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.
In this note, we give a sufficient condition such that a projective variety with Picard number two is a Mori dream space. Using this condition, we obtain examples of Mori dream spaces with Picard number two.
General criteria are given for when an embedding of a Mori dream space into another satisfies certain nice combinatorial conditions on some of their associated cones. An explicit example of such an embedding is studied.
The purpose of this paper is to study the slices of the Okounkov bodies of Mori dream spaces. First, we analyze all the slices of the Okounkov bodies of big divisors on Mori dream spaces associated to some admissible flags. As a byproduct, we obtain their descriptions on Mori dream threefolds. Finally, we consider its application to the rational polyhedrality of them.
We propose a generalisation of Mori dream spaces to stacks. We show that this notion is preserved under root constructions and taking abelian gerbes. Unlike the case of Mori dream spaces, such a stack is not always given as a quotient of the spectrum of its Cox ring by the Picard group. We give a criterion when this is true in terms of Mori dream spaces and root constructions. Finally, we compare this notion with the one of smooth toric Deligne-Mumford stacks.
We show that the Fulton-MacPherson compactification of the configuration space of $n$ distinct labeled points in certain varieties of arbitrary dimension $d$, including projective space, is not a Mori dream space for $n$ larger than $d+8$.
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 codimension of the unstable locus. This includes an explicit description of the Picard number, the pseudoeffective cone, and the Mori chambers in terms of GIT. We apply the results to quiver moduli, to show that they are Mori dream spaces if the quiver contains no oriented cycles, and if stability and semistability coincide. We give a formula for the Picard number in quiver terms. As a second applic…
We study the question of whether the blow-ups of toric surfaces of Picard number one at the identity point of the torus are Mori Dream Spaces. For some of these toric surfaces, the question whether the blow-up is a Mori Dream Space is equivalent to countably many planar interpolation problems. We state a conjecture which generalizes a theorem of Gonzalez and Karu. We give new examples and non-examples of Mori Dream Spaces among these blow-ups.
Mori dream spaces form a large example class of algebraic varieties, comprising the well known toric varieties. We provide a first software package for the explicit treatment of Mori dream spaces and demonstrate its use by presenting basic sample computations. The software package is accompanied by a Cox ring database which delivers defining data for Cox rings and Mori dream spaces in a suitable format. As an application of the package, we determine the common Cox ring for the symplectic resolutions of a certain quotient singularity investigated by Bellamy/Schedler and Donten-Bury/Wiśniewski.
We present an algorithm to compute the automorphism group of a Mori dream space. As an example calculation, we determine the automorphism groups of singular cubic surfaces with general parameters. The strategy is to study graded automorphisms of affine algebras graded by a finitely generated abelian groups and apply the results to the Cox ring. Besides the application to Mori dream spaces, our results could be used for symmetry based computing, e.g. for Gröbner bases or tropical varieties.
In this paper, we suggest a new approach to study minimal surfaces of general type with $p_g=0$ via their Cox rings, especially using the notion of combinatorially minimal Mori dream space introduced by Hausen. First, we study general properties of combinatorially minimal Mori dream surfaces. Then we discuss how to apply these ideas to the study of minimal surfaces of general type with $p_g=0$ which are very important but still mysterious objects. In our previous paper, we provided several examples of Mori dream surfaces of general type with $p_g=0$ and computed their effective cones explicitl…
Building on the work of Goto, Nishida and Watanabe on symbolic Rees algebras of monomial primes, we prove that the moduli space of stable rational curves with n punctures is not a Mori Dream Space for n>133. This answers the question of Hu and Keel.
In this short note, we show that a construction by Ottem provides an example of a rationally connected variety that is not birationally equivalent to a Mori dream space. This answers in the negative (at least in the category of terminal varieties) a question posed by Krylov.
