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 and toroidal emebeddings in the framework of algebraic stacks and prove some fundamental properties. Furthermore, we study the stack-theoretic analogue of toroidal embeddings.

Smooth toric Deligne-Mumford stacks

“…Remark 7.13. In [Iwa07b], Iwanari defined a smooth toric Artin stack over any scheme associated to a stacky fan Σ rig . Remark 7.14.…”

“…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] .…”