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Abstract des Vortrages von Elena Martinengo

Mori dream stacks

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 stacks. 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. We give a criterion under which this is true. Finally, we compare this notion with the one of smooth toric stacks. In a work in progress we use Mori dream stacks to reinterpret a recent construction of Brown-Buczynski to lift a rational map between toric varieties to a multivalued map of Cox rings.