*(English)*Zbl 0960.14002

This is a very useful book for all who wants to know something about schemes but never dared to ask the right questions.

Indeed, it is a successor to the authors’: “Schemes: The language of modern algebra” [*D. Eisenbud* and *J. Harris* (1992; Zbl 0745.14002)]. There, the influence of *D. Mumford* [“The red book of varieties and schemes”, Lect. Notes Math. 1358 (1988; Zbl 0658.14001)] was not to be overseen.

The additions are intended to show schemes at work in a number of topics in classical geometry. For example the authors define blow-ups and study the blow-up of the plane at various non-reduced points. They define duals of plane curves, and study how the dual degenerates as the curve does.

The many examples and explaining pictures make this book recommendable for students how want to get the feeling for the abstract version of curves, surfaces, tangents, etc. to come to moduli spaces. A general method is the functor of points.

The author introduces Fano schemes, a concept not to be found in the book by *R. Hartshorne* [“Algebraic geometry”, Grad. Texts Math. 52 (1977; Zbl 0367.14001)], to explain some very nice classical examples, as are the 27 lines on a smooth cubic surface (which is then given as an exercise to do in detail).

Also, quartic surfaces are considered.