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The real rank two locus of an algebraic variety is the closure of the union of all secant lines spanned by real points. We seek a semi-algebraic description of this set. Its algebraic boundary consists of the tangential variety and the edge variety. Our study of Segre and Veronese varieties yields a characterization of tensors of real rank two. [ABSTRACT FROM AUTHOR]

Hrushovski's generalization and application of Jouanolou (1978) [9] is here refined and extended to the partial differential setting with possibly nonconstant coefficient fields. In particular, it is shown that if X is a differential-algebraic variety over a partial differential field F that is finitely generated over its constant field F 0 , then there exists a dominant differential-rational map from X to the constant points of an algebraic variety V over F 0 , such that all but finitely many codimension one subvarieties of X over F arise as pull-backs of algebraic subvarieties of V over F 0 . As an application, it is shown that the algebraic solutions to a first order algebraic differential equation over C ( t ) are of bounded height, answering a question of Eremenko. Two expected model-theoretic applications to DCF 0 , m are also given: 1) Lascar rank and Morley rank agree in dimension two, and 2) dimension one strongly minimal sets orthogonal to the constants are ℵ 0 -categorical. A detailed exposition of Hrushovski's original (unpublished) theorem is included, influenced by Ghys (2000) [5] . [ABSTRACT FROM AUTHOR]

This note is about the Chow groups of a certain family of smooth cubic fourfolds. This family is characterized by the property that each cubic fourfold X in the family has an involution such that the induced involution on the Fano variety F of lines in X is symplectic and has a K3 surface S in the fixed locus. The main result establishes a relation between X and S on the level of Chow motives. As a consequence, we can prove finite-dimensionality of the motive of certain members of the family. [ABSTRACT FROM AUTHOR]

Differential algebraic geometry seeks to extend the results of its algebraic counterpart to objects defined by differential equations. Many notions, such as that of a projective algebraic variety, have close differential analogues but their behavior can vary in interesting ways. Workers in both differential algebra and model theory have investigated the property of completeness of differential varieties. After reviewing their results, we extend that work by proving several versions of a “differential valuative criterion" and using them to give new examples of complete differential varieties. We conclude by analyzing the first examples of incomplete, finite-rank projective differential varieties, demonstrating a clear difference from projective algebraic varieties. [ABSTRACT FROM PUBLISHER]

A standard observation in algebraic geometry and number theory is that a ramified cover of an algebraic variety X → X over a finite field Fq furnishes the rational points x ∈ X(Fq) with additional arithmetic structure: the Frobenius action on the fiber over x. For Xample, in the case of the Vieta cover of polynomials over Fq this structure describes a polynomial's irreducible decomposition type. Furthermore, the distribution of these Frobenius actions is encoded in the cohomology of X via the Grothendieck-Lefschetz trace formula. This note presents a version of the trace formula that is suited for studying the distribution in the context of representation stability: for certain sequences of varieties (Xn) the cohomology, and therefore the distribution of the Frobenius actions, stabilizes in a precise sense. We conclude by fully working out the example of the Vieta cover of the variety of polynomials. The calculation includes the distribution of cycle decompositions on cosets of Young subgroups of the symmetric group, which might be of independent interest. [ABSTRACT FROM AUTHOR]

The article presents mathematical explanations on Geometric Complexity Theory and Noether's Normalization Lemma (NNL). Topics discussed include role of NNL to construct a finite set of generators for invariant ring, specification differences between the basic and computational algebraic geometry and invariant theory, and use of circuits to specify algebraic varieties in complexity theory.

This is a computational study of bottlenecks on algebraic varieties. The bottlenecks of a smooth variety X ⊆ C n are the lines in C n which are normal to X at two distinct points. The main result is a numerical homotopy that can be used to approximate all isolated bottlenecks. This homotopy has the optimal number of paths under certain genericity assumptions. In the process we prove bounds on the number of bottlenecks in terms of the Euclidean distance degree. Applications include the optimization problem of computing the distance between two real varieties. Also, computing bottlenecks may be seen as part of the problem of computing the reach of a smooth real variety and efficient methods to compute the reach are still to be developed. Relations to triangulation of real varieties and meshing algorithms used in computer graphics are discussed in the paper. The resulting algorithms have been implemented with Bertini [4] and Macaulay2 [17]. [ABSTRACT FROM AUTHOR]

Different types of compactness in the Zariski topology are explored: for instance, equational Noetherianity, equational Artinianity, q-compactness, and u-compactness. Moreover, general results on the Zariski topology over algebras and groups are proved. [ABSTRACT FROM AUTHOR]

