Logarithmic Geometry, Minimal Free Resolutions and Toric Algebraic Stacks

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 embeddings in the framework of algebraic stacks and prove some fundamental properties. Also. We study the stack-theoretic analogue of toroidal embeddings.