In the question On a Hirzebruch surface, I've seen that the $n$-th Hirzebruch surface is isomorphic to a surface of bidegree $(n,1)$ in $\mathbb{P}^1\times \mathbb{P}^2$. I am trying to answer the following connected questions about toric varieties obtained from integral polytopes:
- How can one make sure that the isomorphism from On a Hirzebruch surface induces an isomorphism between the standard real part of the $n$-th Hirzebruch surface and the real part of the considered surface of bidegree $(n,1)$ ?
- Is there a systematic way to find a toric $3$-fold (respectively, a toric $4$-fold) in which a given toric surface (respectively, a given toric $3$-fold) embeds as a a real algebraic hypersurface ?
Do you have any suggestions ?
