Varieties Of Rank Research Articles

Investigating Secant Varieties

A variety is a geometric object that locally looks like the set of points that are the solutions of some polynomial equations. Any given variety gives rise to a sequence of secant varieties indexed by the nonnegative integers. The kth secant variety is the union of all planes spanned by k+1 points on the original variety. Like vector spaces in linear algebra, varieties have an intrinsic dimension. A naive parameter count produces an upper bound on the dimension of the kth secant variety called its expected dimension. A secant variety is called defective if its actual dimension is less than its expected dimension. Despite being well understood in a few interesting cases, the classification of varieties with defective secant varieties in general remains far from complete. For a specific class of varieties called smooth toric varieties, our goal is to determine which ones admit defective secant varieties. Using the computer algebra system Macaulay2, we conducted experiments on this class of varieties and collected data on defectiveness, focusing on varieties of low dimension and Picard rank. Toric varieties have strong connections to combinatorics and convex polytopes; for any toric variety, there is a corresponding polytope that contains data about the variety. From the polytope, we are able to construct a particular kind of matrix that encodes information about the variety and its secants, which allowed us to prove defectivity for various Picard rank 2 toric varieties. Conversely, applying results from a paper by Laface–Massarenti–Rischter (2022) to the polytopes, we were able to prove non-defectivity of all secant varieties of certain toric surfaces with Picard rank 2. These statements together yield a classification of defective secant varieties for a subset of 2-dimensional smooth toric varieties with Picard rank 2.

The Manin–Peyre conjecture for smooth spherical Fano varieties of semisimple rank one

Abstract The Manin–Peyre conjecture is established for a class of smooth spherical Fano varieties of semisimple rank one. This includes all smooth spherical Fano threefolds of type T as well as some higher-dimensional smooth spherical Fano varieties.

Algebraic optimization of sequential decision problems

We study the optimization of the expected long-term reward in finite partially observable Markov decision processes over the set of stationary stochastic policies. In the case of deterministic observations, also known as state aggregation, the problem is equivalent to optimizing a linear objective subject to quadratic constraints. We characterize the feasible set of this problem as the intersection of a product of affine varieties of rank one matrices and a polytope. Based on this description, we obtain bounds on the number of critical points of the optimization problem. Finally, we conduct experiments in which we solve the KKT equations or the Lagrange equations over different boundary components of the feasible set, and we compare the result to the theoretical bounds and to other constrained optimization methods.

Constructing Fano 3-folds from cluster varieties of rank 2

Cluster algebras give rise to a class of Gorenstein rings which enjoy a large amount of symmetry. Concentrating on the rank 2 cases, we show how cluster varieties can be used to construct many interesting projective algebraic varieties. Our main application is then to construct hundreds of families of Fano 3-folds in codimensions 4 and 5. In particular, for Fano 3-folds in codimension 4 we construct at least one family for 187 of the 206 possible Hilbert polynomials contained in the Graded Ring Database.

Finite morphisms and simultaneous reduction of the multiplicity

AbstractLet X be a singular algebraic variety defined over a perfect field k, with quotient field . Let be the highest multiplicity of X and let be the set of points of multiplicity s. If is a regular center and is the blow up at Y, then the highest multiplicity of X1 is less than or equal to s. A sequence of blow ups at regular centers , say , is said to be a simplification of the multiplicity if the maximum multiplicity of is strictly lower than that of X, that is, if is empty. In characteristic zero there is an algorithm which assigns to each X a unique simplification of the multiplicity. However, the problem remains open when the characteristic is positive. In this paper we will study finite dominant morphisms between singular varieties of generic rank (i.e., ). We will see that, when imposing suitable conditions on β, there is a strong link between the strata of maximum multiplicity of X and , say and respectively. In such case, we will say that the morphism is strongly transversal. When is strongly transversal one can obtain information about the simplification of the multiplicity of X from that of and vice versa. Finally, we will see that given a singular variety X and a finite field extension L of of rank , one can construct (at least locally, in étale topology) a strongly transversal morphism , where has quotient field L.

Local finiteness for Green’s relations in semigroup varieties

ABSTRACTA semigroup variety is said to be locally 𝒦-finite, where 𝒦 stands for any of Green’s relations ℋ, ℛ, ℒ, 𝒟, or 𝒥, if every finitely generated semigroup in this variety has only finitely many 𝒦-classes. We characterize locally 𝒦-finite varieties of finite axiomatic rank in the language of “forbidden objects”.

