Movable Cone Research Articles

On Symplectic Birational Self-Maps of Projective Hyperkähler Manifolds of K3[n]-Type

Abstract We prove that projective hyperkähler manifolds of K3$^{[n]}$-type admitting a non-trivial symplectic birational self-map of finite order are isomorphic to moduli spaces of stable (twisted) coherent sheaves on K3 surfaces. Motivated by this result, we analyze the reflections on the movable cone of moduli spaces of sheaves and determine when they come from a birational involution.

All 81 crepant resolutions of a finite quotient singularity are hyperpolygon spaces

We demonstrate that the linear quotient singularity for the exceptional subgroup G G in S p ( 4 , C ) \mathrm {Sp}(4,\mathbb {C}) of order 32 is isomorphic to an affine quiver variety for a 5-pointed star-shaped quiver. This allows us to construct uniformly all 81 projective crepant resolutions of C 4 / G \mathbb {C}^4/G as hyperpolygon spaces by variation of GIT quotient, and we describe both the movable cone and the Namikawa Weyl group action via an explicit hyperplane arrangement. More generally, for the n n -pointed star shaped quiver, we describe completely the birational geometry for the corresponding hyperpolygon spaces in dimension 2 n − 6 2n-6 ; for example, we show that there are 1684 projective crepant resolutions when n = 6 n=6 . We also prove that the resulting affine cones are not quotient singularities for n ≥ 6 n \geq 6 .

The movable cone of Calabi–Yau threefolds in ruled Fano manifolds

We describe explicitly the chamber structure of the movable cone for a general complete intersection Calabi–Yau threefold in a non-split (n+4)-dimensional Pn-ruled Fano manifold of index n+1 and Picard number two. Moreover, all birational minimal models of such Calabi–Yau threefolds are found whose number is finite.

Computational aspects of Calogero–Moser spaces

We present a series of algorithms for computing geometric and representation-theoretic invariants of Calogero–Moser spaces and rational Cherednik algebras associated with complex reflection groups. In particular, we are concerned with Calogero–Moser families (which correspond to the mathbb {C}^times -fixed points of the Calogero–Moser space) and cellular characters (a proposed generalization by Rouquier and the first author of Lusztig’s constructible characters based on a Galois covering of the Calogero–Moser space). To compute the former, we devised an algorithm for determining generators of the center of the rational Cherednik algebra (this algorithm has several further applications), and to compute the latter we developed an algorithmic approach to the construction of cellular characters via Gaudin operators. We have implemented all our algorithms in the Cherednik Algebra Magma Package by the second author and used this to confirm open conjectures in several new cases. As an interesting application in birational geometry we are able to determine for many exceptional complex reflection groups the chamber decomposition of the movable cone of a mathbb {Q}-factorial terminalization (and thus the number of non-isomorphic relative minimal models) of the associated symplectic singularity. Making possible these computations was also a source of inspiration for the first author to propose several conjectures about the geometry of Calogero–Moser spaces (cohomology, fixed points, symplectic leaves), often in relation with the representation theory of finite reductive groups.

Variation of stability for moduli spaces of unordered points in the plane

We study compactifications of the moduli space of unordered points in the plane via variation of GIT-quotients of their corresponding Hilbert scheme. Our VGIT considers linearizations outside the ample cone and within the movable cone. For that purpose, we use the description of the Hilbert scheme as a Mori dream space, and the moduli interpretation of its birational models via Bridgeland stability. We determine the GIT walls associated with curvilinear zero-dimensional schemes, collinear points, and schemes supported on a smooth conic. For seven points, we study a compactification associated with an extremal ray of the movable cone, where stability behaves very differently from the Chow quotient. Lastly, a complete description for five points is given.

Geometric realizations of birational transformations via {{mathbb {C}}}^*-actions

In this paper, we study varieties admitting torus actions as geometric realizations of birational transformations. We present an explicit construction of these geometric realizations for a particular class of birational transformations, and study some of their geometric properties, such as their Mori, Nef and Movable cones.

On Numerical Dimensions of Calabi–Yau Varieties

Abstract Let $X$ be a Calabi–Yau variety of Picard number two with infinite birational automorphism group. We show that the numerical dimension $\kappa ^{\mathbb{R}}_{\sigma }$ of the extremal rays of the closed movable cone of $X$ is $\dim X/2$. More generally, we investigate the relation between the two numerical dimensions $\kappa ^{\mathbb{R}}_{\sigma }$ and $\kappa ^{\mathbb{R}}_{\textrm{vol}}$ for Calabi–Yau varieties. We also compute $\kappa ^{\mathbb{R}}_{\sigma }$ for non-big divisors in the closed movable cone of a projective hyperkähler manifold.

