{"product_id":"resolution-of-curve-and-surface-singularities-in-characteristic-zero-von-k-kiyek-j-l-vicente","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}.\u003cdiv class=\"aw-variant-hidden-subtitle-div\" id=\"aw-variant-subtitle-9789048165735\"\u003e\u003ch3\u003e\u003c\/h3\u003e\u003c\/div\u003e","brand":"Libri","offers":[{"title":"Softcover - 9789048165735","offer_id":39416608424029,"sku":"9789048165735","price":53.49,"currency_code":"EUR","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0940\/0622\/files\/74464307-74ae-4296-b358-b0f6f91db1a4.jpg?v=1772088462","url":"https:\/\/shop.autorenwelt.de\/products\/resolution-of-curve-and-surface-singularities-in-characteristic-zero-von-k-kiyek-j-l-vicente","provider":"Autorenwelt Shop","version":"1.0","type":"link"}