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We discuss various applications of a uniform vanishing result for the graded components of the finite length Koszul module associated to a subspace K ⊆ ⋀ 2 V , where V is a vector space. Previously Koszul modules of finite length have been used to give a proof of Green's Conjecture on syzygies of generic canonical curves. We now give applications to effective stabilization of cohomology of thickenings of algebraic varieties, divisors on moduli spaces of curves, enumerative geometry of curves on K3 surfaces and to skew-symmetric degeneracy loci. We also show that the instability of sufficiently positive rank 2 vector bundles on curves is governed by resonance and give a splitting criterion. [ABSTRACT FROM AUTHOR]

In this paper, we consider the affine variety codes obtained evaluating the polynomials b y = a k x k + ... + a 1 x + a 0 , b , a i ∈ F q r , at the affine F q r -rational points of the Norm-Trace curve. In particular, we investigate the weight distribution and the set of minimal codewords. Our approach, which uses tools of algebraic geometry, is based on the study of the absolute irreducibility of certain algebraic varieties. [ABSTRACT FROM AUTHOR]

We study intersection theory for differential algebraic varieties. Particularly, we study families of differential hypersurface sections of arbitrary affine differential algebraic varieties over a differential field. We prove the differential analogue of Bertini's theorem, namely that for an arbitrary geometrically irreducible differential algebraic variety which is not an algebraic curve, generic hypersurface sections are geometrically irreducible and codimension one. Surprisingly, we prove a stronger result in the case that the order of the differential hypersurface is at least one; namely that the generic differential hypersurface sections of an irreducible differential algebraic variety are irreducible and codimension one. We also calculate the Kolchin polynomials of the intersections and prove several other results regarding intersections of differential algebraic varieties. [ABSTRACT FROM AUTHOR]

Kleinian singularities, that is, the varieties corresponding to the algebras of invariants of Kleinian groups are of fundamental importance for Algebraic geometry, Representation theory, and Singularity theory. The filtered deformations of these algebras of invariants were classified by Brieskorn (the commutative case) and Losev (the general case). To an inclusion of Kleinian groups, there is the corresponding inclusion of algebras of invariants. We classify deformations of these inclusions when the smaller subgroup is normal in the larger. [ABSTRACT FROM AUTHOR]

We develop a purely categorical theory of action filtrations and their associated growth invariants. When specialized to categories of geometric interest, such as the wrapped Fukaya category of a Weinstein manifold, and the bounded derived category of coherent sheaves on a smooth algebraic variety, our categorical action filtrations essentially recover previously studied filtrations of geometric origin. Our approach is built around the notion of a smooth categorical compactification. We prove that a smooth categorical compactification induces well-defined growth invariants, which are moreover preserved under zig-zags of such compactifications. The technical heart of the paper is a method for computing these growth invariants in terms of the growth of certain colimits of (bi)modules. In practice, such colimits arise in both geometric settings of interest. The main applications are: (1) A "quantitative" refinement of homological mirror symmetry, which relates the growth of the Reeb-length filtration on the symplectic geometry side with the growth of filtrations on the algebraic geometry side defined by the order of pole at infinity (often these can be expressed in terms of the dimension of the support of sheaves). (2) A proof that the Reeb-length growth of symplectic cohomology and wrapped Floer cohomology on a Weinstein manifold are at most exponential. (3) Lower bounds for the entropy and polynomial entropy of certain natural endofunctors acting on Fukaya categories. [ABSTRACT FROM AUTHOR]

We study the Rouquier dimension of wrapped Fukaya categories of Liouville manifolds and pairs, and apply this invariant to various problems in algebraic and symplectic geometry. On the algebro-geometric side, we introduce a new method based on symplectic flexibility and mirror symmetry to bound the Rouquier dimension of derived categories of coherent sheaves on certain complex algebraic varieties and stacks. These bounds are sharp in dimension at most $3$. As an application, we resolve a well-known conjecture of Orlov for new classes of examples (e.g. toric $3$ -folds, certain log Calabi–Yau surfaces). We also discuss applications to non-commutative motives on partially wrapped Fukaya categories. On the symplectic side, we study various quantitative questions including the following. (1) Given a Weinstein manifold, what is the minimal number of intersection points between the skeleton and its image under a generic compactly supported Hamiltonian diffeomorphism? (2) What is the minimal number of critical points of a Lefschetz fibration on a Liouville manifold with Weinstein fibers? We give lower bounds for these quantities which are to the best of the authors' knowledge the first to go beyond the basic flexible/rigid dichotomy. [ABSTRACT FROM AUTHOR]