Remarks on nef and movable cones of hypersurfaces in Mori dream spaces

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 isomorphism is X itself, and moreover, the movable cone conjecture holds for X, namely, there exists a rational polyhedral cone which is a fundamental domain for the action of birational automorphisms on the effective movable cone of X. The third result is: Let P:=Pn×⋯×Pn be the N-fold self-product of the n-dimensional projective space. Let X be a general complete intersection of n+1 hypersurfaces of multidegree (1,…,1) in P with dim⁡X≥3. Then X has only finitely many minimal models up to isomorphism, and moreover, the movable cone conjecture holds for X.

Birational automorphism groups and the movable cone theorem for Calabi–Yau complete intersections of products of projective spaces

For a Calabi-Yau manifold X, the Kawamata – Morrison movable cone conjecture connects the convex geometry of the movable cone Mov‾(X) to the birational automorphism group. Using the theory of Coxeter groups, Cantat and Oguiso proved that the conjecture is true for general varieties of Wehler type, and they described explicitly Bir(X). We generalize their argument to prove the conjecture and describe Bir(X) for general complete intersections of ample divisors in arbitrary products of projective spaces. Then, under a certain condition, we give a description of the boundary of Mov‾(X) and an application connected to the numerical dimension of divisors.

On the Numerical Dimension of Calabi–Yau 3-Folds of Picard Number 2

Abstract We show that for any smooth Calabi–Yau threefold $X$ of Picard number $2$ with infinite birational automorphism group, the numerical dimension $\kappa _\sigma $ of the extremal rays of the movable cone of $X$ is $\frac {3}{2}$. Furthermore, we provide new examples of Calabi–Yau threefolds of Picard number $2$ with infinite birational automorphism group.

The movable cone of certain Calabi–Yau threefolds of Picard number two

We describe explicitly the chamber structure of the movable cone for a general smooth complete intersection Calabi-Yau threefold $X$ of Picard number two in certain Pr-ruled Fano manifold and hence verify the Morrison-Kawamata cone conjecture for such $X$. Moreover, all birational minimal models of such Calabi-Yau threefolds are found, whose number is finite up to isomorphism.

ПОСТРОЕНИЕ И СРАВНЕНИЕ РЕГРЕССИОНЫХ МОДЕЛЕЙ ОБОГАЩЕНИЯ ЗОЛОТОСОДЕРЖАЩЕГО СЫРЬЯ В ЦЕНТРОБЕЖНОЙ ОТСАДОЧНОЙ МАШИНЕ

Decreasing ore grades is a common challenge for every gold mining country in the world. Deposits of low-grade and complex ores are increasingly brought into production. These ores are typically treated by hydrometallurgical methods, for example, heap leaching. Meanwhile, the use of cyanide and its alternatives poses a risk for the environment. Thus, the development of gravity methods including jigging in a centrifugal-field is becoming an attractive option of low-grade and complex ore processing. Two mathematical models have been developed using the regression method to predict grade and recovery characteristics of the products of a centrifugal jigging machine (CJM) with variable process parameters including jig chamber rotational speed, pulsation frequency and vibration amplitude of a movable cone, underscreen-water flow rate. The models are based on the results of the field trials of a pilot centrifugal jigging machine designed by JSC Irgiredmet at existing gold-processing plants. Mathematical modelling provided a means of the process parameters ranking according to their impact on the separation process. A model has been identified that ensures a more accurate prediction of the separation process variables (metal recovery in concentrate) when the CJM's adjustable parameters are changed. The object of the study is the centrifugal jigging method of gold-bearing ore separation. The scope of the study is mathematical modelling of gold-bearing ore separation using CJM. The objective of the study is the determination and ranking of the CJM's process parameters according to their impact on the separation process, identification of a model that ensures a more accurate prediction of the separation process variables. The methodology of the study involves finding the correlation between gold grade/gold recovery into the separation products and the CJM's adjustable parameters. The method of the study is regression analysis

Birational geometry of the Mukai system of rank two and genus two

Using the techniques of Bayer–Macrì, we determine the walls in the movable cone of the Mukai system of rank two for a general K3 surface S of genus two. We study the (essentially unique) birational map to S^{[5]} and decompose it into a sequence of flops. We give an interpretation of the exceptional loci in terms of Brill–Noether loci.

Nef Cone of a Generalized Kummer 4-fold

In this note, we calculate the boundary of movable cones and nef cones of the generalized Kummer 4-fold $\mathrm{Km}^2(A)$ attached to an abelian surface $A$ with $\mathrm{rk} \mathrm{NS}(A) = 1$.

On the birational geometry of spaces of complete forms I: collineations and quadrics

Moduli spaces of complete collineations are wonderful compactifications of spaces of linear maps of maximal rank between two fixed vector spaces. We investigate the birational geometry of moduli spaces of complete collineations and quadrics from the point of view of Mori theory. We compute their effective, nef and movable cones, the generators of their Cox rings, and their groups of pseudo-automorphisms. Furthermore, we give a complete description of both the Mori chamber and stable base locus decompositions of the effective cone of the space of complete collineations of the 3-dimensional projective space.

