Given an affine manifold with singularities and a polyhedral decomposition \((B, \mathcal P)\), the first step in constructing the central fiber \(X_0\) of a toric degeneration whose dual intersection is \((B, \mathcal P)\) is to write down a collection of sets which will be glued to form an etale cover of \(X_0\). This is that process.
First note that for a cone \(\sigma \in \mathcal P\), the vector space \(\Lambda_{\sigma, \mathbb R}\) is the stalk of \(\Lambda_{\mathcal P, \mathbb R}\) at any point \(y\in \Int(\sigma)\). Definition 1.31 in “Mirror Symmetry via Log Degeneration Data I” notes that the spaces \((\Lambda_{\mathcal P, \mathbb R})_y\) are all canonically identified by parallel transport for different choices of \(y\in \Int(\sigma)\). The sheaf \(\Lambda_{\mathcal P, \Lambda}\) is a subsheaf of the typical \(\Lambda_{\mathbb R}\) sheaf on \(B_0 = B\setminus \Delta\), defined
\[\Gamma(U, \Lambda_{\mathcal P, \mathbb R}) = \left\{v\in \Gamma(U, \Lambda_{\mathbb R}) ~ \mid ~ \forall y\in U, \sigma \in \mathcal P \text{ with } y\in \sigma, v\text{ is tangent to } \sigma \text{ at }y\right\}.\]
In practice, \(\Lambda_{\mathcal P, \mathbb R}|_{\Int(\sigma)}\) is simply the tangent sheaf of \(\Int(\sigma)\) thought of as an affine manifold in its own right.
We will construct affine toric varieties \(U_\sigma\) and \(V_\sigma\) for every maximal cell \(\sigma \in \mathcal P\).
- Form the cone over \(\sigma\):
\[C(\sigma) = \{(rp, r) ~ \mid ~ r\in \mathbb R_{\geq 0}, p \in \widetilde \sigma\}\subseteq \Lambda_{\sigma, \mathbb R} \oplus \mathbb R\] That is, choose your favorite vertex \(v\in \sigma\), embed \(\sigma\) in \(\mathbb R^{\dim \sigma}\oplus \mathbb R\) at height \(1\) with \(v \mapsto (0,1)\) and take the cone over the resulting embedding.
- Take the dual cone:
\[C(\sigma)^\vee = \left\{m \in \Lambda_{\sigma, \mathbb R}^\vee \oplus \mathbb R ~ \mid ~ \langle m, v\rangle \geq 0, \forall v \in C(\sigma)\right\}\]
- Due to the integral structure of \(B\) we have a lattice inside of \(\Lambda_{\sigma, \mathbb R}^\vee\), which we denote by \(\Lambda_{\sigma}^\vee\). We intersect our cone with it to obtain a monoid in standard toric fashion:
\[P_\sigma = C(\sigma)^\vee \cap (\Lambda_{\sigma}^\vee \oplus \mathbb Z)\]
- We define
\[U_\sigma = \Spec \mathbb Z[P_\sigma].\]
- To define \(V_\sigma\), we have two options.
- We can either form it as a quotient of \(\mathbb Z[P_\sigma]\). To do this, we note that there is an element in \(P_\sigma\) which corresponds to the height projection on \(C(\sigma)\), it’s denoted \(\rho_\sigma:\Lambda_{\sigma}\oplus \mathbb Z\to \mathbb Z\) and is simply projection onto the second factor. In coordinate on \(\Lambda_{\sigma}^\vee\oplus \mathbb Z\), it is the element \((0,...,0,1)\). We then define
\[V_\sigma = \Spec \mathbb Z[P_\sigma]/(z^{\rho_\sigma}).\]
- We can define a new monoid from \(P_\sigma\), denoted \(\partial P_\sigma\), which as a set is the integral points on the boundary of \(C(\sigma)^\vee\) union infinity,
\[\partial P_\sigma = \left(\partial C(\sigma)^\vee \cap (\Lambda_{\sigma}\oplus \mathbb Z)\right)\cup \{\infty\}\] and whose addition is defined by \[ x + y = \begin{cases} x + y & x + y \in \partial C(\sigma)^\vee \cap (\Lambda_{\sigma}\oplus \mathbb Z) \\ \infty & \text{otherwise} \end{cases}. \] Then set \[V_\sigma = \Spec \mathbb Z[\partial P_\sigma].\] In either case we get the same thing.
- Finally, if we wish \(X_0\) to be a scheme over \(S\), then we define our preferred etale cover to be
\[\big\{V_\sigma \times_{\Spec \mathbb Z} S\big\}_{\sigma \in \mathcal P_{\text{max}}}.\]