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In this question - 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.

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    $\begingroup$ No, such hypersurfaces would have Picard number equal to $m$, but there are many simplicial toric varieties (e.g., weighted projective spaces), which have Picard number 1 (but are not hypersurfaces). In fact, most of the threefolds $P(O+O(a)+O(b))$ (over $P^1$, for $a,b$ integers) cannot be embedded in a product of two projective spaces. $\endgroup$
    – byu
    Nov 14, 2017 at 17:24
  • $\begingroup$ @byu In that case, could we instead show that $(\mathbb{C}^N\backslash U/(\mathbb{C^*})^m)$ is isomorphic to a product of $m$ spaces (not necessarily projective) with Picard number 1, or a hypersurface thereof? $\endgroup$
    – Mtheorist
    Jan 22, 2018 at 11:13

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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.

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  • $\begingroup$ Which book are you referring to? Introduction to toric varieties? $\endgroup$
    – Mtheorist
    Nov 15, 2017 at 6:33
  • $\begingroup$ That's right. I'll edit to clarify. $\endgroup$
    – Tippi
    Nov 15, 2017 at 8:37

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