We show that the results we had previously obtained on diagonals of 9- and 10-parameter families of rational functions in three variables x , y , and z , using creative telescoping, yielding modular forms expressed as pullbacked 2 F 1 hypergeometric functions, can be obtained much more efficiently by calculating the j -invariant of an elliptic curve canonically associated with the denominator of the rational functions. These results can be drastically generalized by changing the parameters into arbitrary rational functions of the product p = x y z . In other cases where creative telescoping yields pullbacked 2 F 1 hypergeometric functions, we extend this algebraic geometry approach to other families of rational functions in three or more variables. In particular, we generalize this approach to rational functions in more than three variables when the denominator can be associated to an algebraic variety corresponding to products of elliptic curves, or foliations in elliptic curves. We also extend this approach to rational functions in three variables when the denominator is associated with a genus-two curve such that its Jacobian is a split Jacobian, corresponding to the product of two elliptic curves. We sketch the situation where the denominator of the rational function is associated with algebraic varieties that are not of the general type, having an infinite set of birational automorphisms. We finally provide some examples of rational functions in more than three variables, where the telescopers have pullbacked 2 F 1 hypergeometric solutions, because the denominator corresponds to an algebraic variety that has a selected elliptic curve. [ABSTRACT FROM AUTHOR]

We relate the geometry of curves to the notion of hyperbolicity in real algebraic geometry. A hyperbolic variety is a real algebraic variety that (in particular) admits a real fibered morphism to a projective space whose dimension is equal to the dimension of the variety. We study hyperbolic varieties with a special interest in the case of hypersurfaces that admit a real algebraic ruling. The central part of the paper is concerned with secant varieties of real algebraic curves where the real locus has the maximal number of connected components, which is determined by the genus of the curve. For elliptic normal curves, we further obtain definite symmetric determinantal representations for the hyperbolic secant hypersurfaces, which implies the existence of symmetric Ulrich sheaves of rank one on these hypersurfaces. We also use this to derive better bounds on the size of semidefinite representations for convex hulls of real algebraic curves of genus 1. [ABSTRACT FROM AUTHOR]

Let $ {\mathbb {C}}^{n+1}_o$ denote the germ of $ {\mathbb {C}}^{n+1}$ at the origin. Let $V$ be a hypersurface germ in $ {\mathbb {C}}^{n+1}_o$ and $W$ a deformation of $V$ over $ {\mathbb {C}}_{o}^{m}$. Under the hypothesis that $W$ is a Newton non-degenerate deformation, in this article we prove that $W$ is a $\mu$ -constant deformation if and only if $W$ admits a simultaneous embedded resolution. This result gives a lot of information about $W$ , for example, the topological triviality of the family $W$ and the fact that the natural morphism $(\operatorname {W({\mathbb {C}}_{o})}_{m})_{{\rm red}}\rightarrow {\mathbb {C}}_{o}$ is flat, where $\operatorname {W({\mathbb {C}}_{o})}_{m}$ is the relative space of $m$ -jets. On the way to the proof of our main result, we give a complete answer to a question of Arnold on the monotonicity of Newton numbers in the case of convenient Newton polyhedra. [ABSTRACT FROM AUTHOR]

Let an arbitrary variety of algebras and the category of all free finitely generated algebras in that variety be given. This paper is the second in a series of papers started in [Algebra and Logic, 61, No. 1, 1-15 (2022)] where we deal with automorphisms of the category of free finitely generated algebras. Here we describe in detail a method of verbal operations. The method provides a characterization of automorphisms of the category of all free finitely generated algebras in a given variety. The characterization plays a crucial role in universal algebraic geometry. We supply the reader with illuminating examples which clarify the method. [ABSTRACT FROM AUTHOR]