On Some Families of Smooth Affine Spherical Varieties of Full Rank

Let G be a complex connected reductive group. Losev has shown that a smooth affine spherical G-variety X is uniquely determined by its weight monoid, which is the set of irreducible representations of G that occur in the coordinate ring of X. In this paper we use a combinatorial characterization of the weight monoids of smooth affine spherical varieties to classify: (a) all such varieties for G = SL(2) × ℂ × and (b) all such varieties for G simple which have a G-saturated weight monoid of full rank. We also use the characterization and Knop’s classification theorem for multiplicity free Hamiltonian manifolds to give a new proof of Woodward’s result that every reflective Delzant polytope is the moment polytope of such a manifold.

Real identifiability vs. complex identifiability

A real tensor T of rank r is identifiable if it has a unique decomposition with rank 1 tensors. Sometimes identifiability fails over , for general tensors of fixed rank. This behaviour is peculiar when r is sub-generic. In several cases, the failure is due to an elliptic normal curve passing through general points of the corresponding variety of rank 1 tensors (Segre, Veronese, Grassmann). When this happens, we prove the existence of nonempty euclidean open subsets of some varieties of rank r tensors, whose elements have several complex decompositions, but only one of them is real.

Late Triassic‐Early Jurassic abnormal thermal event constrained by zircon fission track dating and vitrinite reflectance in Xishan coalfield, Qinshui Basin, central North China

Xishan coalfield, Shanxi, is located in the northwest of the Qinshui Basin, central North China. It is notable for its varieties of coal rank ranging from high volatile bituminous coals to anthracite as well as having abundant coalbed methane resources. Zircon fission track (ZFT) analyses were carried out on the zircons in 2 Upper Carboniferous and 5 Lower‐Middle Permian sandstones, and vitrinite reflectance of Late Carboniferous and Early Permian coals were measured to determine the timing of thermal events and maximum paleo‐temperatures, which were responsible for coal maturation and coalbed methane generation. Maximum paleo‐temperatures calculated from vitrinite reflectance values reached to about 232 and 223 °C in Late Carboniferous and Early Permian coals, respectively, and the estimated paleo‐temperature gradient was 11.84 °C/100 m, representing an intensive abnormal thermal event. Results of the ZFT dating indicated that 5 samples failed the χ2‐test and 2 samples passed the test. The decomposition results of the 5 samples divided their age populations into 3 periods: (a) older ages (537, 584, and 802 Ma) than sandstones ages, (b) close to or slightly older than their depositional ages (289, 301, and 331 Ma), and (c) younger than the depositional ages (181–215). The 2 samples that passed χ2‐test yield the central ages of 168 ± 7 Ma and 190 ± 8 Ma, respectively, younger than the deposition age. The close to or older ages than the sandstones depositional ages represent the tectonothermal events occurring in their source areas; the younger ages indicate the existence of the postdepositional tectonothermal event. The agreement of the partly annealing temperature zone (210–300 °C) of zircon fission tracks with the calculated maximum paleo‐temperatures from vitrinite reflectance suggests a Late Triassic‐Early Jurassic abnormal thermal event with the formation time of the present coal rank being 181–215 Ma, rather than a unique intrusion at 95–135 Ma on the western margin of coalfield as previously believed. Combined with other ZFT ages regionally, this abnormal event also occurred in the southern as well as the northern parts of the Qinshui Basin. The Late Triassic‐Early Jurassic intensive extension in the North China Craton is the geodynamic setting of this tectonothermal event.

$$\mathbb {A}^1$$-connected varieties of rank one over nonclosed fields

In this paper, we proved two results regarding the arithmetics of separably $\mathbb{A}^1$-connected varieties of rank one. First we proved over a large field, there is an $\mathbb{A}^1$-curve through any rational point of the boundary, if the boundary divisor is smooth and separably rationally connected. Secondly, we generalize a theorem of Hassett-Tschinkel for the Zariski density of integral points over function fields of curves.

Equivariant vector bundles on complete symmetric varieties of minimal rank

Let X be the wonderful compactification of a complex symmetric space G/H of minimal rank. For a point x ∈ G, denote by Z the closure of BxH/H in X, where B is a Borel subgroup of G. The universal cover of G is denoted by [Formula: see text]. Given a [Formula: see text] equivariant vector bundle E on X, we prove that E is nef (respectively, ample) if and only if its restriction to Z is nef (respectively, ample). Similarly, E is trivial if and only if its restriction to Z is so.

