Peng Zhou

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====== Product vs Convex Hull ======

===== Basic Facts =====
Let $V_1^\Z, V_2^\Z$ be lattices of rank $n_1, n_2$, and $V_i = V_i^\Z \otimes \R$.  
Suppose $\Delta_i$ is a reflexive polytope in $V_i$, with polar dual polytope $\Delta_i^\vee$.

Consider the product space $V_1 \times V_2$ and view $\Delta_1$ as a sub-polytope $\Delta_1 \times 0$, etc.  
Then we have dual reflexive polytopes
$$ (\Delta_1 + \Delta_2)^\vee = Conv(\Delta_1^\vee \cup \Delta_2^\vee) =: \Delta_1^\vee \diamond \Delta_2^\vee. $$

Example: $([-1,1] \times [-1,1])^\vee = Conv( \{(1,0), (0,1, (-1,0), (0,-1)\}) $

===== Implication for Mirror Symmetry of CY double cover =====
Take $\Delta_2 = [-1,1] \In \R$, then $\Delta_2^\vee = [-1,1]$ as well.

Take $\Delta_1$ be any reflexive polytope, $X=X_{\Delta_1}$ be the toric fano variety (DM stack) with moment polytope (of anticanonical bundle) $\Delta_1$. And $X^\vee=X_{\Delta_1^\vee}$ the dual fano. 

Then, $X_{\Delta_1 \times \Delta_2} = X \times \P^1$. Identify its interior with $(\C^*)^n_z \times \C^*_w$. Choose Laurent polynomials $s_+(z),s_-(z)$ on $(\C^*)^n$ with Newton polytope $\Delta_1$. Then define
$$ W(z,w) = w s_+(z) + w^{-1} s_-(z) + 1 $$
Then $W^{-1}(0)$ is extends to a CY hypersurface in $X \times \P^1$, double cover over $X$. The ramification loci is given by $1 - 4s_+(z) s_-(z) = 0$. 

On the mirror side, we have toric stack
$$ X_{ \Delta_1^\vee \diamond \Delta_2^\vee}.  $$
Consider Laurent polynomial with Newton polytope $\Delta_1^\vee \diamond \Delta_2^\vee$, more precisely. Identify the interior of $X_{ \Delta_1^\vee \diamond \Delta_2^\vee}$ with $(\C^*)^n_u \times \C^*_v$,  choose a Laurent polynomial $f(u)$ with NP $\Delta_1^\vee$, then define
$$ W^\vee(u,v) = v + 1/v + f(u) $$
The zero loci of $W^\vee$ is a double cover of $(\C^*)^n_u$. The ramification loci is at $f(u)^2 - 4=0$.