TITLE: Is every variety an image of a smooth variety?
QUESTION [6 upvotes]: Let $X$ be a finite type scheme over a field $k$.
Is it true that there exists a surjective morphism $f : Y \rightarrow X$, where $Y$ is smooth over $k$?
In other words, is every such scheme a quotient of a smooth scheme over $k$?

REPLY [8 votes]: Presumably, you want to say that $X$ is reduced, since $Y$ will be reduced and any map from a reduced scheme lands in $X^{\mathrm{red}}$. Once you've said that, this follows from De Jong's alterations theorem, appearing in

A.J. De Jong, Smoothness, semi-stability and alterations, Publications Mathématiques de l'IHÉS, Tome 83 (1996), pp. 51–93, doi:10.1007/BF02698644,
Numdam.