Jaroslav Buczynski: Maps of Mori Dream Spaces

Abstract: Any rational map between affine spaces or projective spaces can be described in terms of their (homogeneous) coordinates. Toric Varieties and Mori Dream Spaces are classes of algebraic varieties for which there exist a sensible analogue of homogeneous coordinate ring. I will present how to obtain a description of a map of Mori Dream Spaces (or Toric Varieties) in terms of such coordinate rings. More precisely (in the case of regular maps) I will show there exists a finite extension of the coordinate ring of the source, such that the regular map lifts to a morphism from the Cox ring of the target to the finite extension. Moreover the extension only involves roots of homogeneous elements. Such a description of the map can be applied in practical computations.