As the story goes, Euclid was tutoring Ptolemy on geometry out of Elements and the king asked Euclid if there was an easier way to learn geometry than to work through all of these books. Euclid replied simply that “There is no royal road to geometry.” While this story may or may not be apocryphal, Euclid’s pithy response has lent its name to any number of books and articles over the years. Audun Holme has taken up Euclid’s challenge as well, and written a new book A Royal Road To Algebraic Geometry recently published by Springer. To Holme, the idea of a “royal road” is not necessarily an easy path but one that conveys the important concepts and theorems in the subject without going into all of the technical details. So the book sets out on a project to convey the big ideas of algebraic geometry in the 21st century in a way that will be accessible and that will get to get to the big ideas (relatively) quickly.
The first half of Holme’s book deals with curves, and introduces many of the ideas in algebraic geometry in this context. The reader learns about affine space, projective space, singular and nonsingular points, different types of singularities and multiplicities. There are sections about tangents and asymptotes, about conic sections and Bezout’s Theorem. There is a chapter which brings the algebra to the forefront of the discussion, introducing coordinate rings and valuations and the Jacobian and Hessian. And a final chapter generalizes this material somewhat, to projective varieties in projective space, with the final section in the first half of the book introducing the Riemann-Roch theorem for nonsingular curves (deferring the proof until later in the book, as we will discuss momentarily). Given Holme’s stated goal of writing a book that could be used as a quick introduction to the field, this first half does an admirable job. While I might quibble with some of his choices about where to place emphasis and where to skim details, the author does a generally good job of explaining the material — although this reviewer still thinks other books, such as Keith Kendig’s recent Guide to Plane Algebraic Curves, do a better job.
The second half of the book, however, does not do as good a job of living up to the author’s goals. As anyone who has studied modern algebraic geometry knows, much of the subject is now devoted to schemes, which were originally developed by Grothendieck in order to generalize the notion of an algebraic variety. Roughly speaking, a scheme is a locally ringed topological space covered by a collection of open sets each of which is isomorphic to the set of prime ideals of a commutative ring. If this definition sounds daunting to you then you are not alone, as several generations of graduate students have had nightmares about trying to understand the category of schemes. I was one of these graduate students, so I will admit that I was both eager and skeptical about the idea of Holme finding a “royal road” through the material. And I am sorry to say that my skepticism proved correct. Before defining a scheme, Holme spends fifty pages giving a tour of categories, functors, and affine spectra of rings; even this tour will be too fast-paced for many readers. The definition of the category of schemes takes close to another 20 pages, and only then can he move on to stating results about the morphisms of schemes, vector bundles of schemes, and the like. The remainder of the book deals with these properties, as well as cohomology theory on schemes and quick introductions to intersection theory and duality theorems.
Saying that Holme did not entirely succeed at what is likely an impossible task is not a complete condemnation, however, and while I do not think he found the “royal road” he was looking for, his road is in fact paved with good intentions. The topics he chooses do give a good overview of modern algebraic geometry, and I think that a reader who is interested in getting quick statements of theorems will find a lot to like in this book. I am not sure, however, how many readers will want this type of overview of technical material given that they will then need to track down their own references to find complete proofs.
One example that I found typical of Holme’s approach is the chapter on the Riemann-Roch Theorem late in the book. The author begins this section by saying that he will only consider the theory for nonsingular schemes and referring the reader to Fulton’s Intersection Theory if they want details for schemes with singularites. He then states Hirzebruch’s Riemann-Roch Theorem, and briefly sketches how this implies the Riemann-Roch theorems for curves and surfaces. In order to prove Hirzebruch’s version of the theorem, Holme instead states the more general version of Riemann-Roch due to Grothendieck and then gives a paragraph sketching very briefly how it implies the Hirzebruch version. Finally, he “gives one of the basic ideas behind the proof of Grothendieck’s Riemann Roch”, referring the reader to a paper by Borel and Serre for the full proof. I expect that Holme views this approach — giving readers the statements of the big theorems with hints of how proofs work — as what he is trying for, and many readers might agree. This reviewer, however, found it more frustrating than rewarding.
Another thing that I found problematic about Holme’s book was the lack of concrete examples, and even moreso the lack of exercises. Even with a subject as abstract as algebraic geometry can be, I know that I find it essential both as a learner and a teacher to try to work through examples much more than Holme’s book allows. For this and other reasons, I suspect that if I were trying to learn from this book I would spend as much time looking in Hartshorne or a similar book as I did in this book itself. One choice that the author makes that I did like was to include periodic historical asides, and I wish there were more of these. The author also includes portraits of many of the mathematicians whose work appears in the book, all of which he sketched himself.
There is one more issue that I feel I must bring up in the review. When I review books I am generally willing to give the author a great deal of leeway on issues of grammar, punctuation, typos, and spelling. I am far from perfect on these issues in my own writing, after all, and I have found that it is best to give authors the benefit of the doubt as long as it does not interfere with my ability to read the book. Alas, that was not the case with A Royal Road to Algebraic Geometry, as I could rarely make it more than a couple of pages without finding a mistake of one sort or another. While many of these were inconsequential (“can not” instead of “cannot”), others were more difficult to parse (“in” instead of “if”, “deal” instead of “ideal”). And even allowing that much of this book was originally written in Spanish by a Norwegian author, there were some sentence constructions that I found difficult to work through. Put bluntly, this book badly needed a more careful copy-editing. More generally, the book reminded me of a set of lecture notes one might give to one’s students — with amateurish mistakes, hand-drawn portraits, and vague sketches of proofs. Viewed in this context, it is a nice achievement and Holme often brings a new take on the material he is writing about. However for a reader looking for a road to algebraic geometry, I would recommend taking a road more traveled. It could make all the difference.
Darren Glass is an Associate Professor of Mathematics at Gettysburg College. His main mathematical interests include number theory and algebraic geometry, even if he still sometimes gets chills thinking back to his long nights with Chapter III of Hartshorne’s book. He can be reached at email@example.com.