On the minimal model program for the moduli space of curves

In this talk I will report on a recent work with G. Codogni and F. Viviani in which we investigate the first possible steps of the minimal model program for the moduli space of stable pointed curves $M$. In particular, we show that such steps have a modular interpretation and we relate them to the so Hassett–Keel program, which predicts that the log canonical models of $M$(with natural boundaries) have also a modular interpretation.