Many interesting problems arise on the borderline between real algebraic geometry and topology. We focus on 12 open questions. Some of them come from regulous geometry, which emerged as a subfield of real algebraic geometry less than 15 years ago. [ABSTRACT FROM AUTHOR]

In algebraic geometry, the Frobenius map (or called Frobenius action , or Frobenius operator) F ⁎ on cohomology groups plays an important role in the classification of algebraic varieties over a field of positive characteristic. In particular, representation matrices for F ⁎ give rise to many important invariants such as p -rank and a -number. Several methods for computing representation matrices for F ⁎ have been proposed for specific curves. In this paper, we present an algorithm to compute representation matrices for F ⁎ of general projective schemes over a perfect field of positive characteristic. We also propose an efficient algorithm specific to complete intersections; it requires to compute only certain coefficients in a power of a multivariate polynomial. Our algorithms shall derive fruitful applications such as computing Hasse-Witt matrices, and enumerating superspecial curves. In particular, the second algorithm for complete intersections provides a useful tool to judge the superspeciality of an algebraic curve, which is a key ingredient to prove main results in Kudo and Harashita (2017a,b, 2020) on the enumeration of superspecial genus-4 curves. [ABSTRACT FROM AUTHOR]

Every real algebraic variety determines a Voronoi decomposition of its ambient Euclidean space. Each Voronoi cell is a convex semialgebraic set in the normal space of the variety at a point. We compute the algebraic boundaries of these Voronoi cells. [ABSTRACT FROM AUTHOR]

We study the autocovariance functions of moving average random fields over the integer lattice Z d from an algebraic perspective. These autocovariances are parametrized polynomially by the moving average coefficients, hence tracing out algebraic varieties. We derive dimension and degree of these varieties and we use their algebraic properties to obtain statistical consequences such as identifiability of model parameters. We connect the problem of parameter estimation to the algebraic invariants known as euclidean distance degree and maximum likelihood degree. Throughout, we illustrate the results with concrete examples. In our computations we use tools from commutative algebra and numerical algebraic geometry. [ABSTRACT FROM AUTHOR]

Let an arbitrary variety of algebras and the category of all free finitely generated algebras in that variety be given. In universal algebraic geometry over an arbitrary variety of algebras, the group of automorphisms of the category of free finitely generated algebras plays an important role. This paper is first in a series where we will deal with the group mentioned. Here we describe properties of automorphisms of the category of all free finitely generated algebras and distinguish two important subgroups: namely, the subgroup of inner automorphisms and the subgroup of strongly stable automorphisms. [ABSTRACT FROM AUTHOR]

For n ∈ N, we consider the algebraic variety V obtained by intersecting n + 1algebraic curves of degree n in R², when the leading terms of the associated bivariate polynomials are all different. We provide a new proof, based on the Flat Extension Theorem from the theory of truncated moment problems, that the cardinality of V cannot exceed (2n+1). In some instances, this provides a slightly better estimate than the one given by Bézout's Theorem. Our main result contributes to the growing literature on the interplay between linear algebra, operator theory, and real algebraic geometry. [ABSTRACT FROM AUTHOR]

Differential geometry, especially the use of curvature, plays a central role in modern Hodge theory. The vector bundles that occur in the theory (Hodge bundles) have metrics given by the polarizations of the Hodge structures, and the sign and singularity properties of the resulting curvatures have far reaching implications in the geometry of families of algebraic varieties. A special property of the curvatures is that they are 1 st order invariants expressed in terms of the norms of algebro-geometric bundle mappings. This partly expository paper will explain some of the positivity and singularity properties of the curvature invariants that arise in the Hodge theory with special emphasis on the norm property. [ABSTRACT FROM AUTHOR]