Birational geometry of symplectic quotient singularities

For a finite subgroup Gamma subset mathrm {SL}(2,mathbb {C}) and for nge 1, we use variation of GIT quotient for Nakajima quiver varieties to study the birational geometry of the Hilbert scheme of n points on the minimal resolution S of the Kleinian singularity mathbb {C}^2/Gamma . It is well known that X:={{,mathrm{{mathrm {Hilb}}},}}^{[n]}(S) is a projective, crepant resolution of the symplectic singularity mathbb {C}^{2n}/Gamma _n, where Gamma _n=Gamma wr mathfrak {S}_n is the wreath product. We prove that every projective, crepant resolution of mathbb {C}^{2n}/Gamma _n can be realised as the fine moduli space of theta -stable Pi -modules for a fixed dimension vector, where Pi is the framed preprojective algebra of Gamma and theta is a choice of generic stability condition. Our approach uses the linearisation map from GIT to relate wall crossing in the space of theta -stability conditions to birational transformations of X over mathbb {C}^{2n}/Gamma _n. As a corollary, we describe completely the ample and movable cones of X over mathbb {C}^{2n}/Gamma _n, and show that the Mori chamber decomposition of the movable cone is determined by an extended Catalan hyperplane arrangement of the ADE root system associated to Gamma by the McKay correspondence. In the appendix, we show that morphisms of quiver varieties induced by variation of GIT quotient are semismall, generalising a result of Nakajima in the case where the quiver variety is smooth.

On the birational geometry of spaces of complete forms II: Skew-forms

Moduli spaces of complete skew-forms are compactifications of spaces of skew-symmetric linear maps of maximal rank on a fixed vector space, where the added boundary divisor is simple normal crossing. In this paper we compute their effective, nef and movable cones, the generators of their Cox rings, and for those spaces having Picard rank two we give an explicit presentation of the Cox ring. Furthermore, we give a complete description of both the Mori chamber and stable base locus decompositions of the effective cone of some spaces of complete skew-forms having Picard rank at most four.

Positivity functions for curves on algebraic varieties

This is the second part of our work on Zariski decomposition structures, where we compare two different volume type functions for curve classes. The first function is the polar transform of the volume for ample divisor classes. The second function captures the asymptotic geometry of curves analogously to the volume function for divisors. We prove that the two functions coincide, generalizing Zariski's classical result for surfaces to all varieties. Our result confirms the log concavity conjecture of the first named author for weighted mobility of curve classes in an unexpected way, via Legendre-Fenchel type transforms. We also give a number of applications to birational geometry, including a refined structure theorem for the movable cone of curves.

O’Grady’s birational maps and strange duality via wall-hitting

We prove that O’Grady’s birational maps [K. G O’Grady, The weight-two Hodge structure of moduli spaces of sheaves on a K3 surface, J. Algebr. Geom. 6(4) (1997) 599–644] between moduli of sheaves on an elliptic K3 surface can be interpreted as intermediate wall-crossing (wall-hitting) transformations at so-called totally semistable walls, studied by Bayer and Macrì [A. Bayer and E. Macrì, MMP for moduli of sheaves on K3s via wall-crossing: nef and movable cones, Lagrangian fibrations, Inventiones Mathematicae 198(3) (2014) 505–590]. As a key ingredient, we describe the first totally semistable wall for ideal sheaves of [Formula: see text] points on the elliptic [Formula: see text]. As an application, we give new examples of strange duality isomorphisms, based on a result of Marian and Oprea [A. Marian and D. Oprea, Generic strange duality for K3 surfaces, with an appendix by Kota Yoshioka, Duke Math. J. 162(8) (2013) 1463–1501].

On nef subvarieties

In this paper, we first study generalizations of Fujita vanishing theorems for q-ample divisors. We then apply them to study positivity of subvarieties with nef normal bundle in the sense of intersection theory.After Ottem's work on ample subschemes, we introduce the notion of a nef subscheme, which generalizes the notion of a subvariety with nef normal bundle. We show that restriction of a pseudoeffective (resp. big) divisor to a nef subvariety is pseudoeffective (resp. big). We also show that ampleness and nefness are transitive properties.We define the weakly movable cone as the cone generated by the pushforward of cycle classes of nef subvarieties via proper surjective maps. This cone contains the movable cone and shares similar intersection-theoretic properties with it, thanks to the aforementioned properties of nef subvarieties.On the other hand, we prove that if Y⊂X is an ample subscheme of codimension r and D|Y is q-ample, then D is (q+r)-ample. This is analogous to a result proved by Demailly-Peternell-Schneider and Küronya.