists in the same generality as for schemes (e.g., finitely presented and flat with CM fibers) by etale descent from the scheme case since in the scheme case its formation is canonically compatible with etale localization on $X$ as well as with any base change on $S$. $\endgroup$
    – grghxy
    Commented Jun 9, 2015 at 21:54

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