Algebraic Geometry over Lie Algebras

合成代数几何中的Chatelet定理

Chatelet's Theorem in Synthetic Algebraic Geometry

本文在合成代数几何框架下，针对任意基环证明了关于Severi-Brauer簇有理点的Chatelet定理的一个版本。

Recent Trends in Algebraic Geometry -- EuroConference on Algebraic Geometry (Warwick, July 1996)

This file, provided as bibliographical information, contains the table of contents and editors' foreword from the proceedings of the July 1996 Warwick EuroConference on Algebraic Geometry.

Hopf-algebraic techniques applied to super Lie groups over a complete field

We show basic results on super-manifolds and super Lie groups over a complete field of characteristic $\ne 2$, extensively using Hopf-algebraic techniques. The main results are two theorems. The first main theorem shows a category equivalence between super Lie groups and Harish-Chandra pairs, which is applied especially to construct the Hopf super-algebra of all analytic representative functions on a super Lie group. The second constructs homogeneous super-manifolds by a new Hopf-algebraic method, showing their remarkable property.

Differential Galois Theory of Algebraic Lie-Vessiot Systems

In this paper we develop a differential Galois theory for algebraic Lie-Vessiot systems in algebraic homogeneous spaces. Lie-Vessiot systems are non autonomous vector fields that are linear combinations with time-dependent coefficients of fundamental vector fields of an algebraic Lie group action. Those systems are the building blocks for differential equations that admit superposition of solutions. Lie-Vessiot systems in algebraic homogeneous spaces include the case of linear differential equations. Therefore, the differential Galois theory for Lie-Vessiot systems is an extension of the class…

Lectures notes in universal algebraic geometry

Lectures notes in universal algebraic geometry for beginners

奇点理论在应用代数几何与代数统计中的应用

Applications of singularity theory in applied algebraic geometry and algebraic statistics

本文综述了拓扑学与奇点理论在研究应用代数几何和代数统计中具体优化问题的代数复杂度方面的近期应用。

Lie algebras of zero divergence vector fields on complex affine algebraic varieties

For a smooth manifold $X$ equipped with a volume form, let $\dL$ be the Lie algebra of volume preserving smooth vector fields on $X$. A. Lichnerowicz proved that the abelianization of $\dL$ is a finite-dimensional vector space, and that its dimension depends only on the topology of $X$. In this paper we provide analogous results for some classical examples of non-singular complex affine algebraic varieties that admit a nowhere-zero algebraic form of top degree (which plays the role of a volume form).

Scheme theory for groups and Lie algebras

Algebraic geometry for groups and Lie algebraic has been recently defined and studied by many authors on the purpose to study set defined by algebraic equations on abstract groups and Lie algebras. The purpose of this paper is to present a free coordinates approach of this problem. This must be related to the theory of schemes in classical algebraic geometry, and provide a geomotric framework to study finite groups

Real Algebraic Geometry for Geometric Constraints

Real algebraic geometry adapts the methods and ideas from (complex) algebraic geometry to study the real solutions to systems of polynomial equations and polynomial inequalities. As it is the real solutions to such systems modeling geometric constraints that are physically meaningful, real algebraic geometry is a core mathematical input for geometric constraint systems.

超越代数几何

Transcending Algebraic Geometry

本文阐述引导Jacob Tsimerman开创性工作的思想，该工作发展了算术与代数几何中的o-极小可定义性理论。

Contractions of Lie algebras and algebraic groups

Degenerations, contractions and deformations of various algebraic structures play an important role in mathematics and physics. There are many different definitions and special cases of these notions. We try to give a general definition which unifies these notions and shows the connections among them. Here we focus on contractions of Lie algebras and algebraic groups.

Algebraic flows on commutative complex Lie groups

We recover results by Ullmo-Yafaev and Peterzil-Starchenko on the closure of the image of an algebraic variety in a compact complex torus. Our approach uses directed closed currents and allows us to extend the result for dimension 1 flows to the setting of commutative complex Lie groups which are not necessarily compact. A version of the classical Ax-Lindemann-Weierstrass theorem for commutative complex Lie groups is also given.

