The mumford degeneration is an example of a toric degeneration. For references, see Section 1.2 from [An Invitation to Toric Degenerations] or Example 3.6 on page 97 of [Tropical Geometry and Mirror Symmetry]. For the relevant toric varieties constructions see Chapter 7 of [Toric Varieties, Cox Little Schneck].
Start with a \(n+1\)-dimensional convex polyhedron \(\tilde \sigma \subset M_{\mathbb R}\times \mathbb R\) that is closed under positive translation in the last coordinate: \[\tilde\sigma = \tilde\sigma + \{0\}\times \mathbb R.\] Take \(q:M_\mathbb R\times \mathbb R\to M_\mathbb R\) to be the projection and let \[\sigma := q(\tilde \sigma).\] The boundary \(\partial \tilde\sigma\) is a union of \(n\)-dimensional polyhedra intersecting along \((n-1)\)-dimensional facets. There are two different kinds of these boundary components, horizontal components which are mapped homeomorphically to polyhedra in \(M_\mathbb R\) by the projection \(q\) and vertical components which are “parallel to \(\{0\}\times \mathbb R\)” and lose dimension upon projection. The set \(\sigma\) is homeomorphic to the union of the horizontal components of \(\partial \tilde \sigma \) under \(q\).
Furthermore, the horizontal portion of \(\partial \tilde\sigma\) is the graph of a piecewise affine linear function \(\varphi:\sigma \to \mathbb R\) with rational slopes. The domains of linearity of \(\varphi\) give a polyhedral decomposition \(\mathcal P\) of \(\sigma\), which corresponds to the decomposition of the horizontal portion of \(\partial \tilde \sigma\) into polyhedra. From this data we can recover \(\tilde \sigma \), it is the upper convex hull of the graph of \(\varphi\).
\begin{equation*} \left\{\substack{\text{convex polyhedra } \tilde\sigma \text{ in } M_\mathbb R\times \mathbb R \\ \text{ closed under positive translation} \\\text{in the last component}}\right\} \leftrightarrow \left\{\substack{(\sigma, \mathcal P, \varphi) \text{ where } \sigma \subseteq M_\mathbb R, \\ \mathcal P \text{ is a polyhedral decomposition of }\sigma \\ \varphi:\sigma \to \mathbb R\text{ is convex piecewise affine}}\right\} \end{equation*}Anyways, given \(\tilde \sigma \) we can apply the polyhedra toric variety construction to get a \((n+1)\)-dimensional toric variety \(X_{\tilde \sigma}\). It is glued together from affine patches given by \[U_v = \Spec \mathbb C[C_v\cap (M\oplus \mathbb Z)],\] where \(C_v\) is the cone associated to vertex \(v\) in \(\tilde \sigma \) obtained by translating \(\tilde \sigma \) so \(v\) is at the origin and then taking the cone over the entire polyhedron: \(C_v = \cone(\tilde \sigma - v).\) Since \(\tilde \sigma \) is closed under positive translations in the last coordinate, each \(C_v\) contains the ray \(\{0\}\times \mathbb R_{\geq 0}\), giving us a projection to the affine line \[p:X_{\tilde \sigma} \to \mathbb A^1.\] We can say more about this projection. On the algebra level, it is given by the morphism \[\mathbb C[t] = \mathbb C[(\{0\}\times \mathbb R_{\geq 0}) \cap (M\oplus \mathbb Z)] \hookrightarrow \mathbb C[C_v\cap (M\oplus \mathbb Z)]\] sending \(t\) to \(z^{(0,1)}\). The monomial \(z^{(0,1)}\) vanishes precisely on the prime boundary divisors of \(X_{\tilde \sigma}\) corresponding to the horizontal components of \(\partial \tilde \sigma\), or using the other presentation, on the prime boundary divisors isomorphic to to the toric varieties \(X_\tau\) for maximal \(\tau \in \mathcal P\). (To see this claim about the vanishing of \(z^{(0,1)}\) we move over to the fan description of \(X_{\tilde \sigma}\) whose cones are the dual cones \(C_v^\vee \subset N_\mathbb R\times \mathbb R\), and look at the pairing of the character \((0,1) \in M\times \mathbb Z\) with the normal vectors of the top dimensional facets of \(C^\vee_v\).)
This means that the fiber of \(p\) over \(0\) resembles a union of \(n\)-dimensional toric varieties intersecting along prime boundary divisors.
To understand the fibers of \(p\) at \(t\neq 0\), we localize at \(t\). This has the effect of removing the lower boundary of \(\tilde\sigma,\) that is, going over to \(\tilde\sigma + (\{0\}\times \mathbb R) = \sigma\times \mathbb R\). In the cone picture the polyhedron \(\mathbb R\) corresponds to a \(\mathbb G_m\) component, so \[p^{-1}(\mathbb A^1\setminus \{0\}) = X_{\sigma \times \mathbb R} = X_\sigma \times (\mathbb A^1\setminus \{0\}).\] Here, \(X_\sigma\) is the \(n\)-dimensional toric variety corresponding to the polyhedra \(\sigma\), that \(\sigma\) with its polyhedral decomposition forgotten. To summarize this construction:
Summary
- The map \(p:X_{\tilde \sigma} \to \mathbb A^1 \) is called a Mumford degeneration.
- It’s general fiber is a toric variety given by applying the polytope construction to \(\sigma\subset M_\mathbb R\), so \(p^{-1}(\mathbb A^1\setminus \{0\}) = X_\sigma \times (A^1\setminus \{0\})\)
- It’s special fiber looks like the “union of toric varieties encoded by the polyhedral decomposition \(\mathcal P\) of \(\sigma\)”. The piecewise affine function \(\varphi:\sigma \to \mathbb R\) controls the way the boundaries of the irreducible components of \(p^{-1}(0)\) are glued together.