Using Algebraic Geometry pdf epub mobi txt 电子书 下载 2026

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出版者:Springer

作者:David A. Cox

出品人:

页数:520

译者:

出版时间:1998-08-13

价格:USD 44.95

装帧:Paperback

isbn号码:9780387984926

丛书系列:

图书标签:

• 代数几何
• 数学
• Geometry
• Algebraic
• 应用
• 代数几何7
• basis
• Using
• 代数几何
• 代数
• 几何
• 数学
• 抽象代数
• 代数拓扑
• 交换代数
• 代数簇
• 射影几何
• 编码理论

Exploring the Depths of Shapes and Spaces: A Journey into the Realm of Algebraic Geometry This book invites you on an intellectual adventure, delving into the captivating world of algebraic geometry. While the name "Using Algebraic Geometry" might suggest a purely practical application, this exploration ventures deeper, unearthing the fundamental principles and elegant structures that underpin this powerful field. Prepare to transcend mere utility and discover the inherent beauty and profound insights that algebraic geometry offers. Our journey begins with the very essence of algebraic geometry: the intricate dance between algebra and geometry. We will uncover how equations, typically the domain of abstract manipulation, find their most intuitive and vibrant expression as geometric shapes – curves, surfaces, and more complex manifolds. Conversely, we will witness how geometric properties, such as tangency, intersection, and singularity, can be precisely described and analyzed through algebraic means. This duality is not just a curious observation; it forms the bedrock upon which all of algebraic geometry is built, offering a unique lens through which to perceive and understand mathematical reality. We will start by laying a solid foundation, revisiting essential concepts from classical algebraic geometry. This includes an in-depth examination of affine and projective varieties. Imagine the difference between a shape existing in a standard coordinate plane versus one that seamlessly extends to infinity – this distinction is crucial, and we will meticulously explore how algebraic equations define these distinct geometric spaces. Understanding the nuances of these varieties, the sets of points satisfying polynomial equations, will equip you with the fundamental vocabulary and conceptual framework to navigate more advanced topics. We will dissect the properties of these geometric objects, such as their dimension, irreducibility, and the nature of their singularities – points where the geometric object might behave in a less smooth or predictable manner. As we progress, the focus will shift towards the algebraic structures that encode these geometric realities. Polynomial rings, ideals, and quotient rings will emerge not as abstract algebraic constructs, but as powerful tools for characterizing and classifying varieties. We will learn how the algebraic properties of these rings directly translate into the geometric features of the corresponding varieties. This connection will reveal a deep, harmonious relationship, where algebraic operations mirror geometric transformations and algebraic invariants correspond to geometric invariants. This is where the "algebraic" in algebraic geometry truly shines, providing a rigorous and systematic way to study geometric phenomena. A significant portion of our exploration will be dedicated to the concept of schemes. This advanced framework, developed by Alexander Grothendieck, revolutionized algebraic geometry by generalizing the notion of a variety. Schemes allow us to extend the principles of algebraic geometry to a broader range of mathematical objects, including those that might not have a direct geometric interpretation in the classical sense. Think of it as an expansion of our geometric intuition, allowing us to "see" geometric structures in algebraic situations that were previously inaccessible. We will meticulously unpack the definition of a scheme, understanding its local nature and how it elegantly unifies different mathematical perspectives. This will involve delving into concepts like the category of rings and their relationship to the category of schemes, highlighting the power of abstract frameworks in unifying diverse mathematical ideas. Furthermore, we will investigate key concepts within the theory of schemes. The notion of coherent sheaves, for instance, will be explored in detail. These sheaves act as generalized functions or geometric objects defined on schemes, and their study reveals profound information about the underlying geometric structure. We will understand how coherent sheaves provide a sophisticated way to analyze properties like local freeness and the dimension of a scheme. The interplay between schemes and their sheaves will illuminate a rich and interconnected landscape, where algebraic and geometric properties are deeply intertwined. The book will also touch upon certain intrinsic properties of varieties and schemes. Concepts such as the dimension of a variety, its genus (a topological invariant that reveals information about the "holes" in a surface), and the degree of a variety will be presented not just as definitions, but as tools for classification and understanding. We will explore how these properties can be computed and how they provide valuable insights into the fundamental nature of the geometric objects we are studying. Throughout this exploration, the emphasis will be on developing a deep conceptual understanding, rather than on presenting a mere collection of theorems and techniques. We aim to foster an appreciation for the elegance and power of algebraic geometry as a unified mathematical discipline. While the name might suggest "using" it, our goal is to equip you with the foundational knowledge and the intellectual framework to truly comprehend and appreciate the beauty and utility of algebraic geometry, enabling you to then confidently apply its principles to a vast array of problems in mathematics and beyond. This book is an invitation to see the world through a new, mathematically enriched perspective, where abstract equations blossom into tangible shapes and geometric intuition is grounded in rigorous algebraic structure.

