Algebraic geometry is both one of the most classical and one of the most modern areas of mathematics. The study of algebraic varieties, especially curves, goes back at least as far as Newton and L’Hospital. It was one of the central foci of mathematics in the 19th century. On the other hand, over the last half-century the subject was completely refashioned. The long history overlaid by new words and new concepts can make the subject difficult for beginners or for outsiders who only want to use the theory. This is the problem Elena Rubei tries to solve with this book.
Rather than attempting an overall survey of the basics (as, say, in the MAA Guides series), Rubei organizes the concepts alphabetically as a sort of dictionary. This allows quick access to the term one wants to learn about, but also means that some chasing of references might be needed.
The selection of terms shows that the focus is algebraic geometry per se and its connections to the theory of complex manifolds. There are no entries for topics that belong to the more arithmetic side of the subject (e.g., there are entries for “Grauert’s semicontinuity theorem” and “Fubini-Study metric” but not for “formal schemes,” “étale,” or “algebraic stack”).
Let’s look, for example, at the entry for “divisors,” one of the potentially confusing terms a beginner might want to look up. The entry starts with a long list of numerical references to the bibliography; these are both the sources used by the author and the place to look for more information. Then come the two crucial definitions: “Cartier divisor” and “Weil divisor.” Then we get a definition of the order of a regular function at a subvariety of codimension 1, which includes a reference to another article, “Length of a module.” Then we have a theorem relating the two notions: there is a homomorphism from Cartier divisors to Weil divisors, which is injective/bijective under appropriate conditions. There follow the definitions of “effective divisor,” “principal divisor,” “linear equivalence,” “divisor class group,” “Picard group,” the “line bundle associated to a divisor,” and finally we are told that several terms (from “ample” to “b.p.f.” are applied to divisors when the associated line bundle has those properties. This is followed, of course, by a cross-reference to the article on line bundles. Finally, there are other pointers to related articles: “Linear systems,” “equivalence,” etc.
How helpful is this? It won’t provide much in the way of overview or context, and it’s certainly not the place to begin learning the subject. I suspect that it would be quite useful for students who have already taken an introductory course and need to refresh their memory as to the meaning of some concept or the statement of some theorem. And it will be useful for someone like me, whose main focus is elsewhere but occasionally need to find out what, say, “Lefschetz’s (1,1)-theorem” says.
Fernando Q. Gouvêa is Carter Professor of Mathematics at Colby College in Waterville, ME. These days, he is 40% a number theorist, 40% a historian, 30% a writer and editor, and 100% overcommitted.