Let f : X → X be a dominant rational map of a smooth projective variety defined over a characteristic 0 global field K, let δf be the dynamical degree of f, and let hX: X(K̅) → [1, ∞) be a Weil height relative to an ample divisor. We prove that for every ϵ > 0 there is a height bound hX ∘ fn ⪡ (δf + ϵ)nhX, valid for all points whose f-orbit is well-defined, where the implied constant depends only on X, hX, f, and ϵ. An immediate corollary is a fundamental inequality α̅f (P) ≤ δf for the upper arithmetic degree. If further f is a morphism and D is a divisor satisfying an algebraic equivalence f* D ≡ βD for some β> √δf, we prove that the canonical height hf,D = lim β-nhD ∘ fn converges and satisfies hf,D ∘ f = βhf,D and hf,D = hD + O(√hX). We also prove that the arithmetic degree αf(P), if it exists, gives the main term in the height counting function for the f-orbit of P. We conjecture that αIf(P) = δf whenever the f-orbit of P is Zariski dense and describe some cases for which we can prove our conjecture. [ABSTRACT FROM AUTHOR]

Let F be a non-Archimedean local field of characteristic 0 and G be the general linear group GLn over F. Bushnell and Henniart described the essentially tame local Langlands correspondence for G(F) using admissible characters, with their rectifiers, of tamely ramified elliptic maximal tori of G(F). The main result of this article is to relate these rectifiers to χ-data in the theory of endoscopy of Kottwitz, Langlands, and Shelstad. Therefore, we can describe the essentially tame local Langlands correspondence using admissible embeddings of L-tori. [ABSTRACT FROM AUTHOR]

As was shown earlier, for a linear differential-algebraic system A y′ + A y = 0 with a selected part of unknowns (entries of a column vector y), it is possible to construct a differential system ỹ′ = B ỹ, where the column vector ỹ is formed by some entries of y, and a linear algebraic system by means of which the selected entries that are not contained in ỹ can be expressed in terms of the selected entries included in ỹ. In the paper, sizes of the differential and algebraic systems obtained are studied. Conditions are established under the fulfillment of which the size of the algebraic system is determined unambiguously and the size of the differential system is minimal. [ABSTRACT FROM AUTHOR]

The discriminant set of a real polynomial is studied. It is shown that this set has a complex hierarchical structure and consists of algebraic varieties of various dimensions. A constructive algorithm for a polynomial parameterization of the discriminant set in the space of the coefficients of the polynomial is proposed. Each variety of a greter dimension can be geometrically considered as a tangent developable surface formed by one-dimensional linear varieties. The role of the directrix is played by the component of the discriminant set with the dimension by one less on which the original polynomial has a single multiple root and the other roots are simple. The relationship between the structure of the discriminant set and the partitioning of natural numbers is revealed. Various algorithms for the calculation of subdiscriminants of polynomials are also discussed. The basic algorithms described in this paper are implemented as a library for Maple. [ABSTRACT FROM AUTHOR]

It is proved that the problem of determining whether a given morphic sequence is almost periodic is decidable. [ABSTRACT FROM AUTHOR]

We describe the minimal model program in the family of ℚ-Gorenstein projective horospherical varieties, by studying a family of polytopes defined from the moment polytope of a Cartier divisor of the variety we begin with. In particular, we generalize the results on MMP for toric varieties due to M. Reid, and we complete the results on MMP for spherical varieties due to M. Brion in the case of horospherical varieties. [ABSTRACT FROM AUTHOR]

In this note we prove a result comparing rationality of algebraic cycles over the function field of a SL 1 ( A ) -torsor for a central simple algebra A and over the base field. [ABSTRACT FROM AUTHOR]

The index of a 3-fold canonical singularity at a crepant centre is at most 6. [ABSTRACT FROM AUTHOR]

The estimate of integral points in right-angled simplices has many applications in number theory, complex geometry, toric variety and tropical geometry. In [24], [25], [27], the second author and other coworkers gave a sharp upper estimate that counts the number of positive integral points in n dimensional ( [ABSTRACT FROM AUTHOR]

