This is a Dover reprint of a book first published by Princeton University Press in 1953. That edition was itself based on a set of lecture notes first published, in two parts, in 1935 and 1938. Though the 1953 edition was much revised and enlarged, it retains many of the characteristics of a set of lecture notes. Today, it should be considered more for its historical value than as a textbook in algebraic geometry.
The book's main innovation, at the time, seems to have been its purely algebraic point of view. Lefschetz eschews "transcendental" methods, replacing them with the tools of what was then still "modern algebra." In particular, he makes use of rings of formal power series and what he calls "algebroid varieties" over these rings. Fields of prime characteristic are mentioned here and there, but for the most part the theory is developed only for (suitably large) fields of characteristic zero.
Indications of the book's origin as lecture notes appear throughout. Some sentences are unexpectedly informal. My favorite is the opening of the section on projections: "The two classical operations on algebraic varieties are projections and intersections. We deal here and there with intersections and shall now consider projections." Here and there!)
In general, the proofs given are quite terse. Most algebraic results are simply assumed, sometimes without more than a general reference to van der Waerden's Moderne Algebra .
When the book first appeared in 1953, the famous algebraic geometer Beniamino Segre was asked to discuss it in Mathematical Reviews (the link requires a MathSciNet subscription). His review opens as follows:
For about a century, algebraic geometry has been consolidated and enriched in a prodigious way, developing according to multiple — and constrasting — questions and needs relating to algebra, analysis and topology. So much so that it seems utopian to want to give an up to date and sufficiently complete expository introduction in little more than 200 pages. (My own — fairly free — translation from Segre's Italian.)
One cannot help but wonder what Segre made of the Grothendieck revolution that was just around the corner! But in fact the project is somewhat utopian, and the exposition is often quite sketchy. (The book is "un po' stringato," says Segre.) Given that and the dramatic changes in the subject since 1960, this is not the place to learn algebraic geometry. Historians of the subject, however, will be interested in this glimpse of what the subject looked like before the revolution.
Fernando Q. Gouvêa is professor of mathematics at Colby College in Waterville, ME.