# Mori dream spaces and GIT.

@article{Hu2000MoriDS, title={Mori dream spaces and GIT.}, author={Y. Hu and S. Keel}, journal={Michigan Mathematical Journal}, year={2000}, volume={48}, pages={331-348} }

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…

## 408 Citations

Quotients of spectra of almost factorial domains and Mori dream spaces

- Mathematics
- 2015

We prove that a GIT chamber quotient of an affine variety $X=Spec(A)$ by a reductive group $G$, where $A$ is an almost factorial domain, is a Mori dream space if it is projective, regardless of the…

Embedding non-projective Mori Dream Spaces

- Mathematics
- 2018

This paper is devoted to extend some Hu-Keel results on Mori dream spaces (MDS) beyond the projective setup. Namely, $\Q$-factorial algebraic varieties with finitely generated class group and Cox…

On images of Mori dream spaces

- Mathematics
- 2011

The purpose of this paper is to study the geometry of images of morphisms from Mori dream spaces. First we prove that a variety which admits a surjective morphism from a Mori dream space is again a…

A Lefschetz hyperplane theorem for Mori dream spaces

- Mathematics
- 2008

Let X be a smooth Mori dream space of dimension ≥ 4. We show that, if X satisfies a suitable GIT condition which we call small unstable locus, then every smooth ample divisor Y of X is also a Mori…

Embedding non-projective Mori dream space

- MathematicsGeometriae Dedicata
- 2019

This paper is devoted to extend some Hu–Keel results on Mori dream spaces (MDS) beyond the projective setup. Namely, $$\mathbb {Q}$$ Q -factorial algebraic varieties with finitely generated class…

A G ] 3 A ug 2 01 5 QUOTIENTS OF SPECTRA OF ALMOST FACTORIAL DOMAINS AND MORI DREAM

- Mathematics
- 2021

We prove that a GIT chamber quotient of an affine variety X = Spec(A) by a reductive group G, where A is an almost factorial domain, is a Mori dream space if it is projective, regardless of the…

Mori dream spaces and Fano varieties

- Mathematics
- 2012

These lectures have mainly two aims. The first is to explain the notion of Mori dream space, introduced by Hu and Keel, and to show the main geometrical properties, after [HK00]. In particular we…

Geometry of singular Fano varieties and projective bundles over curves

- Mathematics
- 2017

This thesis is devoted to the geometry of Fano varieties and projective vector bundles over a smooth projective curve.
In the first part we study the geometry of mildly singular Fano varieties on…

Geometry of singular Fano varieties and projective vector bundles over curves

- Mathematics
- 2017

This thesis is devoted to the geometry of Fano varieties and projective vector bundles over a smooth projective curve.In the first part we study the geometry of mildly singular Fano varieties on…