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§ o. Some terminology and notation . 196 198 I. Applications to the birational theory of surfaces · . . .. . .. .. . . . . .. . . . 199 § I. Birational behavior of rational singularities . . . . . . . . . . . . • . . . . . . . . . . . . . . . . . . . . . . . . . . 199 § 2. Resolution of singularities by quadratic transformations and normalization (method of Zariski) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ~o I § 3. Conclusion of the proof: a key proposition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203 § 4. Resolution of rational singularities; factorization of proper birational maps into quadratic transformations · . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ~04

Rational singularities, with applications to algebraic surfaces and unique factorization