Imbrex geometries

We introduce an axiom on strong parapolar spaces of diameter 2, which arises naturally in the framework of Hjelmslev geometries. This way, we characterize the Hjelmslev–Moufang plane and its relatives (line Grassmannians, certain half-spin geometries and Segre geometries). At the same time we provide a more general framework for a lemma of Cohen, which is widely used to study parapolar spaces. As an application, if the geometries are embedded in a projective space, we provide a common characterization of (projections of) Segre varieties, line Grassmann varieties, half-spin varieties of low rank, and the exceptional variety E6,1 by means of a local condition on tangent spaces.

Equivariant cobordism of flag varieties and of symmetric varieties

We obtain an explicit presentation for the equivariant cobordism ring of a complete flag variety. An immediate corollary is a Borel presentation for the ordinary cobordism ring. Another application is an equivariant Schubert calculus in cobordism. We also describe the rational equivariant cobordism rings of wonderful symmetric varieties of minimal rank.

Compactification of Drinfeld modular varieties and Drinfeld modular forms of arbitrary rank

We give an abstract characterization of the Satake compactification of a general Drinfeld modular variety. We prove that it exists and is unique up to unique isomorphism, though we do not give an explicit stratification by Drinfeld modular varieties of smaller rank which is also expected. We construct a natural ample invertible sheaf on it, such that the global sections of its k-th power form the space of (algebraic) Drinfeld modular forms of weight k. We show how the Satake compactification and modular forms behave under all natural morphisms between Drinfeld modular varieties; in particular we define Hecke operators. We give explicit results in some special cases.

Fano Symmetric Varieties with Low Rank

The symmetric projective varieties of rank one are all smooth and Fano by a classical result of Akhiezer. We classify the locally factorial (respectively smooth) projective symmetric G -varieties of rank 2 that are Fano. When G is semisimple we also classify the locally factorial (respectively smooth) projective symmetric G -varieties of rank 2 that are only quasi-Fano. Moreover, we classify the Fano symmetric G -varieties of rank 3 obtainable from a wonderful variety by a sequence of blow-ups along G -stable varieties. Finally, we classify the Fano symmetric varieties of arbitrary rank that are obtainable from a wonderful variety by a sequence of blow-ups along closed orbits.

Principally Polarized Semi-Abelic Varieties of Small Torus Rank, and the Andreotti-Mayer Loci

We obtain, by a direct computation, explicit descriptions of all principally polarized semi-abelic varieties of torus rank up to 3. We describe the geometry of their symmetric theta divisors and obtain explicit formulas for the involution and its fixed points. These results allow us to give a new proof of the statements about the dimensions, for small genus, of the loci of ppav with theta divisor containing two-torsion points of multiplicity three. We also prove a result about the closure of this set. Our computations used in our work arXiv:1103.1857 to compute the class of the closure of the locus of intermediate jacobians of cubic threefolds in the Chow ring of the perfect cone compactification of the moduli space of principally polarized abelian fivefolds.

Non-formality of Milnor fibres of line arrangements

The complement M of a complex line arrangement 풜 in ℙ2(ℂ) is formal, and admits a finite cover F → M, where F is the associated global Milnor fibre. An example where F is not formal (not even 1-formal), is given in this article, using the relationship between characteristic varieties of rank one local systems and resonance varieties.

Characteristic varieties and logarithmic differential 1-forms

AbstractWe introduce in this paper a hypercohomology version of the resonance varieties and obtain some relations to the characteristic varieties of rank one local systems on a smooth quasi-projective complex variety M. A logarithmic resonance variety is also considered and, as an application, we determine the first characteristic variety of the configuration space of n distinct labeled points on an elliptic curve. Finally, for a logarithmic 1-form α on M we investigate the relation between the resonance degree of α and the codimension of the zero set of α on a good compactification of M. This question was inspired by the recent work by Cohen, Denham, Falk and Varchenko.

On secant varieties of compact Hermitian symmetric spaces

We show that the secant varieties of rank three compact Hermitian symmetric spaces in their minimal homogeneous embeddings are normal, with rational singularities. We show that their ideals are generated in degree three—with one exception, the secant variety of the 21-dimensional spinor variety in P 63 where we show that the ideal is generated in degree four. We also discuss the coordinate rings of secant varieties of compact Hermitian symmetric spaces.

Equivariant Chow Ring and Chern Classes of Wonderful Symmetric Varieties of Minimal Rank

We describe the equivariant Chow ring of the wonderful compactification X of a symmetric space of minimal rank, via restriction to the associated toric variety Y. Also, we show that the restrictions to Y of the tangent bundle T X and its logarithmic analogue S X decompose into a direct sum of line bundles. This yields closed formulas for the equivariant Chern classes of T X and S X , and, in turn, for the Chern classes of reductive groups considered by Kiritchenko.