We prove that the moduli space of $n$-pointed stable maps $\overline{M}_{0,n}(\mathbb{P}^1,1)$ is a Mori dream space whenever the moduli space $\overline{M}_{0,n+3}$ of $(n+3)$-pointed rational curves is. We also show that $\overline{M}_{0,n}(\mathbb{P}^1,1)$ is a log Fano variety for $n \le 5$.
For every $n\geq 3$, we find a sufficient condition for the blow-up of a weighted projective space $\mathbb{P}(a,b,c,d_1,\cdots,d_{n-2})$ at the identity point not to be a Mori Dream Space. We exhibit several infinite sequences of weights satisfying this condition in all dimensions $n\geq 3$.
Given a smooth Mori dream space $X$ we construct a model dominating all the small $\mathbb{Q}$-factorial modifications via tropicalization. This construction allows us to recover a Minkowski basis for the Newton-Okounkov bodies of divisors on $X$ and hence the movable cone of $X$. The existence of such basis allows us to prove the polyhedrality of the global Newton-Okounkov body and the existence of toric degenerations.
Let X be a hypersurface of a Mori dream space Z. We provide necessary and sufficient conditions for the Cox ring R(X) of X to be isomorphic to R(Z)/(f), where R(Z) is the Cox ring of Z and f is a defining section for X. We apply our results to Calabi-Yau hypersurfaces of toric Fano fourfolds. Our second application is to general degree d hypersurfaces in P^n containing a linear subspace of dimension n-2.
In this paper we study effective, nef and semiample cones of minimal surfaces of general type with $p_g=0.$ We provide examples of minimal surfaces of general type with $p_g=0, 2 \leq K^2 \leq 9$ which are Mori dream spaces. On these examples we also give explicit description of effective cones and all irreducible reduced curves of negative self-intersection. We also present non-minimal surfaces of general type with $p_g=0$ that are not Mori dream surfaces.
Let $H_{a,b}^n$ denote the component of the Hilbert scheme whose general point parameterizes an $a$-plane union a $b$-plane meeting transversely in $\mathbf{P}^n$. We describe the effective and nef cones of $H_{a,b}^n$ and determine when the component is Fano. Moreover, we show that $H_{a,b}^n$ is a Mori dream space for all values of $a,b,n$.
Given a map $ϕ: X \to Y$ of $\mathbb Q$-factorial Mori dream spaces, one can ask whether this map is induced by a homogeneous homomorphism $R(Y) \to R(X)$ of Cox rings. As soon as $Y$ is singular, such a homomorphism needs not to exist, as pulling back Weil divisors is not well-defined. In this article, we prove that there is a unique Cox lift $Φ: \mathcal X \to \mathcal Y$ of Mori dream stacks coming from a homogeneous homomorphism $R(Y) = R(\mathcal Y) \to R(\mathcal X)$, where $\mathcal Y$ is a canonical stack to $Y$ and $\mathcal X$ is obtained from $X$ by root constructions. Moreover, $ϕ$…
In this paper we study the geometry of the $14$ families of K3 surfaces of Picard number four with finite automorphism group, whose Néron-Severi lattices have been classified by È.B. Vinberg. We provide projective models, we identify the degrees of a generating set of the Cox ring and in some cases we prove the unirationality of the associated moduli space.
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 ring, here called \emph{weak} Mori dream spaces (wMDS), are considered. Conditions guaranteeing the existence of a neat embedding of a (completion of a) wMDS into a complete toric variety are studied, showing that, on the one hand, those which are complete and admitting low Picard number are always projective, hence Mori dream spaces in the sense of Hu-Keel. On the other hand, an example of a wMDS doe…
We highlight a relation between the existence of Sarkisov links and the finite generation of (certain) Cox rings. We introduce explicit methods to use this relation in order to prove birational rigidity statements. To illustrate, we complete the birational rigidity results of Okada for Fano complete intersection 3-folds in singular weighted projective spaces.
