Towards twisted stable maps in higher dimensions

Location: 11th Floor Lecture Hall Speaker Dori Bejleri, University of Maryland Session Chair Klaus Hulek, Leibniz-University Hannover Abstract The space of Kontsevich stable maps to a Deligne-Mumford stack is not proper. To compactify it, Abramovich and Vistoli introduced the moduli space of twisted stable maps where the source curves themselves may also acquire stacky structure. In this talk, I will outline a higher dimensional theory of twisted stable maps to Deligne-Mumford stacks. When the target is a scheme, higher dimensional stable maps were defined by Alexeev as a joint generalization of the Kontsevich space and the Koll\'ar--Shepherd-Barron--Alexeev (KSBA) moduli space of stable pairs. As in the 1-dimensional case, when the target is a stack, the source may also acquire stacky structure. Thus we need to extend tools from the minimal model program and KSBA theory to Deligne-Mumford stacks which naturally leads to considering the stacky approach to the KSBA moduli theory furnished by twisted stable pairs.