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We construct a presentation for the Cox ring of the projectivization P E of any rank n irreducible toric vector bundle on P n. We use this presentation to show that P E always satisfies Fujita's freeness and ampleness conjectures. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the notion of a Klyachko diagram for a monomial ideal I in a certain multi-graded polynomial ring, namely the Cox ring R of a smooth complete toric variety, with irrelevant maximal ideal B. We present procedures to compute the Klyachko diagram of I from its monomial generators, and to retrieve the B −saturation Isat of I from its Klyachko diagram. We use this description to compute the first local cohomology module H B 1 (I) . As an application, we find a formula for the Hilbert function of Isat, and a characterization of monomial ideals with constant Hilbert polynomial, in terms of their Klyachko diagram. [ABSTRACT FROM AUTHOR]

Consider the blow-up Y of a weighted projective plane at a point in the open orbit over a field of characteristic 0. We assume that there exists a curve C on Y such that C 2 < 0 and C. E = 1 , where E is the exceptional curve. In this paper we give a (very simple) necessary and sufficient condition for finite generation of the Cox ring of Y (Theorem 1.2). It is an affirmative answer to a conjecture due to He and Kurano-Nishida. [ABSTRACT FROM AUTHOR]

Finite generation of the symbolic Rees ring of a space monomial prime ideal of a 3-dimensional weighted polynomial ring is a very interesting problem. Negative curves play important roles in finite generation of these rings. We are interested in the structure of the negative curve. We shall prove that negative curves are rational in many cases. We also see that the Cox ring of the blow-up of a toric variety at the point (1 , 1 , ... , 1) coincides with the extended symbolic Rees ring of an ideal of a polynomial ring. For example, Roberts' second counterexample to Cowsik's question (and Hilbert's 14th problem) coincides with the Cox ring of some normal projective variety (Remark 2.7). [ABSTRACT FROM AUTHOR]

A criterion for the total coordinate space of a trinomial hypersurface to be a hypersurface is found. An algorithm for calculating the Cox ring in explicit form is proposed, and criteria for the total coordinate space to be rational and factorial are obtained. [ABSTRACT FROM AUTHOR]

By an additive action on an algebraic variety X we mean a regular effective action 𝔾 a n × X → X with an open orbit of the commutative unipotent group  𝔾 a n . In this paper, we give a classification of additive actions on complete toric surfaces. [ABSTRACT FROM AUTHOR]

We classify commutative algebraic monoid structures on normal affine surfaces over an algebraically closed field of characteristic zero. The answer is given in two languages: comultiplications and Cox coordinates. The result follows from a more general classification of commutative monoid structures of rank 0, n − 1 or 𝑛 on a normal affine variety of dimension 𝑛. [ABSTRACT FROM AUTHOR]

We study commutative associative polynomial operations 𝔸 n × 𝔸 n → 𝔸 n with unit on the affine space 𝔸 n over an algebraically closed field of characteristic zero. A classification of such operations is obtained up to dimension 3. Several series of operations are constructed in arbitrary dimension. Also we explore a connection between commutative algebraic monoids on affine spaces and additive actions on toric varieties. [ABSTRACT FROM AUTHOR]

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 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 that does not admit any neat embedded sharp completion (i.e. Picard number preserving) into a complete toric variety is given, on the contrary of what Hu and Keel exhibited for a MDS. Moreover, termination of the Mori minimal model program for every divisor and a classification of rational contractions for a complete wMDS are studied, obtaining analogous conclusions as for a MDS. Finally, we give a characterization of wMDS arising from a small Q -factorial modification of a projective weak Q -Fano variety. [ABSTRACT FROM AUTHOR]

We investigate the class of degenerations of smooth cubic surfaces which are obtained from degenerating their Cox rings to toric algebras. More precisely, we work in the spirit of Sturmfels and Xu who use the theory of Khovanskii bases to determine toric degenerations of Del Pezzo surfaces of degree 4 and who leave the question of classifying these degenerations in the degree 3 case as an open problem. In order to carry out this classification we describe an approach which is closely related to tropical geometry and present partial results in this direction. [ABSTRACT FROM AUTHOR]