Any rational map between affine spaces, projective spaces or toric varieties can be described in terms of their affine, homogeneous, or Cox coordinates. We show an analogous statement in the setting of Mori Dream Spaces. More precisely (in the case of regular maps) we show there exists a finite extension of the Cox ring of the source, such that the regular map lifts to a morphism from the Cox ring of the target to the finite extension. Moreover the extension only involves roots of homogeneous elements. Such a description of the map can be applied in practical computations.
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 Mori dream space. Secondly we introduce a natural fan structure on the effective cone of Mori dream spaces. We show that it encodes the information of Zariski decompositions, which in turn is equivalent to the information of the variation of GIT quotients of their Cox rings. Finally we show that under a surjective morphism between Mori dream spaces, the fan of the target space coincides with the res…
We will give a criterion to assure that an extremal contraction of a K3 surface which is not a Mori Dream Space produces a singular surface which is a Mori Dream Spaces. We list the possible Néron--Severi groups of K3 surfaces with this property and an extra geometric condition such that the Picard number is greater then or equal to 10. We give a detailed description of two geometric examples for which the Picard number of the K3 surface is 3, i.e. the minimal possible in order to have the required property. Moreover we observe that there are infinitely many examples of K3 surfaces with the re…
In this paper we classify weak Fano varieties that can be obtained by blowing-up general points in prime Fano varieties. We also classify spherical blow-ups of Grassmannians in general points, and we compute their effective cone. These blow-ups are, in particular, Mori dream spaces. Furthermore, we compute the stable base locus decomposition of the blow-up of a Grassmannian in one point, and we show how it is determined by linear systems of hyperplanes containing the osculating spaces of the Grassmannian at the blown-up point, and by the rational normal curves in the Grassmannian passing throu…
We investigate nef and movable cones of hypersurfaces in Mori dream spaces. The first result is: Let $Z$ be a smooth Mori dream space of dimension at least four whose extremal contractions are of fiber type of relative dimension at least two and let $X$ be a smooth ample divisor in $Z$, then $X$ is a Mori dream space as well. The second result is: Let $Z$ be a Fano manifold of dimension at least four whose extremal contractions are of fiber type and let $X$ be a smooth anti-canonical hypersurface in $Z$, which is a smooth Calabi--Yau variety, then the unique minimal model of $X$ up to isomorph…
Let X be a smooth Mori dream space of dimension at least 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 dream space. Moreover, the restriction map identifies the Neron-Severi spaces of X and Y, and under this identification every Mori chamber of Y is a union of some Mori chambers of X, and the nef cone of Y is the same as the nef cone of X. This Lefschetz-type theorem enables one to construct many examples of Mori dream spaces by taking "Mori dream hypersurfaces" of an ambient Mori dream space…
We study a class of rational surfaces (considered in [Campillo, Piltant and Reguera, 2005]) associated to curves with one place at infinity and explicitly describe generators of the Cox ring and global sections of line bundles on these surfaces. In particular, we show that their Cox rings are finitely generated, i.e. they are Mori dream spaces. We also compute their "global Zariski semigroups at infinity" (consisting of line bundles which have no base points `at infinity') and "global Enriques semigroups" (generated by closures of curves in C^2). In particular, we show that the global Zariski …
The goal of the present article is to survey the general theory of Mori Dream Spaces, with special regards to the question: When is the blow-up of toric variety at a general point a Mori Dream Space? We translate the question for toric surfaces of Picard number one into an interpolation problem involving points in the projective plane. An instance of such an interpolation problem is the Gonzalez-Karu theorem that gives new examples of weighted projective planes whose blow-up at a general point is not a Mori Dream Space.
We prove that the Cox ring of the projectivization P(E) of a rank two toric vector bundle E, over a toric variety X, is a finitely generated k-algebra. As a consequence, P(E) is a Mori dream space if the toric variety X is projective and simplicial.
We prove that a Mori dream space over a field of characteristic zero is of Calabi-Yau type if and only if its Cox ring has at worst log canonical singularities. By slightly modifying the arguments we also reprove the characterization of the varieties of Fano type by the log terminality of the Cox rings.