Let (S,L) be a smooth primitively polarized K3 surface of genus g and be the fibration defined by a linear pencil in L. For f general and g≥7, we work out the splitting type of the locally free sheaf , where Ψf is the modular morphism associated to f. We show that this splitting type encodes the fundamental geometrical information attached to Mukai’s projection map , where is the stack parameterizing pairs (S,C) with (S,L) as above and C∈L a stable curve. Moreover, we work out conditions on a fibration f to induce a modular morphism Ψf such that the normal sheaf NΨf is locally free. [ABSTRACT FROM AUTHOR]

We give a criterion which ensures that a group generated by Cartan involutions in the automorph group of a rational quadratic form of signature .n - 1; 1/is "thin", namely it is of infinite index in the latter. It is based on a graph defined on the integral Cartan root vectors, as well as Vinberg's theory of hyperbolic reflection groups. The criterion is shown to be robust for showing that many hyperbolic hypergeometric groups for nFn-1 are thin. [ABSTRACT FROM AUTHOR]

Investigated are continuous rational maps of nonsingular real algebraic varieties into spheres. In some cases, necessary and sufficient conditions are given for a continuous map to be approximable by continuous rational maps. In particular, each continuous map between unit spheres can be approximated by continuous rational maps. [ABSTRACT FROM AUTHOR]

We prove that every clone of operations on a finite set A, if it contains a Malcev operation, is finitely related--i.e., identical with the clone of all operations respecting R for some finitary relation R over A. It follows that for a fixed finite set A, the set of all such Malcev clones is countable. This completes the solution of a problem that was first formulated in 1980, or earlier: how many Malcev clones can finite sets support? More generally, we prove that every finite algebra with few subpowers has a finitely related clone of term operations. Hence modulo term equivalence and a renaming of the elements, there are only countably many finite algebras with few subpowers, and thus only countably many finite algebras with a Malcev term. [ABSTRACT FROM AUTHOR]

In [Ch00], Chekanov showed that the Hofer norm on the Hamiltonian diffeomorphism group of a geometrically bounded symplectic manifold induces a nondegenerate metric on the orbit of any compact Lagrangian submanifold under the group. In this paper we consider the orbits of more general submanifolds. We show that, for the Chekanov-Hofer pseudometric on the orbit of a closed submanifold to be a genuine metric, it is necessary for the submanifold to be coisotropic, and we show that this condition is sufficient under various additional geometric assumptions. At the other extreme, we show that the image of a generic closed embedding with any codimension larger than one is "weightless," in the sense that the Chekanov-Hofer pseudometric on its orbit vanishes identically. In particular this yields examples of submanifolds which have zero displacement energy but are not infinitesimally displaceable. [ABSTRACT FROM AUTHOR]

Multivariate sample spaces may be incomplete Cartesian products, when certain combinations of the categories of the variables are not possible. Traditional log-linear models, which generalize independence and conditional independence, do not apply in such cases, as they may associate positive probabilities with the non-existing cells. To describe the association structure in incomplete sample spaces, this paper develops a class of hierarchical multiplicative models which are defined by setting certain non-homogeneous generalized odds ratios equal to one and are named after Aitchison and Silvey who were among the first to consider such ratios. These models are curved exponential families that do not contain an overall effect and, from an algebraic perspective, are non-homogeneous toric ideals. The relationship of this model class with log-linear models and quasi log-linear models is studied in detail in terms of both statistics and algebraic geometry. The existence of maximum likelihood estimates and their properties, as well as the relevant algorithms are also discussed. [ABSTRACT FROM AUTHOR]

We prove that the tetracanonical map of a variety X of maximal Albanese dimension induces the Iitaka fibration. Moreover, if X is of general type, then the tricanonical map is birational. [ABSTRACT FROM PUBLISHER]

In a previous paper, we described some Siegel modular threefolds which admit a Calabi-Yau model. Using a different method we give in this paper an enlarged list of such varieties. Basic for this method is a paper of van Geemen and Nygaard. They study a variety X that is the complete intersection of four quadrics in ℙ7 (ℂ). This is biholomorphic equivalent to the Satake compactification of H2/Γ' for a certain subgroup Γ' ⊂ Sp(2, ℤ) and it will be the starting point of our investigation. It has been pointed out that a (projective) small resolution of this variety is a rigid Calabi-Yau manifold .... Then we will consider the action of quotients of modular groups on X and study possible resolutions that admits a Calabi-Yau model in the category of complex spaces. [ABSTRACT FROM AUTHOR]

It is shown that if 햵 is a local monoidal pseudovariety of semigroups, then 햪 ⓜ 햵, 햣 ⓜ 햵 and are local. Other operators of the form 햹 ⓜ(_) are considered. In the process, results about the interplay between operators 햹 ⓜ(_) and (_) * 햣k are obtained. [ABSTRACT FROM AUTHOR]

