Logarithmic geometry, minimal free resolutions and toric algebraic stacks

TL;DR

This paper introduces a new class of morphisms of log schemes, defines toric algebraic stacks over arbitrary schemes, and explores their properties and relation to toroidal embeddings within the framework of algebraic stacks.

Contribution

It develops a novel framework for toric algebraic stacks over arbitrary schemes using log scheme morphisms, extending classical toric geometry.

Findings

Defined a new notion of toric algebraic stacks

Studied properties of these stacks and their moduli

Explored the stack-theoretic analogue of toroidal embeddings

Abstract

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