Home » Resolution of Curve and Surface Singularities: In Characteristic Zero by K. Kiyek
Resolution of Curve and Surface Singularities: In Characteristic Zero K. Kiyek

Resolution of Curve and Surface Singularities: In Characteristic Zero

K. Kiyek

Published December 1st 2004
ISBN : 9781402020285
Hardcover
483 pages
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 About the Book 

The Curves The Point of View of Max Noether Probably the oldest references to the problem of resolution of singularities are found in Max Noethers works on plane curves [cf. [148], [149]]. And probably the origin of the problem was to have a formulaMoreThe Curves The Point of View of Max Noether Probably the oldest references to the problem of resolution of singularities are found in Max Noethers works on plane curves [cf. [148], [149]]. And probably the origin of the problem was to have a formula to compute the genus of a plane curve. The genus is the most useful birational invariant of a curve in classical projective geometry. It was long known that, for a plane curve of degree n having l m ordinary singular points with respective multiplicities ri, i E {1, . . ., m}, the genus p of the curve is given by the formula = (n - l)(n - 2) _ r. (r. _ 1) P 2 2 L. ., . . Of course, the problem now arises: how to compute the genus of a plane curve having some non-ordinary singularities. This leads to the natural question: can we birationally transform any (singular) plane curve into another one having only ordinary singularities? The answer is positive. Let us give a flavor (without proofs) 2 on how Noether did it To solve the problem, it is enough to consider a special kind of Cremona trans formations, namely quadratic transformations of the projective plane. Let be a linear system of conics with three non-collinear base points r = {Ao, AI, A }, 2 and take a projective frame of the type {Ao, AI, A- U}.