A Survey on Toroidal Embeddings
Date Issued
2014
Date
2014
Author(s)
Huang, Shing-Yeong
Abstract
In this thesis, we will assume basic facts about toric varieties and commutative algebra, and give a survey of [3], chapter II, with detailed proofs of all the theorems. First of all, the idea of equivariant torus embeddings will be generalized to that of so-called toroidal embeddings, which means intuitively "locally similar to some torus embeddings". More precisely, a toroidal embedding is a smooth variety $X$ containing a smooth open subset $U$, such that for every closed point $x in X$, there exists an $T$-equivariant embedding $X_{sigma}$ of some torus $T$, a closed point $t in X_{sigma}$, and an $k$-local algebra isomorphism:[
widehat{mathcal{O}}_{X,x} simeq widehat{mathcal{O}}_{X_{sigma},t}] and the ideal in $widehat{mathcal{O}}_{X,x} $ generated by the ideal of $Xsetminus U$ corresponds to the ideal in $widehat{mathcal{O}}_{X_{sigma},t}$ generated by the ideal of $X_{sigma}setminus T$. Next, we can stratify a toroidal embedding into different components which generalize the idea of orbits. And then we can analyze a toroidal embedding as toric cases and obtain many similar results. The main goal of this generalization is to apply those developed theorems to reduce the proof of semi-stable reduction theorem to a specific combinatorial construction.
Section 1 gives the definition of toroidal embeddings and the stratification of a toroidal embedding, and then consider the two crucial parts: $M^Y$ and $S^U({
m star}! Y)$ for a stratum $Y$ (Lemma 1.1.7 and Definition 1.1.11), which generalize the idea of $T$-invariant Cartier divisors and 1-parameter subgroup of a $T$-equivariant embedding, and we can also define a cone $sigma^Y$ in some euclidean space relative to the stratum $Y$. At the end of this section, we show that a toroidal embedding can be associated to a "polyhedral complex", which is a collection of cones patched together similar to a fan.
Section 2 introduces "canonical morphism" to a fixed toroidal embedding, and shows that this is equivalent to give a sub-polyhedral complex (Theorem 1.2.2). With this theorem, we then generalize theorems of toric varieties by using polyhedral complices instead of fans, including the existence of morphisms, non-singularity of such varieties and blowing-ups (Theorem 1.2.8, Theorem 1.2.9 and Theorem 1.2.16), and eventually show that there exists a non-singular blowing-up.
Section 3 provides concrete methods that we can convert the semi-stable reduction theorem to the construction of some toroidal embeddings, and then use the theorem in cite{Tor}, chapter III to show the semi-stable reduction theorem.
Subjects
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Type
thesis
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