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  • Open Daily: 10am - 10pm
    Alley-side Pickup: 10am - 7pm

    3038 Hennepin Ave Minneapolis, MN
    612-822-4611

Open Daily: 10am - 10pm | Alley-side Pickup: 10am - 7pm
3038 Hennepin Ave Minneapolis, MN
612-822-4611
Resolution of Curve and Surface Singularities: In Characteristic Zero

Resolution of Curve and Surface Singularities: In Characteristic Zero

Kiyek, K.
Vicente, J. L.

Hardcover

Series: Algebra and Applications, Book 4

AlgebraGeneral MathematicsGeometry

ISBN10: 1402020287
ISBN13: 9781402020285
Publisher: Springer Nature
Published: Oct 1 2004
Pages: 486
Weight: 2.40
Height: 1.20 Width: 6.50 Depth: 9.50
Language: English
The Curves The Point of View of Max Noether Probably the oldest references to the problem of resolution of singularities are found in Max Noether's 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}.

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