Log algebraic stacks and moduli of log schemes.

摘要

In an effort to incorporate the log smooth deformation theory developed by K. Kato and F. Kato into the study of moduli, we develop a theory of log algebraic stacks and an analog of M. Artin's method for determining representability by an algebraic stack. We also study the moduli of log structures themselves and prove that there exist universal log structures in the sense of algebraic stacks. Using this result we translate results from the theory of stacks into the language of log geometry and thereby lay the foundations for the study of moduli of log schemes.; In the second half of the thesis, we apply our foundational work to the study of moduli of stable curves. We construct the Deligne-Mumford compactification of the moduli space of curves of genus g ≥ 2 using log geometry as well as the Deligne-Rapoport compactifications of moduli stacks of elliptic curves with level structure. We use K. Kato's work on log finite flat group schemes instead of generalized elliptic curves to define level structure on degenerate elliptic curves. In the last chapter we use K. Kato and S. Usui's work on log Hodge structures to interpret the compactified moduli stack of elliptic curves from the analytic point of view.