Caucher Birkar
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
This is a survey paper about a selection of results in complex algebraic geometry that appeared in the recent and less recent litterature, and in which rational homogeneous spaces play a prominent r{ô}le. This selection is largely arbitrary and mainly reflects the interests of the author.
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.
Lectures notes in universal algebraic geometry for beginners
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 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 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.
We survey applications of Bridgeland stability conditions in algebraic geometry and discuss open questions for future research.
This a collection of about 100 exercises. It could be used as a supplement to the book Kollár--Mori: Birational geometry 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.
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.
Topics in group schemes and surfaces in positive characteristic
该综述基于作者2023年在爱丁堡大学的讲座，介绍正特征下代数曲面中的无穷小群概形、群作用及Kodaira消没定理失效等现象，并辅以显式例子。
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。
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。
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.
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.
This is a survey on the topic explained in the title, for the proceedings on the K-theory 1997 summer institute in Seattle.
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。
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 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.
Convexity and Concavity from Complex Variables to Algebraic Geometry
本文针对复分析中核心的凸性、凹性及包络主题，探索其在代数几何中的对应内容，并呈现了该方向的初步研究结果。
We discuss selected topics on the topology of moduli spaces of curves and maps, emphasizing their relation with Gromov--Witten theory and integrable systems.
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.
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].
We present a complete classification of complex plane algebraic curves, equipped with the induced Euclidean, up to global bilipschitz homeomorphism.
This is a survey paper about a selection of recent results on the geometry of a special class of Fano varieties, which are called of K3 type. The focus is mostly Hodge-theoretical, with an eye towards the multiple connections between Fano varieties of K3 type and hyperkähler 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…
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。
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.
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.
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.
Formulated problems concern the following topics: (1) Birationally nonequivalent linear actions; (2) Cayley degrees of simple algebraic groups; (3) Singularities of two-dimensional quotients.
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…
We characterize K. Kato's log regularity in terms of vanishing of (co)homology of the logarithmic cotangent complex.
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。
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.
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.
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…
Homemade Algebraic Geometry. Celebrating Enrique Arrondo's 60th birthday
本文综述恩里克·阿隆多的职业生涯，回顾其专业历史并汇总其数学研究成果，以庆祝其60岁生日。
We compute the first, second, third, and fifth rational cohomology groups of the moduli space of stable n-pointed genus g curves, for all g and n, using (mostly) algebro-geometric techniques.
The singularities of theta divisors have played an important role in the study of algebraic varieties. This paper surveys some of the recent progress in this subject, using as motivation some well known results, especially those for Jacobians.
This survey paper is devoted to Riemannian manifolds with special holonomy. To any Riemannian manifold of dimension n is associated a closed subgroup of SO(n), the holonomy group; this is one of the most basic invariants of the metric. A famous theorem of Berger gives a complete (and rather small) list of the groups which can appear. Surprisingly, the compact manifolds with holonomy smaller than SO(n) are all related in some way to Algebraic Geometry. This leads to the study of special algebraic varieties (Calabi-Yau, complex symplectic or complex contact manifolds) for which Riemannian geomet…
Some problems at the interface of approximation theory and algebraic geometry
本文概述了2022年9月于科尔托纳举行的“逼近理论与数值分析遇见代数、几何与拓扑”会议上提出但尚未解决的若干交叉领域问题。
A mixture of an historical article, and of a survey of recent developments, containing also a couple of new results.
Real algebraic geometry is the study of semi-algebraic sets, subsets of $\R^k$ defined by Boolean combinations of polynomial equalities and inequalities. The focus of this thesis is to study quantitative results in real algebraic geometry, primarily upper bounds on the topological complexity of semi-algebraic sets as measured, for example, by their Betti numbers. Another quantitative measure of topological complexity which we study is the number of homotopy types of semi-algebraic sets of bounded description complexity. The description complexity of a semi-algebraic set depends on the context,…
The special geometry of calibrated cycles, closely related to mirror symmetry among Calabi--Yau 3-folds, is itself a real form of a new subject, which we call slightly deformed algebraic geometry. On the other hand, both of these geometries are parallel to classical gauge theories and their complexifications. This article explains this parallelism, so that the appearance of invariants of new type in complexified gauge theory (see Donaldson--Thomas [D-T] and Thomas [T]) can be accompanied by analogous invariants in the theory of special Lagrangian cycles, for which the development is at present…
On logarithmic paper some real algebraic curves look like smoothed broken lines. Moreover, the broken lines can be obtained as limits of those curves. The corresponding deformation can be viewed as a quantization, in which the broken line is a classical object and the curves are quantum. This generalizes to a new connection between algebraic geometry and the geometry of polyhedra, which is more straight-forward than the other known connections and gives a new insight into constructions used in the topology of real algebraic varieties.
In this paper we will introduce a certain type of morphisms of log schemes (in the sense of Fontaine, Illusie, and Kato) and investigate their moduli. Then by applying this we define a notion of toric algebraic stacks over arbitrary schemes, which may be regarded as torus embeddings within the framework of algebraic stacks, and study some basic properties. Furthermore, we study the stack-theoretic analogue of toroidal embeddings.
We formulate a number of new results in Algebraic Geometry and outline their derivation from Theorem 2.12 which belongs to Algebraic Combinatorics.
In this paper we introduce elements of algebraic geometry over an arbitrary algebraic structure. We prove Unification Theorems which gather the description of coordinate algebras by several ways.
A short survey on applications of algebraic geometry in topological data analysis.
In this paper, for a given finitely generated algebra (an algebraic structure with arbitrary operations and no predicates) A we study finitely generated limit algebras of A, approaching them via model theory and algebraic geometry. Along the way we lay down foundations of algebraic geometry over arbitrary algebraic structures.
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.
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.
This is a survey paper on Alegbraic Geometry over Lie Algebras
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.
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.
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.
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.
We prove a general result about the geometry of holomorphic line bundles over Kahler manifolds.
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.
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。