Mori dream space

In algebraic geometry, a Mori dream space is a projective variety whose cone of effective divisors has a well-behaved decomposition into certain convex sets called "Mori chambers".^[1] Hu and Keel showed that Mori dream spaces are quotients of affine varieties by torus actions.^[1] The notion is named so because it behaves nicely from the point of view of Mori's minimal model program.

Examples and Properties

Any quasi-smooth projective spherical variety (in particular, any quasi-smooth projective toric variety) as well as any log Fano 3-fold is a Mori dream space.^[1] In general, it is difficult to find a non-trivial example of a Mori dream space, as being a Mori Dream Space is equivalent to all (multi-)section rings being finitely generated.^[2]

It has been shown that a variety which admits a surjective morphism from a Mori dream space is again a Mori dream space.^[3]

See also

• Spherical variety