This book is devoted to the geometric technique of calculating syzygies. This technique originated with George Kempf and was first used successfully by Alain Lascoux for calculating syzygies of determinantal varieties. Since then it has been applied in studying the defining ideals of varieties with symmetries that play a central role in geometry: determinantal varieties, closures of nilpotent orbits, and discriminant and resultant varieties.

The character of the method makes it comparable to the symbolic method in classical invariant theory. It works in only a limited number of cases, but when it does, it gives a complete answer to the problem of calculating syzygies, and this answer is hard to get by other means.

Even though the basic idea is more than 20 years old, this is the first book treating the geometric technique in detail. This happens because authors using the geometric method have usually been interested more in the special cases they studied than in the method itself, and they have used only the aspects of the method they needed. Therefore the basic theorems from chapter 5 stem from the efforts of several mathematicians.

The possibilities offered by the geometric method are not exhausted by the examples treated in the book.