{"product_id":"resolution-of-curve-and-surface-k-kiyek-9781402020285","title":"Resolution of Curve and Surface Singularities: In Characteristic Zero","description":"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}.\u003cbr\u003e\u003cbr\u003e\u003cb\u003eAuthor:\u003c\/b\u003e K. Kiyek,J. L. Vicente\u003cbr\u003e\u003cb\u003eISBN-10:\u003c\/b\u003e 1402020287\u003cbr\u003e\u003cb\u003eISBN-13:\u003c\/b\u003e 9781402020285\u003cbr\u003e\u003cb\u003ePublisher:\u003c\/b\u003e Springer\u003cbr\u003e\u003cb\u003eLanguage:\u003c\/b\u003e English\u003cbr\u003e\u003cb\u003ePublished:\u003c\/b\u003e 10\/01\/2004\u003cbr\u003e\u003cb\u003ePages:\u003c\/b\u003e 486\u003cbr\u003e\u003cb\u003eFormat:\u003c\/b\u003e Hardcover\u003cbr\u003e\u003cb\u003eWeight:\u003c\/b\u003e 2.40lbs\u003cbr\u003e\u003cb\u003eSize:\u003c\/b\u003e 9.50h x 6.50w x 1.20d","brand":"K. Kiyek","offers":[{"title":"Hardcover","offer_id":47520380551423,"sku":"9781402020285","price":54.99,"currency_code":"USD","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0662\/2982\/9887\/files\/img_eb76552f-a780-4669-8a4c-87415e0abf60.jpg?v=1762885206","url":"https:\/\/www.whiterainbookhouse.com\/products\/resolution-of-curve-and-surface-k-kiyek-9781402020285","provider":"WR Book House","version":"1.0","type":"link"}