Algebraic Geometry Research Articles

Stopping Sets of Algebraic Geometry Codes over Hyperelliptic Curves of Genus Two

Stopping sets are useful for analyzing the performance of a linear code under an iterative decoding algorithm over an erasure channel. In this paper, we consider stopping sets of one-point algebraic geometry codes defined by a hyperelliptic curve of genus g=2 defined by the plane model y2=f(x), where the degree of f(x) was 5. We completely classify the stopping sets of the one-point algebraic geometric codes C=CΩ(D,mP∞) defined by a hyperelliptic curve of genus 2 with m≤4. For m=3, we proved in detail that all sets S⊆{1,2,⋯,n} of a size greater than 3 are stopping sets and we give an example of sets of size 2,3 that are not.

An interesting wall-crossing: failure of the wall-crossing/MMP correspondence

We introduce a novel wall-crossing phenomenon in the space of Bridgeland stability conditions: a wall in the stability space of canonical genus 4 curves that induces non-Q-factorial singularities and hence, it cannot be detected as an operation in the Minimal Model Program of the corresponding moduli space, unlike the case for many surfaces. More precisely, we give an example of a wall-crossing in Db(P3)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{D}^{b}(\\mathbb {P}^{3})$$\\end{document} such that the wall induces a small contraction of the moduli space of stable objects associated to one of the adjacent chambers, but a divisorial contraction to the other. This significantly complicates the overall picture in this correspondence to applications of stability conditions to algebraic geometry. The full wall-crossing for canonical genus four curves and the geometry are considered in the published paper (Rezaee in Proc LMS 128(1):e12577, 2024); this article is devoted to describe a particularly interesting wall among the walls in Rezaee (Proc LMS 128(1):e12577, 2024) in full details to explain the novel phenomenon.

EXTENDING TORSORS UNDER QUASI-FINITE FLAT GROUP SCHEMES

Abstract Let R be a discrete valuation ring of field of fractions K and of residue field k of characteristic $p> 0$ . In an earlier work, we studied the question of extending torsors over K-curves into torsors over R-regular models of the curves in the case when the structural K-group scheme of the torsor admits a finite flat model over R. In this paper, we first give a simpler description of the problem in the case where the curve is semistable using recent work in Holmes, Molcho, Orecchia, and Poiret (2023, Journal für die Reine und Angewandte Mathematik [Crelle’s Journal] 230, 115–159) and Molcho and Wise (2022, Compositio Mathematica 158, 1477–1562). Second, if R is assumed to be Henselian and Japanese, we solve the problem of extending torsors by combining our previous work together with results in Antei and Emsalem (2018, Nagoya Mathematical Journal 230, 18–34) and Phung and Dos Santos (2023, Algebraic Geometry 230, 1–40), including the case where the structural group does not admit a finite flat R-model.

Role of Linear Diophantine Equations in RSA Encryption

Abstract. Diophantine equations are mathematical equations that include only integer solutions and two or more unknown variables. To illustrate, the most elementary form of Diophantine equations are linear Diophantine equations when two different one-degree monomials added up to a constant value. Such equations were given the name of the extraordinary Greek mathematician Diophantus who made great contribution to algebra and number theory. In addition, they help in defining concepts in algebraic geometry and contribute to the development of algorithms for cryptography. They work in the same way as public keys do in cryptocurrencies within blockchain technology to curb cyber threats and scams targeting public systems. They serve as a means of securing transactions on a computer network when using cryptocurrencies such as bitcoins. These encryption techniques also permit cryptographers and mathematicians to create new ideas in relation to these number systems, thus making it possible for further cryptographic techniques to be put into place.

Algebraic algorithms for vector bundles over curves

In this paper, we represent vector bundles over a regular algebraic curve as pairs of lattices over the maximal orders of its function field and we give polynomial time algorithms for several tasks: computing determinants of vector bundles, kernels and images of global homomorphisms, isomorphisms between vector bundles, cohomology groups, extensions, and splitting into a direct sum of indecomposables. Most algorithms are deterministic except for computing isomorphisms when the base field is infinite. Some algorithms are only polynomial time if we may compute Hermite forms of pseudo-matrices in polynomial time. All algorithms rely exclusively on algebraic operations in function fields. For applications, we give an algorithm enumerating isomorphism classes of vector bundles on an elliptic curve, and to construct algebraic geometry codes over vector bundles. We implement all our algorithms into a SageMath package.