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这本书在阐述代数几何的“内在联系”方面做得非常出色，它不仅仅是代数和拓扑的简单叠加，而是在不断地提醒读者，这里的每一个代数结构都具有深刻的几何意义。我最欣赏的是它对经典代数几何（如黎曼-罗赫定理）与现代概形理论之间桥梁的构建。作者并没有将两者割裂开来，而是展示了现代语言如何更优雅、更普适地去表达经典的结果。特别是关于布川上同调群（Betti numbers）的讨论部分，它巧妙地将拓扑的贝蒂数与代数环的结构联系起来，展现了数学分支之间那种令人赞叹的统一性。这种跨越不同领域的视野，使得这本书的知识结构非常立体和丰满。它不像有些书籍那样，只停留在某个特定流派的叙事中，而是试图描绘出整个代数几何领域的全景图，尽管这幅图景无疑是宏大而复杂的。阅读这本书的过程，就是不断发现数学世界中隐藏联系的过程。

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阅读体验上，这本书给我的感觉是它更侧重于“解决问题”的工具箱，而非“历史溯源”的散文集。它在介绍关键定理时，往往会直接给出最有效、最精炼的证明路径，很少会花费篇幅去探讨某个概念是如何被历史上的数学家发现或演变的。例如，在阐述希尔伯特零点定理（Hilbert's Nullstellensatz）时，它直接聚焦于代数到几何的精确对应，然后迅速转向如何利用这个工具去研究曲线和曲面的交点。这种务实的态度对于需要快速掌握代数几何来应用到其他领域（比如密码学或理论物理）的读者来说，无疑是极大的福音。它更像是一本精心编写的参考手册，其中包含了大量精心挑选的习题——这些习题的难度分布非常合理，从基础的验证性练习到需要综合多个章节知识的挑战性问题都有涵盖。我特别喜欢它在章节末尾提供的“拓展阅读”建议，这些建议精准地指向了不同方向的进阶主题，帮助读者在学完基础后知道下一步该往哪里深入探索。

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我花了相当长的时间来消化这本书中的某些章节，坦白说，有些地方的难度曲线陡峭得令人心惊。它毫不避讳地深入到非常现代和技术性的细节中去，尤其是在讨论向量丛（vector bundles）和层上同调（sheaf cohomology）的那几部分。这绝不是一本供“泛泛了解”代数几何的入门读物，它更像是为那些已经掌握了基础代数和拓扑知识，并准备向研究领域迈进的读者准备的。我对作者在处理范畴论语言时的处理方式感到既敬佩又头疼。尽管他努力用具体例子来锚定抽象概念，但一旦进入到函子（functors）和自然变换的讨论，那份抽象的密度就瞬间飙升。我发现自己不得不频繁地查阅补充材料，以确认自己对某些关键引理的理解是否到位。这本书的风格是那种“一次到位、拒绝回头”式的论证，一旦某个概念被建立起来，后续的推导就建立在其坚实的基础之上，几乎没有冗余的解释，这要求读者必须保持高度的专注力。可以说，这本书是那种读完后会让你感觉智力被榨干，但同时也收获了坚实基础的典范。

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如果让我用一个词来形容这本书的语言风格，那就是“精准到近乎冷酷”。它在数学表达上几乎不容忍任何模棱两可之处。对于那些习惯了更具文学色彩或更“温暖”的数学写作风格的读者来说，这本书的开场可能会显得有些生硬。它不拐弯抹角，直接切入核心定义和公理。这种风格的优点是效率极高，每一个句子都承载了大量的数学信息；缺点是，如果你在某处跟丢了，它不会回头等你。书中对例子和反例的选择非常具有目的性，每一个被提出的例子，似乎都是为了精确地说明某个定理的边界条件，或者澄清一个常见的误解。我感觉作者非常清楚自己的目标读者——那些对数学精度有极高要求的学习者。总而言之，这本书是一把高精度的手术刀，它能帮你精确地切割和分析代数几何的结构，但你需要自己准备好足够的基础知识来驾驭它提供的强大力量。

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这本关于代数几何的书，从我读者的角度来看，它的叙述方式实在令人耳目一新。作者似乎很擅长将复杂的概念用一种直观、甚至带有几何美感的方式呈现出来。我尤其欣赏它在引入基础知识时所花费的心思，那种层层递进的讲解，让人感觉不是在啃一堆枯燥的定义和定理，而是在进行一次精心设计的“导览”。比如，它在阐述射影空间（projective space）的构建时，那种从仿射空间到无穷远点的过渡，描绘得极为生动，仿佛能看到那些消失的平行线是如何在一个更高的维度上重新交汇。这种对几何直觉的培养，远比单纯的代数计算来得重要。书中对概形（schemes）的引入也处理得非常巧妙，没有一上来就抛出深奥的范畴论术语，而是先通过环谱（spec $ ext{R}$）这个具体的代数对象，来“模拟”几何空间的构造，这种“以代数为几何之名”的思路，使得初学者也能相对平稳地过渡到更抽象的领域。而且，书中的图示质量极高，每张图似乎都经过深思熟虑，不仅仅是装饰，而是关键的思维跳板。对于那些希望真正理解代数几何“为什么”这样工作的人来说，这本书提供了一个绝佳的起点，它成功地平衡了严谨性与可读性之间的微妙关系。

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谁能想到看这书的机缘并不是string theory，而是场论的费曼图计算？！(Grobner basis)

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谁能想到看这书的机缘并不是string theory，而是场论的费曼图计算？！(Grobner basis)

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the polynomial methods

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GTM 185

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the polynomial methods