Reflexive functors in Algebraic Geometry

Reflexive functors of modules naturally appear in Algebraic Geometry. In this paper we define a wide and elementary family of reflexive functors of modules, closed by tensor products and homomorphisms, in which Algebraic Geometry can be developed.

Applications of Algebraic Combinatorics to Algebraic Geometry

We formulate a number of new results in Algebraic Geometry and outline their derivation from Theorem 2.12 which belongs to Algebraic Combinatorics.

The unreasonable effectiveness of wall-crossing in algebraic geometry

We survey applications of Bridgeland stability conditions in algebraic geometry and discuss open questions for future research.

Exercises in the birational geometry of algebraic varieties

This a collection of about 100 exercises. It could be used as a supplement to the book Kollár--Mori: Birational geometry of algebraic varieties.

Geometry on arc spaces of algebraic varieties

This paper is a survey on arc spaces, a recent topic in algebraic geometry and singularity theory. The geometry of the arc space of an algebraic variety yields several new geometric invariants and brings new light to some classical invariants.

Integral Closures In Real Algebraic Geometry

We study the algebraic and geometric properties of the integral closure of different rings of functions on a real algebraic variety : the regular functions and the continuous rational functions.

Parametrizing algebraic varieties using Lie algebras

In the thesis we present a new method for parametrizing algebraic varieties over the field of characteristic zero. The problem of parametrizing is reduced to a problem of finding an isomorphism of algebras. We introduce the Lie algebra of a variety as a Lie algebra related to its group of automorphisms. Constructing an isomorphism of this one and some classical Lie algebra (for example the algebra of matrices of the zero trace) then leads to parametrizing the variety. The problem of finding an isomorphism of Lie algebras is further reduced to trivializing an associative algebra, which means fi…

代数几何漫游

Adventures in algebraic geometry

基于2024年春季罗格斯大学代数几何入门课程，以简化技术细节为代价介绍相对高级的主题。

关于粘合的一个注记：代数几何的支柱

A note on gluing: a pillar of algebraic geometry

本文在一般范畴论框架下考察粘合概念，并阐明其作为代数几何基础支柱的作用。

Semi-algebraic geometry with rational continuous functions

Let X be an affine real algebraic set . We investigate on the theory of algebraically constructible functions on X and the description of the semi-algebraic subsets of X when we replace the polynomial functions on X by some rational continuous functions on X.

Derived Algebraic Geometry

This text is a survey of derived algebraic geometry. It covers a variety of general notions and results from the subject with a view on the recent developments at the interface with deformation quantization.

An algebraic geometry perspective on topological data analysis

A short survey on applications of algebraic geometry in topological data analysis.

线性化函数域的 Riemann–Roch 空间的基及其在广义代数几何码中的应用

Bases for Riemann--Roch spaces of linearized function fields with applications to generalized algebraic geometry codes

该研究通过给出线性化函数域 Riemann–Roch 空间的显式基，建立广义代数几何码最小距离下界，并据此构造参数良好的编码。

Topics in algebraic geometry

Lecture notes of an algebraic geometry graduate course. The topics covered are as follows. Cohomology: ext sheaves and groups, cohomology with support, local cohomology, local duality. Duality: relative duality, Cohen-Macaulay schemes. Morphisms of schemes: flatness and base change and related topics. Hilbert and Quotient schemes.

Formalized proof, computation, and the construction problem in algebraic geometry

An informal discussion of how the construction problem in algebraic geometry motivates the search for formal proof methods. Also includes a brief discussion of my own progress up to now, which concerns the formalization of category theory within a ZFC-like environment.

Tropical algebraic geometry

Tropical algebraic geometry is an active new field of mathematics that establishes and studies some very general principles to translate algebro-geometric problems into purely combinatorial ones. This expository paper gives an introduction to these new techniques with a special emphasis on the recent applications to problems in enumerative geometry.

