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Intersection Theory (Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge A Series of Modern Surveys in Mathematics)

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Although very formal in its presentation, this book gives the reader some insight into the author's approach to intersection theory that he is known for. Readers are expected to be experts in algebraic geometry, at the level of schemes but not necessarily as specialized as the theory of motives. Outside reading is mandatory of course, particularly since there are no exercises that test the readers insight. A summary of the chapters is as follows:

Chapter 1: Rational Equivalence
Only at the end of this chapter does the author remind the reader that rational equivalence is a generalization of the "classical" or "Italian" notion of linear equivalence if attention is restricted to divisors on non-singular divisors. It would have been better for the reader's understanding and appreciation if the author would have begun with a discussion of linear equivalence and through a series of elementary examples arrive at a definition of rational equivalence. But this chapter, along with the rest of the book is written in the style of most contemporary books in mathematics: very formal with definitions merely stated and not motivated, followed by theorems with rigorous proofs. But an in-depth, intuitive understanding and appreciation of the relevant concepts seems at times to be inversely proportional to rigor.

There is one place however in the "Notes and References" that the author gives valuable insight and intuition (perhaps without intending to). This pertains to the reason that a characteristic class is indeed called a "class". This designation has puzzled many (particularly high-energy physicists) who want to learn the theory of characteristic classes. The author does this when he discusses the work of the Italian geometer F. Severi on rational equivalence for cycles with codimension greater than one. The second Chern class is not a number but rather a class, indeed a (rational) equivalence class of zero-cycles. This explanation may seem trivial to some, but for those who demand an historical and intuitive motivation for mathematical concepts, one that promotes real understanding rather than just formal, it is a welcome gift.

There are many places in this chapter where the author gives hints about what is ahead in the book. One of these concerns the action of a finite group on a variety, which when taking the quotient of the variety by the group the physicist reader may recognize as a kind of "orbifold." The author shows that the intersection ring of the quotient variety tensored with the rational numbers is isomorphic to the intersection ring of the G-invariant subgroup of this tensor product. This strategy of rationalization is apparently one that is used to study singular varieties in later chapters in the book. Another place is where the author briefly discusses the K-theory of coherent sheaves, where again the author refers to how rationalization can be used effectively. In this it will give an isomorphism between the intersection ring and the (graded) Grothendiek group. Using K-theory to do intersection theory has been popular with some groups of researchers in intersection theory, as one might expect if one is familiar with the successes of K-theory in the theory of vector bundles.

Chapter 2: Divisors
As expected, this chapter assumes a priori knowledge of Weil and Cartier divisors. Weil divisors are merely cycles of one less dimension than the variety, but Cartier divisors are much more specialized and as such demand more explanation than what the author offers in this chapter. Loosely speaking, they are determined (not uniquely, but up to multiplication by units) by rational functions defined on an open covering of the variety and do not vanish on the intersection of the opens sets of the covering. The order of these rational functions is well defined, and so for a subvariety one can define the order of a Cartier divisor to be the order of the rational function defined over an open subset of the cover that intersects the subvariety nontrivially.

For one of the examples, readers will have to have knowledge of ample line bundles on projective schemes. The author does not discuss ample line bundles in this chapter or any of the appendices. The notion of a pseudo-divisor is introduced in order to get a divisor that is well behaved under pullbacks and still derive a reasonable intersection theory. Pseudo-divisors can always be pulled back: the author's trick is to pick a line bundle and a closed subset of the scheme, and then a nowhere vanishing section of the line bundle on the complement of the closed subset (i.e. a trivialization on the resulting open set). One can then take the inverse image of the closed subset and the pullbacks of the line bundle and the section to define the pullback of the pseudo-divisor. This notion of pullback agrees with that of a Cartier divisor when the image of the map is not contained in the support of the Cartier divisor.

More forthcoming.