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 $\mathcal{P}$?

Examples. $\mathcal{P}$ contains all the classes mentioned above, and in addition, for instance, all rational $k$-varieties.

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.