Let $X$ be a smooth projective variety over an algebraically closed field of characteristic zero $k$.

We call $\mathcal{P}_0$ the class of $k$-varieties whose objects are:

- abelian varieties
- semi-abelian varieties (extension of abelian varieties by a torus)
- smooth hypersurfaces in large projective spaces over $k$
- smooth projective varieties of dimension $\le 3$
- smooth projective toric varieties
- products of the above

Let $\mathcal{P}$ be the class of $k$-varieties whose objects are obtained by blowing up an element of $\mathcal{P}_0$ along a closed $k$-subvariety that is again an element of $\mathcal{P}_0$.

Does there exist a surjective map $Y\to X$ over $k$ with $Y$ in $\mathcal{P}$?

As an example, it might be useful to first try to answer the question with $Y$ a smooth projective toric variety alone. It is not clear to me this shouldn't be possible, since $Y\to X$ is required to only be surjective.