Elena Martinengo

## Title:

Mori Dream stacks

## Abstract:

Toric stacks were introduced in a combinatorial way by Borisov-Chen-Smith. Later Fantechi-Mann-Nironi gave a geometric definition of toric stacks and got a nice classification of them in term of roots over a toric variety. In a work in collaboration with Andreas Hochenegger, we generalize this work introducing the notion of Mori dream stack. We show that such stacks are preserved under root constructions and taking abelian gerbes. Unlike the case of Mori dream spaces, such stacks are not always given as quotients of the spectrum of their Cox rings by the Picard groups. In our work we give a criterion under which this is true.