arXiv subjects
Publications and source records attributed to Juergen Hausen.
Let X->P^(n-1) be an elliptic fibration obtained by resolving the indeterminacy of the projection of a cubic hypersurface Y of P^(n+1) from a line L not contained in Y. We prove that the Mordell-Weil group of the elliptic fibration is finite if and only if the Cox ring of X is finitely generated. We also provide a presentation of the Cox ring of X when it is finitely generated.
We consider a normal complete rational variety with a torus action of complexity one. In the main results, we determine the roots of the automorphism group and give an explicit description of the root system of its semisimple part. The results are applied to the study of almost homogeneous varieties. For example, we describe all almost homogeneous (possibly singular) del Pezzo k*-surfaces of Picard number one and all almost homogeneous (possibly singular) Fano threefolds of Picard number one having a reductive automorphism group with two-dimensional maximal torus.
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.
We consider finitely generated normal algebras over an algebraically closed field of characteristic zero that come with a complexity one grading by a finitely generated abelian group such that the conditions of a UFD are satisfied for homogeneous elements. Our main results describe these algebras in terms of generators and relations. We apply this to write down explicitly the possible Cox rings of normal complete rational varieties with a complexity one torus action.
Notes of an introductory course given at the conference "Torsors: Theory and Applications" in Edinburgh, January 2011.
We determine the Cox ring of the space of complete rank two collineations.
In a first result, we describe all finitely generated factorial algebras over an algebraically closed field of characteristic zero that come with an effective multigrading of complexity one by means of generators and relations. This enables us to construct systematically varieties with free divisor class group and a complexity one torus action via their Cox rings. For the Fano varieties of this type that have a free divisor class group of rank one, we provide explicit bounds for the number of possible deformation types depending on the dimension and the index of the Picard group in the divisor class group. As a consequence, one can produce classification lists for fixed dimension and Picard index. We carry this out expemplarily in the following cases. There are 15 non-toric surfaces with Picard index at most six. Moreover, there are 116 non-toric threefolds with Picard index at most two; nine of them are locally factorial, i.e. of Picard index one, and among these one is smooth, six have canonical singularities and two have non-canonical singularities. Finally, there are 67 non-toric locally factorial fourfolds and two one-dimensional families of non-toric locally factorial fourfolds. In all cases, we list the Cox rings explicitly.
This is the first chapter of an introductory text under construction; further chapters are available via the authors' web pages. Our aim is to provide an elementary access to Cox rings and their applications in algebraic and arithmetic geometry. Any comments and suggestions on this draft will be highly appreciated.
We investigate the Cox ring of a normal complete variety X with algebraic torus action. Our first results relate the Cox ring of X to that of a maximal geometric quotient of X. As a consequence, we obtain a complete description of the Cox ring in terms of generators and relations for varieties with torus action of complexity one. Moreover, we provide a combinatorial approach to the Cox ring using the language of polyhedral divisors. Applied to smooth k*-surfaces, our results give a description of the Cox ring in terms of Orlik-Wagreich graphs. As examples, we explicitly compute the Cox rings of all Gorenstein del Pezzo k*-surfaces with Picard number at most two and the Cox rings of projectivizations of rank two vector bundles as well as cotangent bundles over toric varieties in terms of Klyachko's description.
We study Cox rings of K3-surfaces. A first result is that a K3-surface has a finitely generated Cox ring if and only if its effective cone is polyhedral. Moreover, we investigate degrees of generators and relations for Cox rings of K3-surfaces of Picard number two, and explicitly compute the Cox rings of generic K3-surfaces with a non-symplectic involution that have Picard number 2 to 5 or occur as double covers of del Pezzo surfaces.
We study varieties with a finitely generated Cox ring. In a first part, we generalize a combinatorial approach developed in earlier work for varieties with a torsion free divisor class group to the case of torsion. Then we turn to modifications, e.g., blow ups, and the question how the Cox ring changes under such maps. We answer this question for a certain class of modifications induced from modifications of ambient toric varieties. Moreover, we show that every variety with finitely generated Cox ring can be explicitly constructed in a finite series of toric ambient modifications from a combinatorially minimal one.
We consider actions of reductive groups on a varieties with finitely generated Cox ring, e.g., the classical case of a diagonal action on a product of projective spaces. Given such an action, we construct via combinatorial data in the Cox ring all maximal open subsets such that the quotient is quasiprojective or embeddable into a toric variety. As applications, we obtain an explicit description of the chamber structure of the linearized ample cone and several Gelfand-MacPherson type correspondences relating quotients of reductive groups to quotients of torus actions. Moreover, our approach provides information on the geometry of many of the resulting quotient spaces.
Generalizing the passage from a fan to a toric variety, we provide a combinatorial approach to construct arbitrary effective torus actions on normal, algebraic varieties. Based on the notion of a ``proper polyhedral divisor'' introduced in earlier work, we develop the concept of a ``divisorial fan'' and show that these objects encode the equivariant gluing of affine varieties with torus action. We characterize separateness and completeness of the resulting varieties in terms of divisorial fans, and we study examples like C*-surfaces and projectivizations of (non-split) vector bundles over toric varieties.
Given an action of an algebraic torus on a normal affine variety, we describe all open subsets admitting a complete orbit space.
Given a multigraded algebra $A$, it is a natural question whether or not for two homogeneous components $A_u$ and $A_v$, the product $A_{nu}A_{nv}$ is the whole component $A_{nu+nv}$ for $n$ big enough. We give combinatorial and geometric answers to this question.
Given an algebraic torus action on a normal projective variety with finitely generated total coordinate ring, we study the GIT-equivalence for not necessarily ample linearized divisors, and we provide a combinatorial description of the partially ordered set of GIT-equivalence classes. As an application, we extend in the $\QQ$-factorial case a basic feature of the collection of ample GIT-classes to the partially ordered collection of maximal subsets with a quasiprojective quotient: for any two members there is at most one minimal member comprising both of them. Moreover, we demonstrate in an example, how our theory can be applied for a systematic treatment of ``exotic projective orbit spaces'', i.e., projective geometric quotients that do not arise from any linearized ample divisor.
We provide a complete description of normal affine varieties with effective algebraic torus action in terms of what we call proper polyhedral divisors on semiprojective varieties. Our theory extends classical cone constructions of Dolgachev, Demazure and Pinkham to the multigraded case, and it comprises the theory of affine toric varieties.
We study equivariant embeddings with small boundary of a given homogeneous space $G/H$, where $G$ is a connected, linear algebraic group with trivial Picard group and only trivial characters, and $H \subset G$ is an extension of a connected Grosshans subgroup by a torus. Under certain maximality conditions, like completeness, we obtain finiteness of the number of isomorphism classes of such embeddings, and we provide a combinatorial description the embbeddings and their morphisms. The latter allows a systematic treatment of examples and basic statements on the geometry of the equivariant embeddings of a given homogeneous space $G/H$.