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 $k$?



Is every variety an image of a smooth variety?
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](https://doi.org/10.1007/BF02698644),
[Numdam](http://www.numdam.org/item/?id=PMIHES_1996__83__51_0).