We prove that the atom spectrum, which is a topological space associated to an arbitrary abelian category introduced by Kanda, of the category of finitely presented graded modules over a noetherian graded ring R is given as a union of the homogeneous spectrum of R with some additional points, which we call non-standard points. This description of the atom spectrum helps in understanding the sheafification process in toric geometry: if S is the Cox ring of a normal toric variety X without torus factors, then a finitely presented graded S -module sheafifies to zero if and only if its atom support consists only of points in the atom spectrum of S which either lie in the vanishing locus of the irrelevant ideal of X or are non-standard. [ABSTRACT FROM AUTHOR]

Abstract 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 and 2 ≤ K 2 ≤ 9 which are Mori dream spaces. On these examples we also give explicit description of their effective cones with all negative curves. We also present non-minimal surfaces of general type with p g = 0 that are not Mori dream surfaces. [ABSTRACT FROM AUTHOR]

Given a morphism F:X→Y from a Mori Dream Space X to a smooth Mori Dream Space Y and quasicoherent sheaves F on X and G on Y, we describe the inverse image of G by F and the direct image of F by F in terms of the corresponding modules over the Cox rings graded in the class groups. [ABSTRACT FROM AUTHOR]

In this paper, we consider the invariant subring of the Cox ring by the automorphism group of the projective variety X under some assumption. We prove that the ring is finitely generated if X is a K3 surface. [ABSTRACT FROM AUTHOR]

We provide a new criterion for flexibility of affine cones over varieties covered by flexible affine varieties. We apply this criterion to prove flexibility of affine cones over secant varieties of Segre-Veronese embeddings and over certain Fano threefolds. We further prove flexibility of total coordinate spaces of Cox rings of del Pezzo surfaces. [ABSTRACT FROM AUTHOR]

Abstract: 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 that 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. [ABSTRACT FROM AUTHOR]

The aim of this paper is twofold. First, we determine which blowups of products of projective spaces at general points are varieties of Fano type, and give boundary divisors making these spaces log Fano pairs. Second, we describe generators of the Cox rings of some cases. [ABSTRACT FROM AUTHOR]

To study syzygies of the Cox rings of del Pezzo surfaces, we calculate important syzygetic invariants such as the Hilbert functions, the Green–Lazarsfeld indices, the projective dimensions, and the Castelnuovo–Mumford regularities. Using these computations as well as the natural multigrading structures by the Picard groups of del Pezzo surfaces and Weyl group actions on Picard lattices, we determine the Betti diagrams of the Cox rings of del Pezzo surfaces of degree at most four. [ABSTRACT FROM AUTHOR]

By an additive action on an algebraic variety X of dimension n we mean a regular action Gan × X → X with an open orbit of the commutative unipotent group Gan. We prove that if a complete toric variety X admits an additive action, then it admits an additive action normalized by the acting torus. Normalized additive actions on a toric variety X are in bijection with complete collections of Demazure roots of the fan SX. Moreover, any two normalized additive actions on X are isomorphic. [ABSTRACT FROM AUTHOR]

We consider rational varieties with a torus action of complexity one and extend the combinatorial approach via the Cox ring developed for the complete case in earlier work to the non-complete, e.g. affine, case. This includes in particular a description of all factorially graded affine algebras of complexity one with only constant homogeneous invertible elements in terms of canonical generators and relations. [ABSTRACT FROM AUTHOR]

Let G ⊆ GL ( n ) be a finite group without pseudo-reflections. We present an algorithm to compute and verify a candidate for the Cox ring of a resolution X → C n / G , which is based just on the geometry of the singularity C n / G , without further knowledge of its resolutions. We explain the use of our implementation of the algorithms in Singular . As an application, we determine the Cox rings of resolutions X → C 3 / G for all G ⊆ GL ( 3 ) with the aforementioned property and of order | G | ≤ 12 . We also provide examples in dimension 4. [ABSTRACT FROM AUTHOR]

In this note we introduce Cox rings of singularities and explicitly compute them in the case of du Val singularities Dn , E6 , E7 and E8 .

