Logarithmic Geometry, Minimal Free Resolutions and Toric Algebraic Stacks

Abstract

In this paper we will introduce a certain type of morphisms of log schemes (in the sense of Fontaine, Illusie and Kato) and study their moduli. Then by applying this we define the notion of toric algebraic stacks, which may be regarded as torus emebeddings in the framework of algebraic stacks and prove some fundamental properties. Also, we study the stack-theoretic analogue of toroidal embeddings.

Smooth toric Deligne-Mumford stacks

“…In fact, the stacky fan can be read off the geometry of the smooth toric Deligne-Mumford stack just like the fan can be read off the geometry of the toric variety. Notice that one can deduce the above theorem when X is an orbifold from Theorem 2.5 of [Per08] and Theorem 1.4 of [Iwa07a] and the geometric characterization of Theorem 1.3 in [Iwa07b] .…”

The Chevalley–Shephard–Todd theorem for finite linearly reductive group schemes

“…The key to proving Proposition 3.1 for diagonalizable G is provided by Theorem 3.3 and Proposition 3.4 of [Iw] after we reinterpret them in the language of pseudo-reflections. We refer the reader to [Iw, for the basic definitions concerning monoids. We recall the following definition given in [Iw,Def 2.5].…”

“…We refer the reader to [Iw, for the basic definitions concerning monoids. We recall the following definition given in [Iw,Def 2.5].…”

Canonical Artin stacks over log smooth schemes

“…Then there exists a smooth log smooth Artin stack X over S and a good moduli space morphism f : X → X over S. Moreover, the base change of f to the smooth locus of X is an isomorphism. This is a generalization of [Iw2,Thm 3.3] where the result is proved for X all of whose charts are given by simplicial toric varieties. The stack X has a moduli interpretation in terms of log geometry and agrees with the stack Iwanari constructs when X is as in [Iw2,Thm 3.3].…”