We study extensions of compressive sensing and low rank matrix recovery to the recovery of tensors of low rank from incomplete linear information. While the reconstruction of low rank matrices via nuclear norm minimization is rather well-understand by now, almost no theory is available so far for the extension to higher order tensors due to various theoretical and computational difficulties arising for tensor decompositions. In fact, nuclear norm minimization for matrix recovery is a tractable convex relaxation approach, but the extension of the nuclear norm to tensors is in general NP-hard to compute. In this article, we introduce convex relaxations of the tensor nuclear norm which are computable in polynomial time via semidefinite programming. Our approach is based on theta bodies, a concept from real computational algebraic geometry which is similar to the one of the better known Lasserre relaxations. We introduce polynomial ideals which are generated by the second-order minors corresponding to different matricizations of the tensor (where the tensor entries are treated as variables) such that the nuclear norm ball is the convex hull of the algebraic variety of the ideal. The theta body of order k for such an ideal generates a new norm which we call the θk-norm. We show that in the matrix case, these norms reduce to the standard nuclear norm. For tensors of order three or higher however, we indeed obtain new norms. The sequence of the corresponding unit-θk-norm balls converges asymptotically to the unit tensor nuclear norm ball. By providing the Gröbner basis for the ideals, we explicitly give semidefinite programs for the computation of the θk-norm and for the minimization of the θk-norm under an affine constraint. Finally, numerical experiments for order-three tensor recovery via θ1-norm minimization suggest that our approach successfully reconstructs tensors of low rank from incomplete linear (random) measurements. [ABSTRACT FROM AUTHOR]

Introducing parallelism and exploring its use is still a fundamental challenge for the computer algebra community. In high-performance numerical simulation, on the other hand, transparent environments for distributed computing which follow the principle of separating coordination and computation have been a success story for many years. In this paper, we explore the potential of using this principle in the context of computer algebra. More precisely, we combine two well-established systems: The mathematics we are interested in is implemented in the computer algebra system Singular, whose focus is on polynomial computations, while the coordination is left to the workflow management system GPI-Space, which relies on Petri nets as its mathematical modeling language and has been successfully used for coordinating the parallel execution (autoparallelization) of academic codes as well as for commercial software in application areas such as seismic data processing. The result of our efforts is a major step towards a framework for massively parallel computations in the application areas of Singular, specifically in commutative algebra and algebraic geometry. As a first test case for this framework, we have modeled and implemented a hybrid smoothness test for algebraic varieties which combines ideas from Hironaka's celebrated desingularization proof with the classical Jacobian criterion. Applying our implementation to two examples originating from current research in algebraic geometry, one of which cannot be handled by other means, we illustrate the behavior of the smoothness test within our framework and investigate how the computations scale up to 256 cores. [ABSTRACT FROM AUTHOR]

Modelling the substitution of nucleotides along a phylogenetic tree is usually done by a hidden Markov process. This allows to define a distribution of characters at the leaves of the trees and one might be able to obtain polynomial relationships among the probabilities of different characters. The study of these polynomials and the geometry of the algebraic varieties defined by them can be used to reconstruct phylogenetic trees. However, not all points in these algebraic varieties have biological sense. In this paper, we explore the extent to which adding semi-algebraic conditions arising from the restriction to parameters with statistical meaning can improve existing methods of phylogenetic reconstruction. To this end, our aim is to compute the distance of data points to algebraic varieties and to the stochastic part of these varieties. Computing these distances involves optimization by nonlinear programming algorithms. We use analytical methods to find some of these distances for quartet trees evolving under the Kimura 3-parameter or the Jukes-Cantor models. Numerical algebraic geometry and computational algebra play also a fundamental role in this paper. [ABSTRACT FROM AUTHOR]

The aim of matrix recovery is to recover P ∈ M ⊂ F p × q from L A (P) = (Tr (A 1 T P) , Tr (A 2 T P) , ... , Tr (A N T P)) T with A j ∈ V j ⊂ F p × q , which is raised in many areas. In this paper, we build up a framework for almost everywhere matrix recovery which means L A is almost everywhere injectivity on M. We mainly focus on the following question: how many measurements are needed to recover almost all the matrices in M ? For the case where both M and V j are algebraic varieties, we use the tools from algebraic geometry to study the question and present some results to address it under many different settings. [ABSTRACT FROM AUTHOR]