Abstract: □□□ [Copyright &y& Elsevier]

Abstract: The fundamental construct of numerical algebraic geometry is the representation of an irreducible algebraic set, A, by a witness set, which consists of a polynomial system, F, for which A is an irreducible component of , a generic linear space of complementary dimension to A, and a numerical approximation to the set of witness points, . Given F, methods exist for computing a numerical irreducible decomposition, which consists of a collection of witness sets, one for each irreducible component of . This paper concerns the more refined question of finding a numerical irreducible decomposition of the intersection of two irreducible algebraic sets, A and B, given a witness set for each. An existing algorithm, the diagonal homotopy, computes witness point supersets for , but this does not complete the numerical irreducible decomposition. In this paper, we use the theory of isosingular sets to complete the process of computing the numerical irreducible decomposition of the intersection. [Copyright &y& Elsevier]

We study the task of randomness extraction from sources that are distributed uniformly on an unknown algebraic variety. In other words, we are interested in constructing a function (an extractor) whose output is close to uniform even if the input is drawn uniformly from the set of solutions of an unknown system of low degree polynomials. This problem generalizes the problem of extraction from affine sources which has drawn a considerable amount of interest lately. We present two constructions of explicit extractors for varieties. The first works for varieties of any size (including one-dimensional varieties or curves) and requires field size that is exponential in the overall dimension of the space. Our second extractor allows the field size to be polynomial in the degree of the equations defining the variety, but works only for varieties whose size is at least the square root of the total size of the space. [ABSTRACT FROM AUTHOR]

Abstract: Cox rings of normal prevarieties are factorially graded, i.e. homogeneous elements allow a unique decomposition into homogeneous factors. We study this property from an algebraic point of view and give a criterion which in a sense reduces it to factoriality. This will allow us to detect and construct Cox rings in a purely algebraic manner. [Copyright &y& Elsevier]

Abstract: We give an explicit affine algebraic variety whose coordinate ring is isomorphic (as a W-algebra) with the equivariant cohomology of some Springer fibers. [Copyright &y& Elsevier]

Algebraic variety is the most important subject in classical algebraic geometry. As the zero set of multivariate splines, the piecewise algebraic variety is a kind generalization of the classical algebraic variety. This paper studies the correspondence between spline ideals and piecewise algebraic varieties based on the knowledge of algebraic geometry and multivariate splines. [ABSTRACT FROM AUTHOR]

Abstract: We provide a necessary and sufficient condition for the derived self-intersection of a smooth subscheme inside a smooth scheme to be a fibration over the subscheme. As a consequence we deduce a generalized HKR isomorphism. We also investigate the relationship of our result to path spaces in homotopy theory, Buchweitz–Flenner formality in algebraic geometry, and draw parallels with similar results in Lie theory and symplectic geometry. [Copyright &y& Elsevier]

We study properties of irreducible and completely reducible representations of finitely generated groups Γ into reductive algebraic groups G. In particular, we study the geometric invariant theory of the G action on the space of G-representations of Γ by conjugation. Let XG(Γ) be the G-character variety of Γ. We prove that for every completely reducible, scheme smooth ρ : Γ → G T[ρ] XG(Γ) ≃ T0 (H1(Γ,Adρ)//SΓ), where H1(Γ,Adρ) is the first cohomology group of Γ with coefficients in the Lie algebra g of G twisted by … and SΓ is the centralizer of ρ(Γ) in G. The condition of ρ being scheme smooth is very important as there are groups Γ such that dim T[ρ]XG(Γ) < T0 H1(Γ,Adρ), for a Zariski open subset of points in XG(Γ). We prove, however, that all irreducible representations of surface groups are scheme smooth. Let M be an orientable 3-manifold with a connected boundary F of genus g ≥ 2. Let X9G(F) be the subset of the G-character variety of π1(F) composed of conjugacy classes of good representations ρ : Γ → G, i.e., irreducible representations such that the centralizer of ρ(Γ) is the center of G. By a theorem of Goldman, X9G(F) is a holomorphic symplectic manifold. We prove that the set of good G-representations of π1(F) which extend to representations of π1(M) is a complex isotropic subspace of X9G(F). It is Lagrangian, if these representations correspond to reduced points of the G-character variety of M. It is an open problem whether it is always the case. [ABSTRACT FROM AUTHOR]

