# Embedding proper algebraic spaces

Does every proper algebraic space (over a field, say) admit a closed immersion into a smooth proper algebraic space?

Remark: Of course, if we say "projective" instead of "proper" then the answer is tautologically "yes": we can take the ambient variety to be some projective n-space. But I'm curious about the general case.

• No : there are proper normal surfaces $S$ with trivial Picard group (see for instance [Stefan Schröer, On non-projective normal surfaces]). If you could embed $S$ in a smooth and proper $X$, you could choose a Weil (hence Cartier) effective divisor in $X$ intersecting $S$ but not containing it, showing that the Picard group of $S$ is not trivial. Nov 7 '14 at 15:04
• Are you sure that this exclude the possibility of embedding $S$ into a smooth algebraic space? It seems to me that this argument only works if $X$ is a scheme, or I am missing something? Nov 7 '14 at 15:12
• @FrancescoPolizzi Ah you are absolutely right ! Sorry ! Nov 7 '14 at 15:24