Elements of Nonstandard Algebraic Geometry

Algebraic geometry over algebraic structures III: Equationally Noetherian property and compactness

In this paper we discuss some special generalizations of equationally Noetherian property which naturally arise in the universal algebraic geometry. We introduce weakly equationally Noetherian, qw-compact, uw-compact, and weakly uw-compact algebras and then examine properties of such algebras. Also we consider the connections between five classes: the class of equationally Noetherian algebras, the class of weakly equationally Noetherian algebras, the class of uw-compact algebras, the class of weakly uw-compact algebras, and the class of qw-compact algebras.

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.

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。

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。

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.

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.

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.

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.

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.

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.

Birational geometry of algebraic varieties

This is a report on some of the main developments in birational geometry in the last few years focusing on the minimal model program, Fano varieties, singularities and related topics, in characteristic zero.

Improvement on Parameters of Algebraic-Geometry Codes from Hermitian Curves

Motivated by Xing's method [7], we show that there exist [n,k,d] linear Hermitian codes over F_{q^2} with k+d>=n-3 for all sufficiently large q. This improves the asymptotic bound of Algebraic-Geometry codes from Hermitian curves given in [9,10].

Lipschitz geometry of complex plane algebraic curves

We present a complete classification of complex plane algebraic curves, equipped with the induced Euclidean, up to global bilipschitz homeomorphism.

An algebraic geometry perspective on topological data analysis

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

Newton polytopes and numerical algebraic geometry

We develop a collection of numerical algorithms which connect ideas from polyhedral geometry and algebraic geometry. The first algorithm we develop functions as a numerical oracle for the Newton polytope of a hypersurface and is based on ideas of Hauenstein and Sottile. Additionally, we construct a numerical tropical membership algorithm which uses the former algorithm as a subroutine. Based on recent results of Esterov, we give an algorithm which recursively solves a sparse polynomial system when the support of that system is either lacunary or triangular. Prior to explaining these results, w…

形式导出代数几何

Formal Derived Algebraic Geometry

本文通过建立导出与谱代数几何对象间的紧密联系，构建形式导出代数几何的基础，并据此证明导出情形下的形式 GAGA 定理。

Curve selection lemma in infinite dimensional algebraic geometry and arc spaces

We prove a Curve Selection Lemma valid in infinite-dimensional algebraic geometry and apply it, together with a structure theorem of Grinberg-Kahzdan and Drinfeld, to obtain a Curve Selection Lemma in arc spaces.

Notes on derived algebraic geometry

These are notes on derived algebraic geometry in the context of animated rings. More precisely, we recall the proof of Toën-Vaquié that the derived stack of perfect complexes is locally geometric in the language of $\infty$-categories. Along the way, we recall the necessary notions in derived commutative algebra and derived algebraic geometry. We also analyze the deformation theory and quasi-coherent modules over derived stacks.

Semiorthogonal decompositions in algebraic geometry

In this review we discuss what is known about semiorthogonal decompositions of derived categories of algebraic varieties. We review existing constructions, especially the homological projective duality approach, and discuss some related issues such as categorical resolutions of singularities.

Problems for the problem session of Workshop "Affine Algebraic Geometry", Oberwolfach, January 7--14, 2007

Formulated problems concern the following topics: (1) Birationally nonequivalent linear actions; (2) Cayley degrees of simple algebraic groups; (3) Singularities of two-dimensional quotients.

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.

A short survey on Newton polytopes, tropical geometry and ring of conditions of algebraic torus

The purpose of this note is to give an exposition of some interesting combinatorics and convex geometry concepts that appear in algebraic geometry in relation to counting the number of solutions of a system of polynomial equations in several variables over complex numbers. The exposition is aimed for a general audience in mathematics and we hope to be accessible to undergraduate as well as advance high school students. The topics discussed belong to relatively new, and closely related branches of algebraic geometry which are usually referred to as tropical geometry and toric geometry. These ar…

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.

Homological characterization of regularity in Logarithmic Algebraic Geometry

We characterize K. Kato's log regularity in terms of vanishing of (co)homology of the logarithmic cotangent complex.

代数磁性：通过$\mathbb{F}_1$-几何的吸引子的$X$-积

Algebraic Magnetism: $X$-products of attractors via $\mathbb{F}_1$-geometry

本文利用$\mathbb{F}_1$-几何给出了吸引子的$X$-积的公式。

Topology of nonarchimedean analytic spaces and relations to complex algebraic geometry

This note surveys basic topological properties of nonarchimedean analytic spaces, in the sense of Berkovich, including the recent tameness results of Hrushovski and Loeser. We also discuss interactions between the topology of nonarchimedean analytic spaces and classical algebraic geometry.

On the geometry of algebraic groups and homogeneous spaces

Given a connected algebraic group G over an algebraically closed field and a G-homogeneous space X, we describe the Chow ring of G and the rational Chow ring of X, with special attention to the Picard group. Also, we investigate the Albanese and the "anti-affine" fibrations of G and X.

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.

Augmented Homotopical Algebraic Geometry

We develop the framework for augmented homotopical algebraic geometry. This is an extension of homotopical algebraic geometry, which itself is a homotopification of classical algebraic geometry. To do so, we define the notion of augmentation categories, which are a special class of generalised Reedy categories. For an augmentation category, we prove the existence of a closed Quillen model structure on the presheaf category which is compatible with the Kan-Quillen model structure on simplicial sets. Moreover, we use the concept of augmented hypercovers to define a local model structure on the c…

Gromov-Hausdorff limits of Kahler manifolds and algebraic geometry

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

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。

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.

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…

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.

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.

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.

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。