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 -8 of arbitrarily large Picard number. [ABSTRACT FROM AUTHOR]

We determine the tropicalizations of very affine surfaces over a valued field that are obtained from del Pezzo surfaces of degree 5, 4 and 3 by removing their ( − 1 ) -curves. On these tropical surfaces, the boundary divisors are represented by trees at infinity. These trees are glued together according to the Petersen, Clebsch and Schläfli graphs, respectively. There are 27 trees on each tropical cubic surface, attached to a bounded complex with up to 73 polygons. The maximal cones in the 4-dimensional moduli fan reveal two generic types of such surfaces. [ABSTRACT FROM AUTHOR]

We consider surfaces X defined by plane divisorial valuations ν of the quotient field of the local ring R at a closed point p of the projective plane P 2 over an arbitrary algebraically closed field k and centered at R . We prove that the regularity of the cone of curves of X is equivalent to the fact that ν is non-positive on O P 2 ( P 2 ∖ L ) , where L is a certain line containing p . Under these conditions, we characterize when the characteristic cone of X is closed and its Cox ring finitely generated. Equivalent conditions to the fact that ν is negative on O P 2 ( P 2 ∖ L ) ∖ k are also given. [ABSTRACT FROM AUTHOR]

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, specifically how a non-toric del Pezzo surface embeds into a toric variety in this manner. [ABSTRACT FROM AUTHOR]

Abstract: Let G be a connected reductive group and a spherical homogeneous space. We show that the ideal of relations between a natural set of generators of the Cox ring of a G-embedding of can be obtained by homogenizing certain equations which depend only on the homogeneous space. Using this result, we describe some examples of spherical homogeneous spaces such that the Cox ring of any of their G-embeddings is defined by one equation. [Copyright &y& Elsevier]

Let X be an affine toric variety. The total coordinates on X provide a canonical presentation $$\bar X \to X$$ of X as a quotient of a vector space $$\bar X$$ by a linear action of a quasitorus. We prove that the orbits of the connected component of the automorphism group Aut( X) on X coincide with the Luna strata defined by the canonical quotient presentation. [ABSTRACT FROM AUTHOR]

Abstract: Given an action of an affine algebraic group with only trivial characters on a factorial variety, we ask for categorical quotients. We characterize existence in the category of algebraic varieties. Moreover, allowing constructible sets as quotients, we obtain a more general existence result, which, for example, settles the case of a finitely generated algebra of invariants. As an application, we provide a combinatorial GIT-type construction of categorical quotients for actions of not necessarily reductive groups on, e.g. complete varieties with finitely generated Cox ring via lifting to the characteristic space. [Copyright &y& Elsevier]

We consider the space of complete rank two collineations. Starting from its description as a limit of GIT-quotients of a Grassmanian G(2, n) by a certain \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathbb {C}^*$\end{document}-action, we determine the Cox ring by means of toric ambient modifications. [ABSTRACT FROM AUTHOR]

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 dream space. Moreover, the restriction map identifies the Néron-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, provided that it satisfies the GIT condition. To facilitate this, we then show that the GIT condition is stable under taking products and taking the projective bundle of the direct sum of at least three line bundles, and in the case when X is toric, we show that the condition is equivalent to the fan of X being 2-neighborly. [ABSTRACT FROM AUTHOR]

Abstract: 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 -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 -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. [Copyright &y& Elsevier]

We explore the circle of ideas connecting finite generation of the Cox ring, Mori dream spaces and invariant theory. [ABSTRACT FROM AUTHOR]

The generalized Cox construction associates with an algebraic variety a remarkable invariant—its total coordinate ring, or Cox ring. In this note, we give a new proof of the factoriality of the Cox ring when the divisor class group of the variety is finitely generated and free. The proof is based on the notion of graded factoriality. We show that if the divisor class group has torsion, then the Cox ring is again factorially graded, but factoriality may be lost. [ABSTRACT FROM AUTHOR]

Abstract: Let be the homogeneous coordinate ring of the blow-up of in r general points, i.e., a smooth Del Pezzo surface of degree . We prove that for , can be embedded into , where is an algebraic group with root system given by the primitive Picard lattice of and is a certain maximal parabolic subgroup. [Copyright &y& Elsevier]

We present a new eigenvalue method for solving a system of Laurent polynomial equations defining a zero-dimensional reduced subscheme of a toric compactification X of (C ∖ { 0 }) n. We homogenize the input equations to obtain a homogeneous ideal I in the Cox ring of X and generalize the eigenvalue, eigenvector theorem for root finding in affine space to compute homogeneous coordinates of the solutions. Several numerical experiments show the effectiveness of the resulting method. In particular, the method outperforms existing solvers in the case of (nearly) degenerate systems with solutions on or near the torus invariant prime divisors. [ABSTRACT FROM AUTHOR]