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 Poincare'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
David Eisenbud is Professor of Mathematics at the University of California, Berkeley, and currently serves as Director of the Mathematical Sciences Research Institute. He is also a Director at Math for America, a foundation devoted to improving mathematics teaching.
Joe Harris is Professor of Mathematics at Harvard University.
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