Seven Lectures on the Universal Algebraic Geometry

Some notions of algebraic geometry can be defined for arbitrary varieties of algebras. This leads to universal algebraic geometry. The main idea of the presented theory is to consider interactions between algebra, logic and geometry in algebras from a given variety of algebras.

What can be computed in algebraic geometry?

This paper is a survey of computational issues in algebraic geometry, with particular attention to the theory of Grobner bases and the regularity of an algebraic variety. 1. A geometric introduction to Grobner bases. 2. An algebraic introduction to Grobner bases. 3. Bounds in algebraic geometry, and regularity and complexity questions. 4. Applications.

Positive characteristic algebraic geometry

These are notes for the Bootcamp volume for the 2015 AMS Summer Institute in Algebraic Geometry. They are based on earlier notes for the "Positive Characteristic Algebraic Geometry Workshop" held at University of Illinois at Chicago in March 2014.

A note on the cotangent complex in derived algebraic geometry

This note is supposed to answer some questions on deformation theory in derived algebraic geometry. We show that derived algebraic geometry allows for a geometrical interpretation of the full cotangent complex and gives a natural setting for deformation and obstruction theories.

变种对称李代数及其在代数统计中的应用

Symmetry Lie Algebras of Varieties with Applications to Algebraic Statistics

本文通过分析与不可约射影簇相关的李代数，提出了一种检测非toric统计模型的方法，并提供了相关计算算法和实例。

On Algebraic-Geometry Approach to Ribaucour Transformations

We develop the idea of using an algebraic-geometry approach to classical differential geometry problems. Consider an orthogonal net constructed according to algebraic-geometric data we obtain a set of smooth orthogonal nets that are Ribaucour transformations of the initial orthogonal net.

Introduction to tropical algebraic geometry

This is an expository introduction to tropical algebraic geometry based on my lectures at the Workshop on Tropical Geometry and Integrable Systems in Glasgow, July 4-8, 2011, and at the ELGA 2011 school on Algebraic Geometry and Applications in Buenos Aires, August 1-5, 2011.

谱代数几何中拟凝聚层的代数模型

Algebraic Models for Quasi-Coherent Sheaves in Spectral Algebraic Geometry

本文通过适配同调理论，证明了稳定同伦论和谱代数几何中特定非连通几何栈上拟凝聚层存在代数模型。

Algebraic supergroups with Lie superalgebras of classical type

We show that every connected affine algebraic supergroup defined over a field k, with diagonalizable maximal torus and whose tangent Lie superalgebra is a k-form of a complex simple Lie superalgebra of classical type is a Chevalley supergroup, as it is defined and constructed explicitly in [R. Fioresi, F. Gavarini, "Chevalley Supergroups", Memoirs of the AMS 215 (2012), no. 1014].

Algebraic geometry of Lie bialgebras defined by solutions of the classical Yang-Baxter equation

This paper is devoted to algebro-geometric study of infinite dimensional Lie bialgebras, which arise from solutions of the classical Yang-Baxter equation. We regard trigonometric solutions of this equation as twists of the standard Lie bialgebra cobracket on an appropriate affine Lie algebra and work out the corresponding theory of Manin triples, putting it into an algebro-geometric context. As a consequence of this approach, we prove that any trigonometric solution of the classical Yang-Baxter equation arises from an appropriate algebro-geometric datum. The developed theory is illustrated by …

Gromov-Hausdorff limits of Kahler manifolds and algebraic geometry

We prove a general result about the geometry of holomorphic line bundles over Kahler manifolds.

On group automorphisms in universal algebraic geometry

In this paper, we study group equations with occurrences of automorphisms. We describe equational domains in this class of equations. Moreover, we solve a number of open problem posed in universal algebraic geometry.

Gromov-Hausdorff limits of Kahler manifolds and algebraic geometry, II

In this paper we continue to study Gromov-Hausdorff limits of Kahler manifolds and algebraic geometry. Our main focus is on the algebro-geometric meaning of Riemannian tangent cones and rescaled limits.

Rationality in Differential Algebraic Geometry

Parametric Cartan theory of exterior differential systems, and explicit cohomology of projective manifolds reveal united rationality features of differential algebraic geometry.

