TITLE: Embedding of a proper scheme into a smooth one
QUESTION [7 upvotes]: Let $\mathbb{K}$ be any field and let $X$ be a proper $\mathbb{K}$-scheme. Does it exist a smooth, proper $\mathbb{K}$-scheme $Y$ and a closed immersion from $X$ to $Y$? This is tautological if $X$ is projective, but what about the general case?

REPLY [14 votes]: For a counterexample over $\mathbb{C}$, it sufficies to take a proper scheme $X$ with trivial Picard group (there are examples among toric varieties).  
Such a scheme cannot be embedded in any smooth variety $Y$; in particular, $X$ is necessarily singular. 
In fact, assume the contrary and take a dense affine subset $U$ of $Y$. Then we can choose an effective divisor $D_U$ on $U$, whose closure would give a non-trivial Cartier (since $Y$ is smooth) divisor $D_Y$ on $Y$, hence a non-trivial Cartier divisor $D_X$ on $X$, against the assumption $\textrm{Pic}\,X=\{\mathcal{O}_X \}$.