代数几何入门 内容简介

本书旨在深层次讲述代数几何原理、20世纪的一些重要进展和数学实践中正在探讨的问题。该书的内容对于对代数几何不是很了解或了解甚少，但又想要了解代数几何基础的数学工作者是非常有用的。目次：仿射代数变量；代数基础；射影变量；quasi射影变量；经典结构；光滑；双有理几何学；映射到射影空间。
读者对象：本书适用于数学专业高年级本科生、研究生和与该领域有关的工作者。

代数几何入门 本书特色

《代数几何入门(英文版)》是由世界图书出版公司出版的。

代数几何入门 目录

notes for the second printing
preface
acknowledgments
index of notation
1 affine algebraic varieties
1.1 definition and examples
1.2 the zariski topology
1.3 morphisms of affine algebraic varieties
1.4 dimension
2 algebraic foundations
2.1 a quick review of commutative ring theory
2.2 hilbert's basis theorem
2.3 hilbert's nuustellensatz
2.4 the coordinate ring
2.5 the equivalence of algebra and geometry
2.6 the spectrum of a ring
3 projective varieties
3.1 projective space
3.2 projective varieties
3.3 the projective closure of an affine variety
3.4 morphisms of projective varieties
3.5 automorphisms of projective space
4 quasi-projective varieties
4.1 quasi-projective varieties
4.2 a basis for the zariski topology
4.3 regular functions
5 classical constructions
5.1 veronese maps
5.2 five points determine a conic
5.3 the segre map and products of varieties
5.4 grassmannians
5.5 degree
5.6 the hilbert function
6 smoothness
6.1 the tangent space at a point
6.2 smooth points
6.3 smoothness in families
6.4 bertini's theorem
6.5 the gauss mapping
7 birational geometry
7.1 resolution of singularities
7.2 rational maps
7.3 birational equivalence
7.4 blowing up along an ideal
7.5 hypersurfaces
7.6 the classification problems
8 maps to projective space
8.1 embedding a smooth curve in three-space
8.2 vector bundles and line bundles
8.3 the sections of a vector bundle
8.4 examples of vector bundles
8.5 line bundles and rational maps
8.6 very ample line bundles
a sheaves and abstract algebraic varieties
a.1 sheaves
a.2 abstract algebraic varieties
references
index

代数几何入门 节选

《代数几何入门(英文版)》旨在深层次讲述代数几何原理、20世纪的一些重要进展和数学实践中正在探讨的问题。该书的内容对于对代数几何不是很了解或了解甚少，但又想要了解代数几何基础的数学工作者是非常有用的。目次：仿射代数变量；代数基础；射影变量；Quasi射影变量；经典结构；光滑；双有理几何学；映射到射影空间。 读者对象：《代数几何入门(英文版)》适用于数学专业高年级本科生、研究生和与该领域有关的工作者。

代数几何入门 相关资料

插图：The remarkable intuition of the turn-of-the-century algebraic geometerseventually began to falter as the subject grew beyond its somewhat shakylogical foundations. Led by David Hilbert, mathematical culture shiftedtoward a greater emphasis on rigor, and soon algebraic geometry fell outof favor as gaps and even some errors appeared in the subject. Luckily,the spirit and techniques of algebraic geometry were kept alive, primarilyby Italian mathematicians. By the mid-twentieth century, with the effortsof mathematicians such as David Hilbert and Emmy Noether, algebra wassufficiently developed so as to be able once again to support this beautifuland important subject.In the middle of the twentieth century, Oscar Zariski and Andr Weilspent a good portion of their careers redeveloping the foundations of alge-braic geometry on firm mathematical ground. This was not a mere processof filling in details left unstated before, but a revolutionary new approach,based on analyzing the algebraic properties of the set of all polynomial func-tions on an algebraic variety. These innovations revealed deep connectionsbetween previously separate areas of mathematics, such as number the-ory and the theory of Riemann surfaces, and eventually allowed AlexanderGrothendieck to carry algebraic geometry to dizzying heights of abstrac-tion in the last half of the century. This abstraction has simplified, unified,and greatly advanced the subject, and has provided powerful tools usedto solve difficult problems. Today, algebraic geometry touches nearly everybranch of mathematics.An unfortunate effect of this late-twentieth-century abstraction is that ithas sometimes made algebraic geometry appear impenetrable to outsiders.Nonetheless, as we hope to convey in this Invitation to Algebraic Geome-try, the main objects of study in algebraic geometry, affine and projectivealgebraic varieties, and the main research questions