Mori dream spaces and GIT.

By Y. Hu, S. Keel. The Michigan mathematical journal, 2000

547 citations recorded by the source corpus.

The main goal of this paper is to study varieties with the best possible Mori theoretic properties (measured by the existence of a certain decomposition of the cone of effective divisors). We call such a variety a Mori Dream Space. There turn out to be many examples, including quasi-smooth projective toric (or more generally, spherical) varieties, many GIT quotients, and log Fano 3-folds. We characterize Mori dream spaces as GIT quotients of affine varieties by a torus in a manner generalizing Cox's construction of toric varieties as quotients of affine space. Via the quotient description, the chamber decomposition of the cone of divisors in Mori theory is naturally identified with the decomposition of the G-ample cone from geometric invariant theory. In particular every rational contraction of a Mori dream space comes from GIT, and all possible factorizations of a rational contraction can be read off from the chamber decomposition.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/846f33e9cb8c6a72e0ae4657cbb3868ffddb7c62

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Citations in the corpus

Existence of minimal models for varieties of log general type (2006, unknown): “Finally, it was shown in [16] that if one has a Mori dream space, then the Cox Ring is finitely generated.”
Birational Geometry of 3-fold Mori Fibre Spaces (2003, methodology): “This property is one of the characteristics of Mori dream spaces, as introduced by Keel–Hu [ HK00 ].”
Threefold thresholds (2003, result): “As we assuming the MMP in dimension n, it follows, by the results of either [12] or [28], that there is a decomposition of the relative effective cone of divisors into finitely many chambers, such that the end result of running a sequence of flops only depends on the chamber to which the effective d”
Derived equivalence for quantum symplectic resolutions (2011, background): “There is a nice class of algebraic varieties called Mori dream spaces [36], which includes flag varieties, smooth projective toric varieties and (more generally) spherical varieties.”
Some non-finitely generated Cox rings (2014, background): “[7] of whether the Cox ring of M0,n is finitely generated now only remains unsettled for 7 ≤ n ≤ 12.”
Alternate Compactifications of Moduli Spaces of Curves (2010, background): “s decomposition of the pseudoeective cone E(X) is called the Mori chamber decomposition. If a Mori chamber decomposition exists for the entire pseudoeective cone E(X), we call Xa Mori dream space [HK00]. Such a chamber decomposition may fail to exist, however, for two reasons. First, Xmay have innit”
Moduli of surfaces with an anti-canonical cycle (2012, background): “ 99K Y′}, the collection of nef cones (the closure of the ample cones) in Pic(Y) ⊗Z R of all Y′ isomorphic to Y oﬀ of a codimension two subset. This collection of cones is a fan; for more details see [HK]. In general the Mori fan can be very diﬃcult to control, but for a universal family we can do s”
On blowing up the weighted projective plane (2016, unknown): “Finally, from [11] we know that X is a Mori dream surface if and only if the semiample cone equals the nef cone and is polyhedral in ClQ(X).”
Derived categories of toric varieties III (2006, background): “uivalence Db(X) ∼= Db(X′) for the smooth Deligne-Mumford stack X′ associated to X′ ([4]). By replacing X by X′ if necessary, we may assume that φis surjective in codimension 1. By the paragraph after [1] Corollary 2.4, X is a Mori dream space. By Deﬁnition 1.10 of loc. cit., all birational maps φ fr”
Towards the ample cone of \overline{}_{,} (2000, background): “d that M0,n is the inverse limit of all GIT quotients Q(varying the polarization) for SL2 acting diagonally on (P1)n. In particular each Qis a contraction of M0,n, and so inherits a stratiﬁcation. In [HuKeel00] F1(Q) and F1(Q) are proved. One can also consider the contractions M0,n → Ln−2, introduce”
ON THE GEOMETRY OF SPHERICAL VARIETIES (2012, background): “ceptual reason for the MMP to run perfectly can be stated as follows. Theorem 4.1.1. A projective spherical Q-factorial variety is a Mori dream space. Mori dream spaces were introduced by Hu and Keel [71] who proved that for these varieties, the MMP works perfectly in all characteristics (see in par”
The Sarkisov program (2009, background): “3) implies that (1-2) always hold for W and (1-4) hold for V and so (3) and (4) clearly hold for W ∈ U .”
Cox Rings and Combinatorics II (2008, background): “arious cones of divisors arising in this context, like the eﬀective cone, the closure of the ample cone, and the Mori cone are all polyhedral. In the present paper, we extend the picture developed in [17] to certain normal, not necessarily (quasi)projective varieties X. The main aim then is to provi”