作者：William
页数：83
出版社：高等教育出版社
出版日期：2018
ISBN：9787040534870
电子书格式：pdf/epub/txt
内容简介
本书介绍了现代相交理论的一些主要思想，追溯了它们在古典几何中的起源，并描绘了一些典型的应用。本书只需要很少的技术背景：数学研究生可以读懂大部分内容。本书涉及许多主题，很重要的是介绍了作者和R.MacPherson发明的一个强大的新方法。这是根据1983年6月27日至7月1日在George Mason大学举行的美国国家科学基金会支持的CBMS会议上所作的讲座撰写的。 本书介绍了利用法锥几何地构造和计算相交积的方法。在相交簇情形，产生了Smuael相交重数；在另一个特别，给出正规丛的Chern类的自相交公式；一般来说，作者和R.MacPherson提出了过分相交公式。在所提出的应用中，有退化轨迹、剩余交点和多点轨迹的公式；相交积的动态解释；Schubert演算和计数几何问题的解；Riemann-Roch定理等。
目录
Preface
Chapter 1.Intersections of Hypersurfaces
1.1.Early history (Bezout,Poncelet)
1.2.Class of a curve (Plicker)
1.3.Degree of a dual surface (Salmon)
1.4.The problem of five conics
1.5.A dynamic formula (Severi,Lazarsfeld)
1.6.Algebraic multiplicity,resultants
Chapter 2.Multiplicity and Normal Cones
2.1.Geometric multiplicity
2.2.Hilbert polynomials
2.3.A refinement of Bezout’s theorem
2.4.Samuel’s intersection multiplicity
2.5.Normal cones
2.6.Deformation to the normal cone
2.7.Intersection products:a preview
Chapter 3.Divisors and Rational Equivalence
3.1.Homology and cohomology
3.2.Divisors
3.3.Rational equivalence
3.4.Intersecting with divisors
3.5.Applications
Chapter 4.Chern Classes and Segre Classes
4.1.Chern classes of vector bundles
4.2.Segre classes of cones and subvarieties
4.3.Intersection forumulas
Chapter 5.Gysin Maps and Intersection Rings
5.1.Gysin homomorphisms
5.2.The intersection ring of a nonsingular variety
5.3.Grassmannians and flag varieties
5.4.Enumerating tangents
Chapter 6.Degeneracy Loci
6.1.A degeneracy class
6.2.Schur polynomials
6.3.The determinantal formula
6.4.Symmetric and skew-symmetric loci
Chapter 7.Refinements
7.1.Dynamic intersections
7.2.Rationality of solutions
7.3.Residual intersections
7.4.Multiple point formulas
Chapter 8.Positivity
8.1.Positivity of intersection products
8.2.Positive polynomials and degeneracy loci
8.3.Intersection multiplicities
Chapter 9.Riemann-Roch
9.1.The Grothendieck-Riemann-Roch theorem
9.2.The singular case
Chapter 10.Miscellany
10.1.Topology
10.2.Local complete intersection morphisms
10.3.Contravariant and bivariant theories
10.4.Serre’s intersection multiplicity
References
Notes(1983-1995)
Chapter 1.Intersections of Hypersurfaces
1.1.Early history (Bezout,Poncelet)
1.2.Class of a curve (Plicker)
1.3.Degree of a dual surface (Salmon)
1.4.The problem of five conics
1.5.A dynamic formula (Severi,Lazarsfeld)
1.6.Algebraic multiplicity,resultants
Chapter 2.Multiplicity and Normal Cones
2.1.Geometric multiplicity
2.2.Hilbert polynomials
2.3.A refinement of Bezout’s theorem
2.4.Samuel’s intersection multiplicity
2.5.Normal cones
2.6.Deformation to the normal cone
2.7.Intersection products:a preview
Chapter 3.Divisors and Rational Equivalence
3.1.Homology and cohomology
3.2.Divisors
3.3.Rational equivalence
3.4.Intersecting with divisors
3.5.Applications
Chapter 4.Chern Classes and Segre Classes
4.1.Chern classes of vector bundles
4.2.Segre classes of cones and subvarieties
4.3.Intersection forumulas
Chapter 5.Gysin Maps and Intersection Rings
5.1.Gysin homomorphisms
5.2.The intersection ring of a nonsingular variety
5.3.Grassmannians and flag varieties
5.4.Enumerating tangents
Chapter 6.Degeneracy Loci
6.1.A degeneracy class
6.2.Schur polynomials
6.3.The determinantal formula
6.4.Symmetric and skew-symmetric loci
Chapter 7.Refinements
7.1.Dynamic intersections
7.2.Rationality of solutions
7.3.Residual intersections
7.4.Multiple point formulas
Chapter 8.Positivity
8.1.Positivity of intersection products
8.2.Positive polynomials and degeneracy loci
8.3.Intersection multiplicities
Chapter 9.Riemann-Roch
9.1.The Grothendieck-Riemann-Roch theorem
9.2.The singular case
Chapter 10.Miscellany
10.1.Topology
10.2.Local complete intersection morphisms
10.3.Contravariant and bivariant theories
10.4.Serre’s intersection multiplicity
References
Notes(1983-1995)
