Reprinted with permission from CHOICE, © American Library Association.

Mauro C. Beltrametti (University of Genova, Italy)
Ettore Carletti (University of Genova, Italy)
Dionisio Gallarati (University of Genova, Italy)
Giacomo Monti Bragadin (University of Genova, Italy)

EMS Textbooks in Mathematics
ISBN 978-3-03719-064-7
DOI 10.4171/064
August 2009, 506 pages, hardcover, 16.5 x 23.5 cm.
58.00 Euro

Among mathematical subjects, algebraic geometry arguably stands as both most ancient and most modern (with number theory, now its intimate ally, the only rival on either front). But such a long, active history creates pedagogical challenges at the undergraduate level, where the subject currently suffers systematic neglect despite its venerable centrality. Any approach that does not seem old-fashioned and naïve to some runs the risk of seeming overly sophisticated, abstruse, and remote to others, and vice versa. M. Reid's Undergraduate Algebraic Geometry (CH, Dec'89, 27-2141) strikes perhaps a perfect balance, but that short, breathless book functions as only an invitation or sampler. Here, one has a systematic development, unique in aiming at the same audience as Reid, but providing thorough preparation for eventual graduate-level study out of treatises such as the ones by Phillip Griffiths and Joseph Harris or Robin Hartshorne. Though Beltrametti and colleagues (all, Univ. of Genova, Italy) concentrate on classical topics such as Cremona transformations and omit modern tools such as Grothendieck's schemes, experts will readily detect how the modern viewpoint really informs these authors' approach, the better to smooth any future transition for students first entering the subject.

Summing Up: Recommended. Upper-division undergraduate mathematics collections.