The Theorems

We would like to understand the proof of the following ``semistable reduction theorem'' of Kempf, Knudsen, Mumford and Saint-Donat [KKMSD73]:

Theorem 1   Let $k$ be an algebraically closed field of characteristic 0 (e.g. $k={\mathbb{C}}$). Let $f:X\to C$ be a surjective morphism from a $k$-variety $X$ to a non-singular curve $C$ and assume there exists a closed point $z\in C$ such that $f_{\vert X\setminus f^{-1}(z)}: X\setminus f^{-1}(z)\to C\setminus\{z\}$ is smooth. Then we find a commutative diagram

$$\xymatrix{ X\ar[d]_f & X\times_C C'\ar[l]\ar[d] & X'\ar[l]_-{p}\ar[dl]^{f'}\\ C & C'\ar[l]^{\pi} }$$

with the following properties

1. $\pi:C'\to C$ is a finite map, $C'$ is a non-singular curve and $\pi^{-1}(z)=\{z'\}$.
2. $p$ is projective and is an isomorphism over $C'\setminus \{z'\}$.
3. $X'$ is non-singular and ${f'}^{-1}(z')$ is a reduced divisor with simple normal crossings, i.e., we can write ${f'}^{-1}(z')=\sum_i E_i$ where the $E_i$ are 1-codimensional subvarieties (i.e., locally they are defined by the vanishing of a single equation), which are smooth and, for all $r$, all the intersections $E_{i_1}\cap\ldots\cap E_{i_r}$ are smooth and have codimension $r$.

The essential point here is, that ${f'}^{-1}(z')=\sum_i E_i$, i.e., there are no multiplicities in front of the $E_i$. If we drop this condition the above can be achieved even with $\pi={\rm id}$, using Hironaka's resolution of singularities (which we will use as a black-box in the seminar). There are also generalizations of Theorem 1, where the base does not need to be a curve and the field $k$ is not assumed to be algebraically closed (but of course ${\rm char}(k)=0$). See [ADK]. See [IT, Thm 3.10] for the most general version there is. Unfortunately we don't have the time to discuss these in the seminar.

The interest for us - besides the usefulness of the statement itself - is that the proof is a beautiful interplay of algebraic and discrete geometry. Let us give a toy example to see where this connection is coming from: by Hironaka we can assume that $X$ is smooth and that $f^{-1}(z)=\sum_i m_i E_i$ with $m_i\in {\mathbb{N}}$ and $\sum_i E_i$ a simple normal crossings divisor. In this situation the map $f:X\to C$ looks very locally around $z$ (meaning formal analytically or étale locally) like this

$${\mathbb{A}}^n_k={\rm Spec \,}k[x_1,\ldots, x_n] \to {\mathbb{A}}^1_k = {\rm Spec \,}k[t],$$   given by $$t=\prod_i x_i^{m_i}.$$

Now take $d$ with $lcm(m_i)\vert d$ and consider $\pi_d: {\mathbb{A}}^1_k\to {\mathbb{A}}^1_k$ given by $t\mapsto t^{\frac{1}{d}}$. (This should become the finite map $\pi$.) If $gcd(m_i)=1$, then the normalization of ${\mathbb{A}}^n_k\times_{{\mathbb{A}}^1_k, \pi_d} {\mathbb{A}}^1_k$ is the toric variety given by

$$Y={\rm Spec \,}[M\cap({\mathbb{R}}_+)^n],$$   with  $$M=\mathbb{Z}^n+(\tfrac{m_1}{d},\ldots,\tfrac{m_n}{d})\cdot\mathbb{Z}$$

and one can check that the inverse image of $\sum_i m_i E_i$ in $Y$ is reduced. But $Y$ does not need to be non-singular. It remains to find a birational map $X'\to Y$, which is an isomorphism over the non-singular locus of $Y$ such that $X'$ is smooth and the inverse image of $E_i$ in $X'$ is still reduced. The main point in the proof of the above theorem is that such birational maps are described by certain fans inside the convex rational polyhedral cone dual to $M\cap({\mathbb{R}}_+)^n$. In this way the above theorem is (with the help of Hironaka) reduced to the following theorem due to Knudsen and Mumford:

Theorem 2   Let $P \subset {\mathbb{R}}^n$ be a polytope with integral vertices. Then there exists an integer $c\ge 1$ and a triangulation of the dilation $cP$ into unimodular simplices.

Here, a simplex is unimodular if its vertices form an affine basis of the lattice $\mathbb{Z}^n$. In order to guarantee that $p$ is projective in Theorem 1$(ii)$ we require, in addition, that the simplices in our triangulation be the domains of linearity of a convex function on $cP$.

One problem is that the general situation looks only very locally around a given point as above. This forces us to work with toroidal embeddings instead of toric varieties. A toroidal embedding is an open immersion $U\hookrightarrow X$ where $X$ is a normal variety and $U$ is non-singular and this embedding looks very locally around each point of $X$ (i.e., formal analytically or étale locally) like a torus embedding. There are much more varieties $X$ which have the structure of a toroidal embedding than toric varieties, e.g. take a simple normal crossings divisor $E$ on any non-singular variety $X$, then $U:=X\setminus E\hookrightarrow X$ defines a toroidal embedding. ¹Therefore there is no hope to classify all these in terms of discrete geometry. But given a fixed toroidal embedding $U\hookrightarrow X$ one can classify all (allowable) toroidal embeddings $U\hookrightarrow Z$ which are birational over $U\hookrightarrow X$ in terms of decompositions of certain polyhedral complexes with integral structure. This is exactly what we need for Theorem 1.

In the seminar we will first discuss the basics of toric varieties, i.e., their connection with semigroups and fans of rational convex polyhedral cones with the final goal to find an interpretation of canonical resolutions of singularities of a toric variety in terms of certain subdivisions of the fans. Then we will introduce the notion of toroidal embeddings and see how birational maps to a fixed such embedding can be described in terms of subdivisions of fans. Then we will see how Theorem 1 is reduced to Theorem 2. Finally we will discuss the proof of Theorem 2 in purely polyhedral terms.

In the seminar we will mainly follow [CLS11] for the toric crash course, [KKMSD73] for the reduction of Theorem 1 to Theorem 2 and [HPPS14] for Theorem 2. Notice that some of the terms used have different names in the different sources, for example:

1. equivariant (affine) embeddings of a torus in [KKMSD73] = (affine) toric variety.
2. convex (rational) polyhedral cones (which do not contain any linear subspace) in [KKMSD73] = (strongly) convex (rational) polyhedral cones.
3. finite rational partial polyhedral decomposition= f.r.p.p. decomposition in [KKMSD73] = fan.

Here is a detailed list with the content of each talk. If not said different, all numbers and paragraphs refer to [KKMSD73]. Since the terminology in [KKMSD73] is a little bit out dated and the proofs and explanations are also a bit short, from time to time you might want to use - at least for the toric part- also some alternative literature such as: [CLS11], [Ful93], see also the overview article [Cox].