We give a numerical algorithm computing Euler obstruction functions using maximum likelihood degrees. The maximum likelihood degree is a well-studied property of a variety in algebraic statistics and computational algebraic geometry. In this article we use this degree to give a new way to compute Euler obstruction functions. We define the maximum likelihood obstruction function and show how it coincides with the Euler obstruction function. With this insight, we are able to bring new tools of computational algebraic geometry to study Euler obstruction functions. [ABSTRACT FROM AUTHOR]

The Isbell duality tells us a dual equivalence between spatial frames (aka. locales) and sober spaces; it is induced by the prime spectrum functor on frames. In the present paper, we give another dual equivalence induced by the maximal spectrum functor. The Isbell duality subsumes all sober spaces, but not all T 1 spaces; the duality shown in this paper subsumes all T 1 spaces, but not all sober spaces. Non-sober T 1 spaces are particularly important in classical algebraic geometry; they include, inter alia, algebraic varieties in the traditional sense, the points of which can be recovered from their open set frames via the maximal spectrum functor (and cannot via the prime spectrum functor). The duality in this paper is particularly useful for those spaces in algebraic geometry. In addition to the duality induced by maximal spectra, we give a dual adjunction lurking behind it, and an algebraic characterization of having enough points in terms of maximal spectra. Communicated by Jason P. Bell [ABSTRACT FROM AUTHOR]

I In memory of Jesús Begoña Aina, excellent teacher and person, whom we used to call affectionately "El Lennon" i We say that a subset of HT ht is a semialgebraic set if it can be described by a finite Boolean combination of polynomial equalities and inequalities. I Then i HT ht I is a semialgebraic set of i HT ht . Under what conditions on the HT ht does there exist an algebraic set HT ht whose orthogonal projection on each HT ht is HT ht ? It is clear that Question 1 can be restated in terms of the I n i -dimensional space HT ht and its I n i standard orthogonal projections. [Extracted from the article]

This paper studies the Lagrangian geometry of algebraic varieties. Given a smooth compact simply-connected algebraic variety, we construct a family of finite-dimensional Kähler manifolds whose elements are the equivalence classes of Lagrangian submanifolds satisfying our new D-exactness condition. In connection with the theory of Weinstein structures, these moduli spaces turn out related to the special Bohr-Sommerfeld geometry constructed by the author previously. This enables us to extract from the moduli spaces some stable components and conjecture that they are not only Kähler but also algebraic. [ABSTRACT FROM AUTHOR]

Given a complex variety X , a linear algebraic group G and a representation ρ of the fundamental group π 1 (X , x) into G , we develop a framework for constructing a functorial mixed Hodge structure on the formal local ring of the representation variety of π 1 (X , x) into G at ρ using mixed Hodge diagrams and methods of L ∞ algebras. We apply it in two geometric situations: either when X is compact Kähler and ρ is the monodromy of a variation of Hodge structure, or when X is smooth quasi-projective and ρ has finite image. [ABSTRACT FROM AUTHOR]

Abstract We prove that the set of non-degenerate second order maximally superintegrable systems in the complex Euclidean plane carries a natural structure of a projective variety, equipped with a linear isometry group action. This is done by deriving an explicit system of homogeneous algebraic equations. We then solve these equations and give a detailed analysis of the algebraic geometric structure of the corresponding projective variety. This naturally associates a unique planar line triple arrangement to every superintegrable system, providing a geometric realisation of this variety and an intrinsic labelling scheme. In particular, our results confirm the known classification by independent, purely algebraic means. [ABSTRACT FROM AUTHOR]

Rapoport–Zink spaces, or more generally local Shimura varieties, are expected to provide geometric realization of the local Langlands correspondence via their l‐adic cohomology. Along this line is a conjecture by Harris and Viehmann, which roughly says that when the underlying local Shimura datum is not basic, the l‐adic cohomology of the local Shimura variety is parabolically induced. We verify this conjecture for Rapoport–Zink spaces which are Hodge type and Hodge–Newton reducible. The main strategy is to embed such a Rapoport–Zink space into an appropriate space of EL type, for which the conjecture is already known to hold by the work of Mantovan. [ABSTRACT FROM AUTHOR]

