Renato Vidal Martins,
Ana-Maria Castravet (Ohio State University),
Mori Dream Spaces and moduli spaces of stable rational curves |
Resumo. Mori Dream Spaces form an ideal class of algebraic varieties in which, as the name suggests, the Minimal Model Program works very well. Introduced by Hu and Keel, Mori Dream Spaces can be algebraically characterized as varieties whose total coordinate ring (or Cox ring) is finitely generated.
Cox rings appear naturally in the context of Hilbert's 14th Problem. One can use the geometry of algebraic varieties to answer questions about finite generation of invariant rings (work of Nagata and Mukai, among others). In their original paper, motivated by questions about the birational geometry of moduli spaces of stable curves, Hu and Keel asked whether the Grothendieck-Knudsen moduli space of stable, $n$-pointed, rational curves is a Mori Dream Space. Along with the Fulton-Faber conjecture, this question was one of the main open problems in the area, generating a flurry of activity in the last couple of years.
In this talk I will explain the connections between Cox rings and invariant rings, introduce the Grothendieck-Knudsen moduli space and the main questions about its geometry, and finally, present joint work with Jenia Tevelev, answering negatively Hu and Keel's question for $n$ large.