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Resumen:
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This monograph presents a local theory of planar and space curve singularities both from an algebraic and geometric point of view, being motivated by possible extensions of the Zariski equisingularity theory to the case of space curves and by making links to the theory of arc spaces, resolution of singularities, valuations. The book covers the following topics. The first chapter is devoted to branches at points, defined in several different ways and characterized by parameterizations, notably the Hamburger-Noether and Puiseux expansions. The second chapter treats the geometric features and numerical invariants extracted from local rings, and among them semigroup of values, Arf closures (multiplicity sequences by successive blow-ups), saturation (invariants of plane projection), including their computation through parameterizations. The third chapter introduces sequences of infinitely near points and their representation via Hamburger-Noether matrices. In chapters four and five, infinitely near points are studied geometrically by means of the divisorial theory of singularities. Many results hold over arbitrary perfect fields, which makes sense from the arithmetic and computational point of view. In general, the presentation is clear and self-contained, and satisfies most of readers' requirements
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