Algebraic Varieties (London Mathematical Society Lecture Note Series Book 172) by G. Kempf (PDF)

Description

In this book Professor Kempf gives an introduction to the theory of algebraic functions on varieties from a sheaf theoretic standpoint. By taking this view he is able to give a clean and lucid account of the subject which will be easily accessible to all newcomers to algebraic varieties, graduate students or experts from other fields alike. Anyone who goes on to study schemes will find that this book is an ideal preparatory text.

User’s Reviews

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐Great book

⭐I found this book quite opaque in general, and not a good place to learn algebraic geometry as a subject, although the discussion of cohomology was relatively good.Kempf assumes familiarity with classical algebraic geometry and defines an algebraic variety as something obtained by glueing together (finitely many) classical varieties. The advantage of this approach is that it avoids working with schemes, and quite a lot of properties of varieties (particularly properties of cohomology) can be established without using schemes. Moreover most of the proofs carry over easily to the case of schemes (or at least the scheme theoretic definition of variety). The disadvantage, in my eyes, is that this definition is both fiddly and unintuitive. Additionally, even though this approach allows one to carry over much of the properties of cohomology to schemes, it doesn’t really give you any intuition for many other important properties of schemes, such as non-reducedness and generic points.Finally, even though this books seems like it is trying to fill the gap between classical and scheme-theoretic algebraic geometry, it doesn’t actually make any connections between the material covered and either of these two areas. Furthermore I found the presentation of the material quite frequently both terse and unclear.

⭐George Kempf was a brilliant mathematician who mastered the most abstract tools invented by Grothendieck, and could use them skillfully to treat every topic in algebraic geometry. Here he presents his own brief tour of cohomology theory on algebraic varieties from scratch, with applications through the Riemann Roch theorem for curves and a bit for surfaces. Although most treatments use Cech cohomology for computations, George’s skill was such that he could compute also with derived functor cohomology and does so here, before showing the relation with Cech theory. The proof he gives of the Riemann Roch theorem is his own original one, the result of his attempt to understand how Galois might have arrived at the results on abelian integrals he mentions in his famous letter, from a memoir apparently lost. Fields medalist David Mumford recommended this proof as required reading for every graduate student in algebraic geometry.He understands his material so deeply that he discusses the most abstract topics in an elementary style, but there are still plenty of gaps for the reader to fill, and the going is far from easy. One reason I like this book is that it emphasizes cohomology rather than schemes, since I believe cohomology is much more important for most geometers than is scheme theory. I.e. in chapters 2 and 3 of Hartshorne’s standard book cohomology is presented only in the context of schemes, making it that much harder, since there one must learn schemes before having the tool of cohomology available. Serre’s approach was to introduce cohomology theory for varieties, and perhaps Serre is still easier to read than Kempf, but Kempf is a bit more elementary and uses the presheaf approach to sheaves as opposed to Serre’s e’tale’ space approach, which seems easier.I agree with others it would be wise to begin say with Harris’ or even Miles Reid’s books to get a feel for the objects of algebraic geometry, but when one is ready to learn the essential and powerful tool of cohomology, this is a beautiful place to explore. There are some serious typographical errors, such as in formulas for the coboundary of cech cohomology and elsewhere, so check everything yourself, but this is a remarkable and unique book.

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