Birational contractions of \(\overline{\mathrm{M}}_{g,n}\) and their dependence on the characteristic

In this talk, we discuss the existence and nonexistence of certain birational contractions of \(\overline{\mathrm{M}}_{g,n}\). Somewhat surprisingly, this depends on the characteristic of the base field: many such contractions exist only in positive characteristic. We present a precise form of this phenomenon and discuss two examples that highlight the difference between characteristic zero and positive characteristic. The first is a simple and explicit contraction that exists only in positive characteristic, and the second is a modular interpretation of the morphisms associated with psi classes on \(\overline{\mathrm{M}}_{1,n}\). We also offer some speculation on why such characteristic-dependent phenomena arise.