Peng Zhou

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====== Double Cover CY ======

Start with dual nef partitions on $\Delta$ and $\nabla$. 


==== Attempt 1 ====
Now, pick generic section $f_i \in O(2 \Delta_i)$, and $g_i \in O(2 \nabla_i)$. 

Put bundle $\oplus_i O(-\Delta_i)$ on $X_\Delta$, do LG model
$$ W_\Delta = \sum_i ev_{O(-\Delta_i)} (f_i) $$

Put bundle $\oplus_i O(-\nabla_i)$ on $X_\nabla$, do LG model
$$ W_\Delta = \sum_i ev_{O(-\nabla_i)} (g_i) $$

Now, there is no hope this gonna work, simply because one sides has too much selection of monomials, and the other side has the same selection of divisors. 

===== Stacky Fano =====
How to get more divisors? We can do 'stacky Fano'. 

For example, consider A-side, we use superpotential 
$$ W = z^2 + 1/z^2 $$
Then B-side, we consider compactification of $\C^*$, where at $0$ and $\infty$, we put in $\mu_2$ orbifold points. 

Before we even talk about BB, we can talk about Batyrev construction, just upgrade CY hypersurface to something else. 

===== Batyrev Mirror and double cover =====
Suppose $\Delta$ is a smooth Fano polytope, and $\nabla = \Delta^\vee$. 

One side, we have $Y_\Delta = Tot(O(-\Delta)) \to X_{\Delta}$, with $W_\Delta = ev(s)$, where $s \in \Gamma(O(\Delta))$ is a section in anti-canonical bundle. 
  * divisors: $1$ plus vertices of $\nabla$. 
  * monomials: lattice points in $\Delta$. 

On the other side
  * divisors: $1$ plus vertices of $\Delta$. Same as lattice points in $\Delta$
  * monomials: lattice points in $\nabla$. Same as $1$ plus vertices in $\nabla$. 

Suppose we do geometric double cover, what we do? 
  * take a section $f \in \Gamma(O(2\Delta))$. 
  * write down an equation in the total space $Y$ of $O(\Delta)$, fiber coord $y$,  where we write $ y^2 = f $
  * If we want to do an LG realization, we look at the defining 'section', $y^2 - f$, of some line bundle $L$ over $Y$,  and we view it as a function $W$ on $Tot_Y(L^{-1})$. 

Say, in the case we have $X = \P^1$, and $f \in O(4H)$, $Y$ is $Tot(O(2H))$, $y^2 - f$ is a section of $\pi^*(O(4H)).