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

References

  1. ^ a b c Hu, Yi; Keel, Sean (2000). "Mori dream spaces and GIT". The Michigan Mathematical Journal. 48 (1): 331–348. arXiv:math/0004017. doi:10.1307/mmj/1030132722. ISSN 0026-2285. MR 1786494.
  2. ^ Castravet, Ana-Maria (2018). "Mori dream spaces and blow-ups". Proceedings of Symposia in Pure Mathematics. 97 (1).
  3. ^ Okawa, Shinnosuke (2016). "On images of Mori dream spaces". Mathematische Annalen. 364 (3–4): 1315–1342. arXiv:1104.1326. doi:10.1007/s00208-015-1245-5. MR 3466868.
This article is issued from Wikipedia. The text is available under Creative Commons Attribution-Share Alike 4.0 unless otherwise noted. Additional terms may apply for the media files.