# 3264 and All That: A Second Course in Algebraic Geometry

Cambridge University Press, 4/26/2016

EAN 9781107602724, ISBN10: 1107602726

Paperback, 632 pages, 24.7 x 17.4 x 3.6 cm

Language: English

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 PoincarÃƒÂ©'s development of topology, and for many subsequent theories, so that intersection theory is now a central topic of modern mathematics.

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 GrothendieckÃ¢â‚¬â€œRiemannÃ¢â‚¬â€œRoch 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.