GIT Constructions of Log Canonical Models of M_g
  The purpose of this article is to give an overview of the construction of moduli spaces of curves from the viewpoint of the log minimal model program for M_g by providing an update of recent developments and discussing future problems. This survey is distinguished from the recent articles of Fedorchuk-Smyth and Morrison in its focus on low degree Hilbert stability of curves. 
Jarod Alper and Donghoon Hyeon
math.AG