Intersection Theory (Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics)

The notion of singularity is basic to mathematics. In algebraic geometry, the resolution of singularities by simple algebraic mappings is truly a fundamental problem. It has a complete solution in characteristic zero and partial solutions in arbitrary characteristic. The resolution of singularities in characteristic zero is a key result used in many subjects besides algebraic geometry, such as differential equations, dynamical systems, number theory, the theory of $\mathcal{D}$-modules, topology, and mathematical physics. This book is a rigorous, but instructional, look at resolutions. A simplified proof, based on canonical resolutions, is given for characteristic zero. There are several proofs given for resolution of curves and surfaces in characteristic zero and arbitrary characteristic. Besides explaining the tools needed for understanding resolutions, Cutkosky explains the history and ideas, providing valuable insight and intuition for the novice (or expert). There are many examples and exercises throughout the text Rational Equivalence.- Divisors.- Vector Bundles.- Cones and Segre Classes.- Deformations to the Normal Cone.- Intersection Products.- Intersection Multiplicites.- Intersections on Non-singular Varieties.- Excess and Residual Intersections.- Families of Algebraic Cycles.- Dynamic Intersections.- Positivity.- Rationality.- Degeneracy Loci and Grassmannians.- Riemann-Roch for Non-singular Varieties.- Correspondences.- Bivariant Intersections Theory.- Riemann-Roch for Singular Varieties.- Algebraic: Homological and Numerical Equivalence.- Generalizations From the ancient origins of algebraic geometry in the solutions of polynomial equations, through the triumphs of algebraic geometry during the last two centuries, intersection theory has played a central role. The aim of this book is to develop the foundations of this theory, and to indicate the range of classical and modern applications. Although a comprehensive history of this vast subject is not attempted, the author points out some of the striking early appearances of the ideas of intersection theory. A suggested prerequisite for the reading of this book is a first course in algebraic geometry. Fulton's introduction to intersection theory has been well used for more than 10 years. It is still the only existing complete modern treatise of the subject and received the Steele Prize for best exposition in August 1996.

Intersection theory has played a central role in mathematics, from the ancient origins of algebraic geometry in the solutions of polynomial equations to the triumphs of algebraic geometry during the last two centuries. This book develops the foundations of the theory and indicates the range of classical and modern applications. The hardcover edition received the prestigious Steele Prize in 1996 for best exposition.

This text seeks to develop the foundations of intersection theory, and to indicate the range of classical and modern applications. The author does not attempt a full history of this vast subject, but points out some of the early appearances of the ideas of the theory. Volume 2 in the Series of Modern Surveys in Mathematics. Its aim is to develop the foundations of intersection theory, and to indicate the range of classical and modern applications.