Mori dream space

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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.

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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. 1 2 3 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.
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