Singular points of plane curves /

Few topics could offer a better culmination for an undergraduate program in pure mathematics than the theory of plane curves, a subject at once conceptual and computational. Sadly, few undergraduates now ever meet it. To call this classical but intensely active topic "interdisciplinary" belies the great extent to which its own development actually spawned or spurred many of the disciplines--projective geometry, topology, commutative ring theory, complex analysis, knot theory--that now appear to converge for its modern treatment. Compared with the slow development E. Brieskorn and H. Knorrer give it in their charmingly discursive 1986 Plane Algebraic Curves (which itself cries out for a contemporary edition), Wall (Univ. of Liverpool) seems quite driven. Nevertheless, undergraduates will find the accessible first third of the book a nice semester's work. Note that Wall treats only the local theory, meaning the geometry of a curve nearby a particular point, so this compares with only the last half of Brieskorn and Knorrer's volume. Wall, however, goes deeper, including recent developments. Highly recommended. "Singularities" refer to those points of a curve or surface or higher dimensional object that resist linearization. Resolving an object's singularities means finding another curve or surface with no singularities, yet with the "same" geometry, at least apart from a small set. In a very long, complex, celebrated 1964 paper, H. Hironaka proved, at least in the context of geometry over the complex numbers, that one can always resolve singularities in any dimension. Although many important theorems depend on Hironaka's work, very few mathematicians will study Hironaka's proof; indeed, mathematicians accord alternative approaches the status of breakthroughs. Kiyek (Univ. of Paderborn, Germany) and Vicente (Universidad de Sevilla, Spain) attest to the complexity even of resolving surface singularities, albeit known long before Hironaka. Since the subject has received very little exposition save for that aimed at experts, one must welcome this careful, self-contained book, replete with book-length appendixes setting out all geometrical and algebraic prerequisites. That said, this highly formal, algebraic treatment dwells little on intuition and contains no diagrams, so only the most intrepid undergraduate will get very far with it. Those who dare to try will vastly improve their chances if they start with Wall's book. Some books earn a place on the shelf by making hard subjects easier; this one makes an impossible subject merely possible. ^BSumming Up: Wall, highly recommended; Kiyek, recommended. Both books, upper-division undergraduates through faculty. D. V. Feldman University of New Hampshire