We study blowups of weighted projective planes at a general point, and more generally blowups of toric surfaces of Picard number one. Based on the positive characteristic methods of Kurano and Nishida, we give a general method for constructing examples of Mori Dream Spaces and non-Mori Dream Spaces among such blowups. Compared to previous constructions, this method uses the geometric properties of the varieties and applies to a number of cases. We use it to fully classify the examples coming from two families of negative curves.
The effective cone of a Mori dream space admits two wall-and-chamber decompositions called Mori chamber and stable base locus decompositions. In general the former is a non trivial refinement of the latter. We investigate, from both the geometrical and the combinatorial viewpoints, the differences between these decompositions. Furthermore, we provide a criterion to establish whether the two decompositions coincide for a Mori dream space of Picard rank two, and we construct an explicit example of a Mori dream space of Picard rank two for which the decompositions are different, showing that our …
We present algorithms for basic computations with monoids in finitely generated abelian groups such as monoid membership testing and computing an element of the conductor ideal. Applying them to Mori dream spaces, we obtain algorithms to test whether a Weil divisor class of a given Mori dream space is base point free, to compute generators of the monoid of base point free Cartier divisor classes and to test whether a Mori dream space with known canonical class fulfills Fujita's base point free conjecture or not.
This paper is devoted to a study of the relative version of a Mori dream space (MDS for short), which was first introduced by Andreatta and Wiśnewski and will be called Mori dream morphism (MDM) in this paper. An MDM is defined to be an algebraic fiber space between normal quasi-projective varieties $ X \to U$ such that $\operatorname{Pic}(X/U)_{\mathbb{Q}}=\operatorname{N} ^1(X/U)_{\mathbb{Q}}$ and the (relative) movable cone is decomposed into the semi-ample cones of finitely many small $\mathbb{Q}$-factorial modifications, each of which is assumed to be rational polyhedral. An MDS is an MDM…
We construct Mori Dream Spaces as fine moduli spaces of representations of bound quivers, thereby extending results of Craw--Smith \cite{CrawSmith} beyond the toric case. Any collection of effective line bundles $\mathscr{L}=(\mathscr{O}_X, L_1,..., L_r)$ on a Mori Dream Space $X$ defines a bound quiver of sections and a map from $X$ to a toric quiver variety $|\mathscr{L}|$ called the multigraded linear series. We provide necessary and sufficient conditions for this map to be a closed immersion and, under additional assumptions on $\mathscr{L}$, the image realises $X$ as the fine moduli space…
We prove that the redundant blow-up preserves the finite generation of the Cox ring of a rational surface under a suitable assumption, and we study the birational structure of Mori dream rational surfaces via redundant blow-ups. It turns out that the redundant blow-up completely characterizes birational morphisms of Mori dream rational surfaces with anticanonical Iitaka dimension $0$. As an application, we construct new Mori dream rational surfaces with anticanonical Iitaka dimension $0$ and $-\infty$ of arbitrarily large Picard number.
We consider torus actions on Mori dream spaces and ask whether the associated Chow quotient is again a Mori dream space and, if so, what does its Cox ring look like. We provide general tools for the study of these problems and give solutions for k*-actions on smooth quadrics.
Given a morphism $F : X \rightarrow Y$ from a Mori Dream Space $X$ to a smooth Mori Dream Space $Y$ and quasicoherent sheaves $\mathcal{F}$ on $X$ and $\mathcal{G}$ on $Y$ , we describe the inverse image of $\mathcal{G}$ by $F$ and the direct image of $\mathcal{F}$ by $F$ in terms of the corresponding modules over the Cox rings graded in the class groups.
We study the birational properties of hypersurfaces in products of projective spaces. In the case of hypersurfaces in P^m x P^n, we describe their nef, movable and effective cones and determine when they are Mori dream spaces. Using these results, we give new simple examples of non-Mori dream spaces and analogues of Mumford's example of a strictly nef line bundle which is not ample.
