Mori dream space
{{Short description|Geometric projective variety}}
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". {{harvtxt|Hu|Keel|2000}} showed that Mori dream spaces are quotients of affine varieties by torus actions. The notion is named so because it behaves nicely from the point of view of Mori's minimal model program.
Properties
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.{{Cite journal |last=Castravet |first=Ana-Maria |date=2018 |title=Mori dream spaces and blow-ups |url=http://dx.doi.org/10.1090/pspum/097.1/01671 |journal=Proceedings of Symposia in Pure Mathematics |volume=97 |issue=1}}
It has been shown that a variety which admits a surjective morphism from a Mori dream space is again a Mori dream space.{{cite journal
| last = Okawa | first = Shinnosuke
| arxiv = 1104.1326
| doi = 10.1007/s00208-015-1245-5
| issue = 3-4
| journal = Mathematische Annalen
| mr = 3466868
| pages = 1315–1342
| title = On images of Mori dream spaces
| volume = 364
| year = 2016}}
See also
References
{{Reflist}}
- {{cite journal | last1=Hu | first1=Yi | last2=Keel | first2=Sean | title=Mori dream spaces and GIT | doi=10.1307/mmj/1030132722 |mr=1786494 | year=2000 | journal=The Michigan Mathematical Journal | issn=0026-2285 | volume=48 | issue=1 | pages=331–348| arxiv=math/0004017 }}
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