Reflection groups in algebraic geometry

This is a survey on appearances of reflection groups, real and complex, in algebraic geometry. We also include a brief introduction into the theory of reflection groups.

Counting Algebraic Curves with Tropical Geometry

Tropical geometry is a piecewise linear "shadow" of algebraic geometry. It allows for the computation of several cohomological invariants of an algebraic variety. In particular, its application to enumerative algebraic geometry led to significant progress. In this survey, we give an introduction to tropical geometry techniques for algebraic curve counting problems. We also survey some recent developments, with a particular emphasis on the computation of the degree of the Severi varieties of the complex projective plane and other toric surfaces as well as Hurwitz numbers and applications to rea…

nilpotent Lie algebras的gradings与smooth complex algebraic varieties的nilpotent fundamental groups相关

Gradings on nilpotent Lie algebras associated with the nilpotent fundamental groups of smooth complex algebraic varieties

本文研究了nilpotent Lie代数的gradings与smooth complex algebraic varieties的nilpotent fundamental groups之间的关系，并给出了晶格同构于基本群的必要条件。

什么是……随机代数几何？

What is... Random Algebraic Geometry?

本文综述随机代数几何，该领域将概率视角引入实代数几何，为希尔伯特第十六问题等经典难题提供现代解决途径。

Erratum to: Deformation Quantization in Algebraic Geometry

This note contains a correction of the proofs of the main results of the paper [A. Yekutieli, Deformation quantization in algebraic geometry, Adv. Math. 198 (2005), 383-432]. The results are correct as originally stated.

Algebraic Geometry over model categories (a general approach to derived algebraic geometry)

For a (semi-)model category M, we define a notion of a ''homotopy'' Grothendieck topology on M, as well as its associated model category of stacks. We use this to define a notion of geometric stack over a symmetric monoidal base model category; geometric stacks are the fundamental objects to "do algebraic geometry over model categories". We give two examples of applications of this formalism. The first one is the interpretation of DG-schemes as geometric stacks over the model category of complexes and the second one is a definition of etale K-theory of E_{\infty}-ring spectra. This first versi…

Numerical Algebraic Geometry for Macaulay2

Numerical Algebraic Geometry uses numerical data to describe algebraic varieties. It is based on the methods of numerical polynomial homotopy continuation, an alternative to the classical symbolic approaches of computational algebraic geometry. We present a package, the driving idea behind which is to interlink the existing symbolic methods of Macaulay2 and the powerful engine of numerical approximate computations. The core procedures of the package exhibit performance competitive with the other homotopy continuation software.

Quantum entanglement, Calabi-Yau manifolds, and noncommutative algebraic geometry

We relate SLOCC equivalence classes of qudit states to moduli spaces of Calabi-Yau manifolds equipped with a collection of line bundles. The cases of 3 qutrits and 4 qubits are also related to noncommutative algebraic geometry.

Algebraic geometry over free semilattice

We deal with equations over free semilattice of infinite rank and prove that any infinite consistent system of equations is equivalent to its finite subsystem. Moreover, we describe irreducible algebraic sets and solve some algorithmic problems.

Algebraic Geometry of Bayesian Networks

We study the algebraic varieties defined by the conditional independence statements of Bayesian Networks. A complete algebraic classification is given for Bayesian Networks on at most five random variables. Hidden variables are related to the geometry of higher secant varieties.

On cyclic algebraic-geometry codes

In this paper we initiate the study of cyclic algebraic geometry codes. We give conditions to construct cyclic algebraic geometry codes in the context of algebraic function fields over a finite field by using their group of automorphisms. We prove that cyclic algebraic geometry codes constructed in this way are closely related to cyclic extensions. We also give a detailed study of the monomial equivalence of cyclic algebraic geometry codes constructed with our method in the case of a rational function field.

An Introduction to Algebraic Geometry codes

We present an introduction to the theory of algebraic geometry codes. Starting from evaluation codes and codes from order and weight functions, special attention is given to one-point codes and, in particular, to the family of Castle codes.