Flat comodules and contramodules as directed colimits, and cotorsion periodicity

This paper is a follow-up to Positselski and Št’ovíček (Flat quasi-coherent sheaves as directed colimits, and quasi-coherent cotorsion periodicity. Electronic preprint arXiv:2212.09639 [math.AG]). We consider two algebraic settings of comodules over a coring and contramodules over a topological ring with a countable base of two-sided ideals. These correspond to two (noncommutative) algebraic geometry settings of certain kind of stacks and ind-affine ind-schemes. In the context of a coring C\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {C}}$$\\end{document} over a noncommutative ring A, we show that all A-flat C\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {C}}$$\\end{document}-comodules are ℵ1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\aleph _1$$\\end{document}-directed colimits of A-countably presentable A-flat C\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {C}}$$\\end{document}-comodules. In the context of a complete, separated topological ring R\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathfrak {R}}$$\\end{document} with a countable base of neighborhoods of zero consisting of two-sided ideals, we prove that all flat R\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathfrak {R}}$$\\end{document}-contramodules are ℵ1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\aleph _1$$\\end{document}-directed colimits of countably presentable flat R\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathfrak {R}}$$\\end{document}-contramodules. We also describe arbitrary complexes, short exact sequences, and pure acyclic complexes of A-flat C\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {C}}$$\\end{document}-comodules and flat R\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathfrak {R}}$$\\end{document}-contramodules as ℵ1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\aleph _1$$\\end{document}-directed colimits of similar complexes of countably presentable objects. The arguments are based on a very general category-theoretic technique going back to an unpublished 1977 preprint of Ulmer and rediscovered in Positselski (Notes on limits of accessible categories. Electronic preprint arXiv:2310.16773 [math.CT]). Applications to cotorsion periodicity and coderived categories of flat objects in the respective settings are discussed. In particular, in any acyclic complex of cotorsion R\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathfrak {R}}$$\\end{document}-contramodules, all the contramodules of cocycles are cotorsion.

Smooth connectivity in real algebraic varieties

AbstractA standard question in real algebraic geometry is to compute the number of connected components of a real algebraic variety in affine space. This manuscript provides algorithms for computing the number of connected components, the Euler characteristic, and deciding the connectivity between two points for a smooth manifold arising as the complement of a real hypersurface of a real algebraic variety. When considering the complement of the set of singular points of a real algebraic variety, this yields an approach for determining smooth connectivity in a real algebraic variety. The method is based upon gradient ascent/descent paths on the real algebraic variety inspired by a method proposed by Hong, Rohal, Safey El Din, and Schost for complements of real hypersurfaces. Several examples are included to demonstrate the approach.

Log-Cholesky filtering of diffusion tensor fields: Impact on noise reduction

Diffusion tensor imaging (DTI) is a powerful neuroimaging technique that provides valuable insights into the microstructure and connectivity of the brain. By measuring the diffusion of water molecules along neuronal fibers, DTI allows the visualization and study of intricate networks of neural pathways.DTI is a noise-sensitive method, where a low signal-to-noise ratio (SNR) results in significant errors in the estimated tensor field. Tensor field regularization is an effective solution for noise reduction.Diffusion tensors are represented by symmetric positive-definite (SPD) matrices. The space of SPD matrices may be viewed as a Riemannian manifold after defining a suitable metric on its tangent bundle. The Log-Cholesky metric is a recently developed concept with advantages over previously defined Riemannian metrics, such as the affine-invariant and Log-Euclidean metrics. The utility of the Log-Cholesky metric for tensor field regularization and noise reduction has not been investigated in detail.This manuscript provides a quantitative investigation of the impact of Log-Cholesky filtering on noise reduction in DTI. It also provides sufficient details of the linear algebra and abstract differential geometry concepts necessary to implement this technique as a simple and effective solution to filtering diffusion tensor fields.