The concept of implicit operation on pseudovarieties of semigroups goes back to Eilenberg and Schutzenberger [1]. The author in [2-5] generalized this concept to other classes of algebras and established a connection between these operations and positively conditional termal functions in the case of uniform local finiteness of the algebras of the class in question. In this article we put forth the concept of an implicit operation for an arbitrary universal algebra, not necessarily locally finite, and establish a connection between these operations and infinite analogs of positively conditional terms, as well as ∞-quasi-identities arising in the algebraic geometry of universal algebras. We also consider conditions for implicit equivalence of algebras to lattices, semilattices, and Boolean algebras. [ABSTRACT FROM AUTHOR]

Abstract: The main goal of this article is to relate asymptotic geometric properties on a tower of coverings of a non-compact Kähler manifold of finite volume with reasonable geometric assumptions to its universal covering. Examples to which our findings are applicable include moduli spaces of hyperbolic punctured Riemann surfaces and Hermitian locally symmetric spaces of finite volume. [Copyright &y& Elsevier]

Let X be an algebraic variety, f a regular function, $${j:U\hookrightarrow X}$$ the complement to the locus of vanishing of f, and M a holonomic D-module on U. Consider the D[ s]-module $${M\otimes ``{f^{s}}''}$$ . The goal of this note is to describe all D[ s] submodules $${N\hookrightarrow j_*(M\otimes ``{f^{s}}'')}$$ such that $${j^*(N)\simeq M\otimes ``{f^{s}}''}$$ . [ABSTRACT FROM AUTHOR]

We prove the isogeny conjecture for A-motives over finitely generated fields K of transcendence degree ≤1. This conjecture says that for any semisimple A-motive M over K, there exist only finitely many isomorphism classes of A-motives M′ over K for which there exists a separable isogeny M′→ M. The result is in precise analogy to known results for abelian varieties and for Drinfeld modules and will have strong consequences for the ${\mathfrak {p}}$-adic and adelic Galois representations associated to M. The method makes essential use of the Harder-Narasimhan filtration for locally free coherent sheaves on an algebraic curve. [ABSTRACT FROM AUTHOR]

We complete the study [1] of smooth ℤ3-quotients of complete intersection Calabi-Yau threefolds by discussing the six new manifolds that admit free ℤ3 actions that were discovered in [2]. These manifolds were missed in [1] and complete the web of smooth ℤ3-quotients in a nice way. We discuss the transitions between these manifolds and include also the other manifolds of the web. This leads to the conclusion that the web of ℤ3-free quotients of complete intersection Calabi-Yau threefolds is connected by conifold transitions. [ABSTRACT FROM AUTHOR]

Abstract: A homogeneous set of monomials in a quotient of the polynomial ring is called Gotzmann if the size of this set grows minimally when multiplied with the variables. We note that Gotzmann sets in the quotient arise from certain Gotzmann sets in . Secondly, we prove a combinatorial result about the deletion of a variable in a Gotzmann set in . [Copyright &y& Elsevier]

Abstract: We present a treatment of the algebraic description of the Jacobian of a generic genus two plane curve which exploits an equivariance and clarifies the structure of Flynn’s 72 defining quadratic relations. The treatment is also applied to the Kummer variety. [Copyright &y& Elsevier]

In a manner similar to the construction of the fundamental group of a connected graph, this article introduces the construction of a fundamental semigroup associated with a bipartite graph. This semigroup is a 0-direct union of idempotent generated completely 0-simple semigroups. The maximal nonzero subgroups are the corresponding fundamental groups of the connected components. Adding labelled edges to the graph leads to a more general completely 0-simple semigroup. The basic properties of such semigroups are examined and they are shown to have certain universal properties as illustrated by the fact that the free completely simple semigroup on n generators and its idempotent generated subsemigroup appear as special cases. [ABSTRACT FROM AUTHOR]

Abstract: We compute the Chow ring of the classifying space in the sense of Totaro using the fibration and a computation of the Chow ring of in a previous paper. We find this Chow ring is generated by Chern classes and a characteristic class defined by Edidin and Graham which maps to times the Euler class under the usual class map from the Chow ring to ordinary cohomology. Moreover, we show this class represents times the nth Chern class of the representation of whose highest weight vector is twice that of the half-spin representation. [Copyright &y& Elsevier]

Abstract: We show this Chow ring is . We do this by partitioning the space into 2n subvarieties each of which is fibered over . [Copyright &y& Elsevier]