In this question - [On a Hirzebruch surface.](https://mathoverflow.net/questions/122952/on-a-hirzebruch-surface) , the Hirzebruch surface is shown to be isomorphic to a hypersurface in $\mathbb{P}^1\times \mathbb{P}^2$. 


My question is, does such an isomorphism exist for all toric varieties (or at least simplicial ones)? To be precise, given a toric variety 
$$
(\mathbb{C}^N \backslash U)/(\mathbb{C}^\*)^m,
$$ 
can we show that it is isomorphic to a hypersurface of a product of $m$ projective spaces?


At least for complete intersection Calabi-Yaus, this seems to be true, based on <https://arxiv.org/abs/0805.2875>.



Can any simplicial toric variety be embedded in a product of projective spaces?
There are well known examples of smooth (hence simplicial) complete toric varieties which are not projective. See for example p. 71 of Fulton's book *Introduction to Toric Varieties*. Any such variety gives a counterexample.