On the closure of the Airault–Mckean–Moser locus for elliptic KdV potentials via Darboux transformations

We study the general elliptic KdV potentials, which can be expressed (up to adding a constant) as q_{\mathbf{p}}(z)\coloneq\sum_{j=1}^{n}m_{j}(m_{j}+1)\wp(z-p_{j}),\quad m_{j}\in\mathbb{N}. We give an elementary proof of the theorem that the singularity \mathbf{p}=( \overset{m_{1}( m_{1}+1) /2}{\overbrace{p_{1},\dots ,p_{1}}},\dots,\overset{m_{n}(m_{n}+1) /2}{\overbrace{p_{n},\dots,p_{n}}% }) is contained in the closure of the elliptic Airault–Mckean–Moser locus, which was proved previously by Treibich and Verdier in the late 1980s using purely algebro-geometric methods. Our proof is based on Darboux transformations and does not use algebraic geometry. This solves an open problem posed by Gesztesy, Unterkofler, and Weikard [Trans. Amer. Math. Soc. 358 (2006), 603–656]. Some applications are also given.

Pure Rickart Modules

This paper focuses on the study of pure Rickart modules (or PR-modules for short), which are a class of modules over a commutative ring with identity. The main objective is to investigate the properties and characterizations of these modules, as well as their relationship with other classes of modules such as free, projective, and flat modules. This paper also explores the connections between PR-modules and various algebraic structures such as rings. The results obtained in this study provide a deeper understanding of the structure and behavior of PR-modules, which can have important applications in algebraic geometry, representation theory, and other areas of mathematics. Some results about PR-module have been investigated in this paper, for example we demonstrate a module is PR-module if and only if for every , is pure submodule of (or for short). Also, some kind of generalization of these rings have been constructed and demonstrated in term of PR-modules.

Intersections of Algebraic Geometry and Coding Theory: A Study of Error-Correcting Codes

Intersections of Algebraic Geometry and Coding Theory: A Study of Error-Correcting Codes

On Non-Commutative Multi-Rings with Involution

The primary motivation for this work is to develop the concept of Marshall’s quotient applicable to non-commutative multi-rings endowed with involution, expanding upon the main ideas of the classical case—commutative and without involution—presented in Marshall’s seminal paper. We define two multiplicative properties to address the involutive case and characterize their Marshall quotient. Moreover, this article presents various cases demonstrating that the “multi” version of rings with involution offers many examples, applications, and relatives in (multi)algebraic structures. Therefore, we established the first steps toward the development of an expansion of real algebra and real algebraic geometry to a non-commutative and involutive setting.

Exploring Ground and Excited States Via Single Reference Coupled-Cluster Theory and Algebraic Geometry.

The exploration of the root structure of coupled cluster (CC) equations holds both foundational and practical significance for computational quantum chemistry. This study provides insight into the intricate root structures of these nonlinear equations at both the CCD and CCSD level of theory. We utilize computational techniques from algebraic geometry, specifically the monodromy and parametric homotopy continuation methods, to calculate the full solution set. We compare the computed CC roots against various established theoretical upper bounds, shedding light on the accuracy and efficiency of these bounds. We hereby focus on the dissociation processes of four-electron systems such as (H2)2 in both D2h and D∞h configurations, H4 symmetrically distorted on a circle, and lithium hydride. We moreover investigate the ability of single-reference CC solutions to approximate excited state energies. We find that multiple CC roots describe energies of excited states with high accuracy. Our investigations reveal that for systems like lithium hydride, CC not only provides high-accuracy approximations to several excited state energies but also to the states themselves.

Discussion on optical solitons, sensitivity and qualitative analysis to a fractional model of ion sound and Langmuir waves with Atangana Baleanu derivatives

Abstract This research explores new soliton solutions to the Atangana–Baleanu derivative (ABD) fractional system of equations for ion sound and Langmuir waves (FISLW). We utilize the fractional ABD operator to convert our system into an ordinary differential equations. In recent years, machine learning (ML) evolves significantly in the context of data analysis and computing different solutions, which typically enables systems to operate wisely. Now, we are going to use numerous ML tools including matplotlib. pyplot as plt, scipy.integrate, mpl toolkits.mplot3d, and Axes3D to generate various types of optical solutions by using complete discriminant of the polynomial method. We will also analyze solutions for the hyperbolic function, trigonometric function, Jacobian elliptic function (JEF), and other solitary wave solutions. Solitons have extensive uses in pure and applied mathematics, including nonlinear partial differential equations: the Boussinesq equation, the nonlinear Schrödinger equation, and the sine-Gordon equation, Lie groups, Lie algebras, and differential and algebraic geometry. In addition, we study the chaotic behaviour, i.e., 2D, 3D, time series, Poincarè maps, and sensitivity analysis of our governing model. Sensitivity analysis explores how changes in a system’s variables affects its behaviour.

