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A survey, thorough and timely, of the singularities of two-dimensional normal complex analytic varieties, the volume summarizes the results obtained since Hirzebruch's thesis (1953) and presents new contributions. First, the singularity is resolved and shown to be classified by its resolution; then, resolutions are classed by the use of spaces with nilpotents; finally, the spaces with nilpotents are determined by means of the local ring structure of the singularity.

Algebraic geometry --- Analytic spaces --- SINGULARITIES (Mathematics) --- 512.76 --- Singularities (Mathematics) --- Geometry, Algebraic --- Spaces, Analytic --- Analytic functions --- Functions of several complex variables --- Birational geometry. Mappings etc. --- Analytic spaces. --- Singularities (Mathematics). --- 512.76 Birational geometry. Mappings etc. --- Birational geometry. Mappings etc --- Analytic function. --- Analytic set. --- Analytic space. --- Automorphism. --- Bernhard Riemann. --- Big O notation. --- Calculation. --- Chern class. --- Codimension. --- Coefficient. --- Cohomology. --- Compact Riemann surface. --- Complex manifold. --- Computation. --- Connected component (graph theory). --- Continuous function. --- Contradiction. --- Coordinate system. --- Corollary. --- Covering space. --- Dimension. --- Disjoint union. --- Divisor. --- Dual graph. --- Elliptic curve. --- Elliptic function. --- Embedding. --- Existential quantification. --- Factorization. --- Fiber bundle. --- Finite set. --- Formal power series. --- Hausdorff space. --- Holomorphic function. --- Homeomorphism. --- Homology (mathematics). --- Intersection (set theory). --- Intersection number (graph theory). --- Inverse limit. --- Irreducible component. --- Isolated singularity. --- Iteration. --- Lattice (group). --- Line bundle. --- Linear combination. --- Line–line intersection. --- Local coordinates. --- Local ring. --- Mathematical induction. --- Maximal ideal. --- Meromorphic function. --- Monic polynomial. --- Nilpotent. --- Normal bundle. --- Open set. --- Parameter. --- Plane curve. --- Pole (complex analysis). --- Power series. --- Presheaf (category theory). --- Projective line. --- Quadratic transformation. --- Quantity. --- Riemann surface. --- Riemann–Roch theorem. --- Several complex variables. --- Submanifold. --- Subset. --- Tangent bundle. --- Tangent space. --- Tensor algebra. --- Theorem. --- Topological space. --- Transition function. --- Two-dimensional space. --- Variable (mathematics). --- Zero divisor. --- Zero of a function. --- Zero set. --- Variétés complexes --- Espaces analytiques