Algebraic geometry is such a vast and deep subject that the list of monographs and textbooks devoted to it is not a short one. But it is so fundamental that there is always a place for a new textbook, sometimes with little overlap with the existing literature. Taking into account the overwhelming amount of basic background needed to understand the basics of the modern Grothendieck-style approach that the average student has to grasp to understand the current literature and begin doing research, one understands why many of these excellent books leave out the examples and results that kept the subject alive before there were sheaves and schemes.
It is true that the good students would eventually find and work out these examples by themselves, perhaps with the help of some books that kept them close and rewrote them in modern language; Shafarevich’s Basic Algebraic Geometry Vol. 1 and Vol. 2 is one that comes to mind immediately. But the older literature, rich with examples and calculations, sometimes is out of reach, not only for its dispersion but also for its outdated language and different standards of rigor.
Fortunately, in recent years some authors have taken the task of presenting the modern reader, in a contemporary language, the concrete examples and theorems filled with intuitive ideas that stimulated the subject and were the springboard for most of the current developments.
The beautiful book under review surveys some parts of the vast landscape of classical algebraic geometry using the tools and language of modern algebraic geometry. By a tasteful selection of a few vistas, the author focuses on some classical constructions and results that make the subject alive in our natural language. I must remark that this is not a historical treatise. It is a mathematical time-travel book where the reader talks to Cremona, del Pezzo, Hesse, and Cayley in contemporary mathematical language.
To give the reader an idea of the contents and level of the book, it is enough to look at the first chapter: This chapter starts with a few pages of linear algebra. Then we are led to the ideas surrounding the theory of polarity: the polar pairing, polar hypersurfaces and polar quadrics. In this chapter we find some results by Hesse, Steiner, Cayley, and other classical geometers, but everything is formulated, written and proved with 20th century methods. What this looks like can be seen starting in page 4, where one is reminded that the projective space comes equipped with the tautological invertible sheaf whose global sections are identified with the dual of the underlying vector space.
This sets the tone for the whole book: cohomological methods are used when needed, for example to prove results on del Pezzo surfaces in Chapter eight. The chosen vistas and stopovers include curves and quadric surfaces, plane cubics, determinantal varieties, plane quartics, Cremona transformations, del Pezzo and cubic surfaces, Grassmannians of lines, ruled surfaces and theta characteristics.
As the author points out in its introduction, the beautiful textbook by Beltrametti, Carletti, Gallarati and Bragadin, Lectures on Curves, Surfaces and Projective Varieties. A Classical View of Algebraic Geometry (EMS, 2009) covers a similar sample of classical varieties, but with a more modest mathematical background from the reader. The books complement each other; together they give an overview, with a modern approach, of a subject rich with fascinating examples that one can miss in the mist of the formidable machinery that one now has to learn to enter this area of mathematics.
Felipe Zaldivar is Professor of Mathematics at the Universidad Autonoma Metropolitana-I, in Mexico City. His e-mail address is firstname.lastname@example.org.