{"product_id":"resolution-of-curve-and-surface-singularities-in-characteristic-zero-9781402020285","title":"Resolution of Curve and Surface Singularities: In Characteristic Zero","description":"\u003cp\u003e • Author(s): K. Kiyek | J. L. Vicente\u003cbr\u003e • Publisher: Springer\u003cbr\u003e • Publisher Imprint: Springer\u003cbr\u003e • BISAC: Geometry - Algebraic\u003c\/p\u003e\u003cp\u003eThe 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}.\u003c\/p\u003e","brand":"Springer","offers":[{"title":"Hardcover","offer_id":47613726326935,"sku":"9781402020285","price":3672.0,"currency_code":"INR","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0666\/3471\/1191\/files\/9781402020285.webp?v=1775087413","url":"https:\/\/atlanticbooks.com\/products\/resolution-of-curve-and-surface-singularities-in-characteristic-zero-9781402020285","provider":"Atlantic Books","version":"1.0","type":"link"}