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Introduction to Algebraic Geometry
Publisher: Cambridge University Press (2007)
Details: 252 pages, Paperback
Topics: Algebraic Geometry
MAA Review[Reviewed by P. N. Ruane, on 06/09/2008]
Those, like myself, seeking deeper insights into the arcane world of algebraic geometry soon encounter the grim reality encapsulated in the following comment from John Silvester :
A distressing aspect of modern mathematics is the extent to which mathematicians, having ascended to a new level of understanding in the subject, then pull the ladder up after them. Algebraic geometers are more guilty of this than most, and in consequence their subject has a fearsome reputation, even among mathematicians.
This, of course, depends on what’s meant by ‘algebraic geometry’, and one finds strangely varying descriptions, from one author to another:
On the other hand, Dieudonné provides an historical description , broken down into seven chronological periods, of which the first four are:
These, in fact, are what other historians describe as the early stages of algebraic geometry, and anyone operating within such parameters might therefore claim to be an algebraic geometer (if only!).
Subsequent stages in Dieudonné’s historical analysis range from the Brill-Noether theory of the late 19th century, to the appearance of sheaves and schemes in the 1950s. Such developments, of course, go hand-in-hand with the emergence of commutative algebra, topology, Lie theory and cohomology. etc. But Dieudonné portrays the astonishing complexity of innovations in the field, up to the 1970s, by which time it was drawing upon almost every other part of mathematics for the purpose of sustaining its remarkable growth.
However, Silvester’s metaphor of ‘ladders’ is meaningless without a clear idea of the levels of abstraction from which they start, and to which they ascend. For example, in terms of undergraduate mathematics, his book  constitutes the basis of an excellent foundation geometry course beginning with Euclid, then progressing through transformation geometry, projective geometry and culminating with a survey of early stages of algebraic geometry (plane algebraic curves).
Obviously, there are many books that separately provide fuller treatment of these and other mathematical themes, all of which form a series of ‘ladders’ reaching up to the lower platforms of generality outlined by Dieudonné. However, there is an alternative route into algebraic geometry that is outlined in this innovative book by Brendan Hassett.
Described as a text for advanced undergraduate and postgraduate students, the pre-requisites are said to be a good working knowledge of linear algebra, along with ‘some familiarity’ with the basic concepts of abstract algebra (rings, ideals and factorisation). And yet, typical of the modern era, this is one of many books with the word ‘geometry’ emblazoned in its title, but with no stated expectation of pre-requisite geometrical knowledge. This, of course, is just as well, because high school students are likely to have little geometric knowledge anyway; and geometry, if it appears in the undergraduate programme at all, is usually offered as a dispensable option.
Anyway, this book is said to form a ‘common sense introduction’ to algebraic geometry; and instead of invoking the heavy machinery outlined by Dieudonné, the methodology is largely computational, thereby encouraging the use of computer algebra systems (Maple or Macaulay II). This philosophy is implemented by an early introduction to Gröbner bases and the development of algorithms for almost every technique that is subsequently derived.
Within its 12 chapters and 250 pages, the central themes are: resultants and elimination theory, affine and projective varieties (Grassmann, Veronese and Segre), morphisms, rational and birational maps. There is also coverage of standard topics such as the Nullstellensatz, Projective Geometry, Parametrizing Linear Subspaces, Hilbert polynomials and Bézout's Theorem.
An attractive feature of the book is the organisation of the material and the precision of the narrative. Most chapters commence with a brief outline of their scope and purpose, and the ideas are introduced alongside many worked examples. Moreover, the exercise sets at the end of each chapter are many and varied, and they serve to consolidate understanding of the theory that precedes them.
Nonetheless, despite being described as an introduction to algebraic geometry, this book would be a very stiff challenge for those opting to work through it on an unsupported basis, and I found myself referring to other sources for clarification of certain ideas (e.g., lexicographical ordering). Morover, there are no solutions to the exercises and the exposition is occasionally rather condensed. For example, the very brief introduction to Zariski topology will be hard to digest for those with no initial background in topology. Also, the introduction to projective geometry, being heavily algebraic, gives no clue as to its place in the Kleinian pedagogical hierarchy. Again, having been advised to come equipped with ‘some familiarity’ with abstract algebra, readers should bear in mind that this should include good prior knowledge of Noetherian rings and other topics.
But this is the problem facing anyone seeking to write a book on this theme because the entire undergraduate mathematics curriculum could be filled with courses that collectively form a mathematical pathway leading to this pinnacle of abstraction known as algebraic geometry. Being based upon lecture notes, this book is more of a teaching text than a self-study manual. Also, despite its high aesthetic appeal from the algebraic point of view, I found the bulk of the material to be visually unmotivated, because many of the basic geometric concepts, such as intersection multiplicity and singularities appear at the end of the book and are stated in rather general form.
Such observations apart, it has to be said that Brendan Hassett has written a book of high mathematical integrity and those teaching introductory algebraic geometry well may choose it as a basis of a taught course.
 Geometry Ancient and Modern, by John R Silvester (OUP, 2001)
 ‘The Historical Development of Algebraic Geometry’, J Dieudonné, American Mathematical Monthly 79 (1972), pp. 827-866
Peter Ruane has taught non-symbolic algebraic geometry to children as young as six years old.