Embedding Method by Real Numerical Algebraic Geometry for Structurally Unamenable Differential-Algebraic Equations

Embedding Method by Real Numerical Algebraic Geometry for Structurally Unamenable Differential-Algebraic Equations

On Łojasiewicz inequalities and the effective Putinar's Positivstellensatz

The representation of positive polynomials on a semi-algebraic set in terms of sums of squares is a central question in real algebraic geometry, which the Positivstellensatz answers. In this paper, we study the effective Putinar's Positivestellensatz on a compact basic semi-algebraic set S and provide a new proof and new improved bounds on the degree of the representation of positive polynomials. These new bounds involve a parameter ε measuring the non-vanishing of the positive function, the constant c and exponent L of a Łojasiewicz inequality for the semi-algebraic distance function associated to the inequalities g=(g1,…,gr) defining S. They are polynomial in c and ε−1 with an exponent depending only on L. We analyse in details the Łojasiewicz inequality when the defining inequalities g satisfy the Constraint Qualification Condition. We show that, in this case, the Łojasiewicz exponent L is 1 and we relate the Łojasiewicz constant c with the distance of g to the set of singular systems.

Motives

Making a survey of recent constructions of universal cohomologies we suggest a new framework for a theory of motives in algebraic geometry.

A Mathematical World Woven by Number Theory, Algebraic Geometry, and Representation Theory

A Mathematical World Woven by Number Theory, Algebraic Geometry, and Representation Theory

Weil cohomology theories and their motivic Hopf algebroids

In this paper we discuss a general notion of Weil cohomology theories, both in algebraic geometry and in rigid analytic geometry. We allow our Weil cohomology theories to have coefficients in arbitrary commutative ring spectra. Using the theory of motives, we give three equivalent viewpoints on Weil cohomology theories: as a cohomology theory on smooth varieties, as a motivic spectrum and as a realization functor. We also associate to every Weil cohomology theory a motivic Hopf algebroid generalizing the construction we gave in Ayoub (2014) for the Betti cohomology. Exploiting results and constructions from Ayoub (2020), we are able to prove that the motivic Hopf algebroids of all the classical Weil cohomology theories are connective. In particular, they give rise to motivic Galois groupoids which are spectral affine groupoid schemes.

A positive/tropical critical point theorem and mirror symmetry

Call a Laurent polynomial W ‘complete’ if its Newton polytope is full-dimensional with zero in its interior. Suppose W is a Laurent polynomial with coefficients in the positive part of the field of (generalised) Puiseaux series. Here a Puiseaux or generalised Puiseux series (with exponents in R) is called ‘positive’ if the coefficient of its leading term is in R>0. We show that W has a unique positive critical point pcrit, i.e. all of whose coordinates are positive, if and only if W is complete. For any complete, positive Laurent polynomial W in r variables we also obtain from its positive critical point pcrit a canonically associated ‘tropical critical point’ dcrit∈Rr by considering the valuations of the coordinates of pcrit. Moreover we give a finite recursive construction of dcrit in terms of a generalisation of the Newton polytope that we call the ‘Newton datum’ of W.We show that this result is compatible with a general form of mutation, so that it can be applied in a cluster varieties setting. We also show that our theorem carries over to the case where the exponents of monomials appearing in W are not integral but in R, even though W is then no longer Laurent.Finally, we describe applications to both algebraic and symplectic toric geometry inspired by mirror symmetry. On the one hand, in the algebraic context of a complete toric variety XΣ we apply our results to obtain for any divisor class [D] satisfying a certain integrality property, a canonical choice of torus-invariant representative. This generalises the standard toric boundary divisor of XΣ to divisor classes other than the anti-canonical class. On the other hand, our result generalises a result of [11] and relates to the construction of canonical non-displaceable Lagrangian tori for toric symplectic orbifolds using [13,37].