JOINS AND INTERSECTIONS

The central topic of the book is refined Intersection Theory and its applications, the basic tool of investigation being the Stückrad-Vogel Intersection Algorithm, based on the join construction. This algorithm is used to present a general version of Bezout's Theorem, in classical and refined form. Connections with the Intersection Theory of Fulton-MacPherson are treated, using work of van Gastel employing Segre classes. Bertini theorems and connectedness Theorems form another major theme, as do various measures of multiplicity. Local algebraic techniques, such as e.g. the theory of residual intersections, are mixed with more geometrical methods to present a wide range of geometrical and algebraic applications and illustrative examples. The book incorporates methods from commutative algebra and algebraic geometry. The hope is that it will inform algebraists of important methods from algebraic geometry and widen the interest of geometers in recent relevant advances in commutative algebra.