We introduce a new construction of towers of algebraic curves over finite fields and provide a simple example of an optimal tower. [ABSTRACT FROM AUTHOR]

Motivated by the Bloch-Beilinson conjectures, Voisin has formulated a conjecture about 0-cycles on self-products of surfaces of geometric genus one. We verify Voisin's conjecture for the family of Todorov surfaces with K2 = 2 and fundamental group Z/2Z.As a by-product,we prove that certainTodorov surfaces have finite-dimensional motive. [ABSTRACT FROM AUTHOR]

Let f (q) be a Coleman family of cusp forms of tame level N. Let k0 be the classical weight at which the specialization of f (q) is new. By the Kohnen-Shintani correspondence, we associate to every even classical weight k, a half-integral weight form (for k ≠ k0) gk = ... c(D, k)qD ∈ Sk+1/2 (Γ0(4N)) and gk0 = ... c(D, k)qD ∈ Sk+1/2 (Γ0(4Np)). We first prove that the Fourier coefficients c(D, k) for k ∈ 2Z>0 can be interpolated by a p-adic analytic function ...(D, κ) with κ varying in a neighbourhood of k0 in the p-adic weight space. For discriminants D such that ...(D, k0) = 0, which we call Type II, we show that d/dκ [...(D, κ)]k=k0 is related to certain algebraic cycles associated to the motiveMk0 attached to the space of cusp forms of weight Sk0 (Γ0 (Np )). These algebraic cycles appear in the theory of Darmon cycles. [ABSTRACT FROM AUTHOR]

Let (X,D) be a log smooth pair of dimension n, where D is a reduced effective divisor such that the log canonical divisor KX +D is pseudoeffective. Let G be a connected algebraic subgroup of Aut(X, D). We show that G is a semi-abelian variety of dimension = min{n- (V), n} with V := X \D. In the dimension two, Iitaka claimed in his 1979 Osaka J. Math. paper that dimG = q(V) for a log smooth surface pair with (V) = 0 and pg(V) = 1. We (re-)prove and generalize this classical result for all surfaces with = 0 without assuming Iitaka's classification of logarithmic Iitaka surfaces or logarithmic K3 surfaces. [ABSTRACT FROM AUTHOR]

The weak units of strict monoidal 1- and 2-categories are defined respectively in [15] and [14]. In this paper, we define them for group-like 1- and 2-stacks. We show that they form a contractible Picard 1- and 2-stack, respectively. We give their cohomological description which provides for these stacks a representation by complexes of sheaves of groups. Later, we extend the discussion to the monoidal case. We consider the (2-)substack of cancelable objects of a monoidal 1-(2-)stack. We observe that this (2-)substack is trivially group-like, its weak units are the same as the weak units of the monoidal 1-(2-)stack, and therefore we can recover the contractibility results in [15] and [14] by analysing it. [ABSTRACT FROM AUTHOR]

Making use of Freyd's free abelian category on a preadditive category we show that if T: D→A is a representation of a quiver D in an abelian category A then there is an abelian category A(T), a faithful exact functor FT: A(T)→A and an induced representation T: D→A(T) such that F T T = T universally. We then can show that T-motives as well as Nori's motives are given by a certain category of functors on definable categories. [ABSTRACT FROM AUTHOR]

The conventional ways to solve optimization problems on low-rank matrix sets which appear in a great number of applications tend to ignore its underlying structure of an algebraic variety and existence of singular points. This leads to the appearance of inverses of singular values in algorithms and since they could be close to 0 it causes certain problems. We tackle this problem by utilizing ideas from algebraic geometry and show how to desingularize these sets. Our main result is an algorithm which uses only bounded functions of singular values and hence does not suffer from the issue described above. [ABSTRACT FROM AUTHOR]

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]

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]

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]

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]

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.

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]