3264 and All That
3264 and All That
This book can form the basis of a second course in algebraic geometry. As motivation, it takes concrete questions from enumerative geometry and intersection theory, and provides intuition and technique, so that the student develops the ability to solve geometric problems. The authors explain key ideas, including rational equivalence, Chow rings, Schubert calculus and Chern classes, and readers will appreciate the abundant examples, many provided as exercises with solutions available online. Intersection is concerned with the enumeration of solutions of systems of polynomial equations in several variables. It has been an active area of mathematics since the work of Leibniz. Chasles' nineteenth-century calculation that there are 3264 smooth conic plane curves tangent to five given general conics was an important landmark, and was the inspiration behind the title of this book. Such computations were motivation for Poincaré's development of topology, and for many subsequent theories, so that intersection theory is now a central topic of modern mathematics.
- Explores intersection theory - a central topic for everyone interested in algebraic geometry, from number theorists to theoretical physicists
- Ideal for graduate students and for individual mathematicians wishing to learn the ideas and techniques of algebraic geometry
- Contains more than 360 exercises with solutions available online
Reviews & endorsements
'… the book covers an important part of classical algebraic geometry with a modern point of view. It is indeed highly recommendable for a second (or a third) course in algebraic geometry| and more generally, for every mathematician interested in concrete algebraic geometry.' Arnaud Beauville, MathSciNet
Product details
April 2016Paperback
9781107602724
603 pages
253 × 176 × 35 mm
1.14kg
80 b/w illus. 360 exercises
Available
Table of Contents
- Introduction
- 1. Introducing the Chow ring
- 2. First examples
- 3. Introduction to Grassmannians and lines in P3
- 4. Grassmannians in general
- 5. Chern classes
- 6. Lines on hypersurfaces
- 7. Singular elements of linear series
- 8. Compactifying parameter spaces
- 9. Projective bundles and their Chow rings
- 10. Segre classes and varieties of linear spaces
- 11. Contact problems
- 12. Porteous' formula
- 13. Excess intersections and the Chow ring of a blow-up
- 14. The Grothendieck–Riemann–Roch theorem
- Appendix A. The moving lemma
- Appendix B. Direct images, cohomology and base change
- Appendix C. Topology of algebraic varieties
- Appendix D. Maps from curves to projective space
- References
- Index.
