## Resolution of Singularities: A research textbook in tribute to Oscar Zariski Based on the courses given at the Working Week in Obergurgl, Austria, September 7–14, 1997Herwig Hauser, Joseph Lipman, Frans Oort, Adolfo Quiros In September 1997, the Working Week on Resolution of Singularities was held at Obergurgl in the Tyrolean Alps. Its objective was to manifest the state of the art in the field and to formulate major questions for future research. The four courses given during this week were written up by the speakers and make up part I of this volume. They are complemented in part II by fifteen selected contributions on specific topics and resolution theories. The volume is intended to provide a broad and accessible introduction to resolution of singularities leading the reader directly to concrete research problems. |

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### Contents

1 | |

Contributions | 15 |

A computer program for the resolution of singularities 231 | 29 |

Resolving singularities of plane analytic branches | 31 |

Classes of the Working Week | 32 |

Alterations and resolution of singularities | 39 |

The alteration theorem | 45 |

Moduli of curves | 80 |

Transformation of flags under blowup | 359 |

Construction of the induction invariant | 364 |

Transformation of iax under monomial blowup | 367 |

Reduction to monomial blowup | 369 |

Proof of Theorem 1 1 | 371 |

References | 372 |

A J de Jong An application of alterations to Dieudonné modules | 375 |

The application of alterations | 376 |

Existence of tautological families | 96 |

Moduli automorphisms and families | 100 |

References | 104 |

J M Aroca Reduction of singularities for differential equations | 109 |

Singular foliations codimension one | 111 |

Reduction of singularities in dimension two | 115 |

Existence of integral curves | 123 |

References | 126 |

J M Aroca Puiseux solutions of singular differential equations | 129 |

The Newton polygon of a differential equation | 133 |

Solutions for first order first degree equations | 139 |

References | 144 |

S Encinas O Villamayor A course on constructive desingularization and equivariance 147 | 146 |

First definitions | 150 |

Pairs | 153 |

Constructive desingularization | 156 |

Basic objects | 157 |

Monomial case | 165 |

Compatibility of the function twith induction | 170 |

Equivariant desingularization | 179 |

Change of base field and generalization to the nonhypersurface case | 193 |

Proofs | 200 |

Order of ideals and uppersemicontinuity | 208 |

Blowups | 210 |

Desingularization | 212 |

On basic objects | 213 |

A doityourself help guide of theorem 7 10 | 219 |

Nonembedded constructive desingularization Compatibility with group actions and formal isomorphisms | 221 |

References | 225 |

Cossart | 239 |

D | 259 |

B van Geemen F Oort | 285 |

T Geisser | 299 |

with one toric morphism | 315 |

Puiseux expansion and the semigroup of a curve | 317 |

Deforming curves | 319 |

Resolution using toric morphisms | 321 |

Existence of a toric resolution | 325 |

Simultaneous resolution | 331 |

The transforms of plane curves | 334 |

An example | 335 |

References | 339 |

H Hauser Excellent surfaces and their taut resolution | 341 |

Preliminaries | 346 |

Definition of the centers of blowup | 348 |

Transformation of equimultiple locus under blowup | 353 |

Essential surjectivity up to isogeny | 378 |

References | 380 |

F V Kuhlmann Valuation theoretic and model theoretic aspects of local uniformization | 381 |

Local uniformization and the Implicit Function Theorem | 388 |

Hensels Lemma | 389 |

A crash course in ramification theory | 394 |

A valuation theoretical interpretation of local uniformization | 399 |

Inertial generation and Abhyankar places | 401 |

The defect | 404 |

Maximal immediate extensions | 408 |

A quick look at Puiseux Series fields | 410 |

The tame and the wild valuation theory | 412 |

Some notions and tools from model theoretic algebra | 415 |

Saturation and embedding lemmas | 421 |

The Generalized GrauertRemmert Stability Theorem | 424 |

Relative local uniformization | 428 |

Local uniformization for Abhyankar places | 432 |

NonAbhyankar places and the Henselian Rationality of immediate function fields | 433 |

Bad places | 437 |

The role of the transcendence basis and the dimension | 439 |

The space of all places of FK | 443 |

Fet | 446 |

Local uniformization vs AxKochenErshov | 448 |

Back to local uniformization in positive characteristic | 451 |

References | 452 |

T Lé Les singularités Sandwich | 457 |

Normalisation et éclatements | 458 |

Singularités Sandwich | 460 |

Approche combinatoire | 462 |

Singularités primitives singularités minimales | 472 |

Courbes polaires des singularités rationnelles | 475 |

Transformation de Nash des singularités minimales | 477 |

La résolution des singularités des surfaces via la transformée de Nash | 479 |

Références | 481 |

482 | |

Equisingularity and simultaneous resolution of singularities 485 | 484 |

Stratifying conditions | 488 |

The Zariski stratification | 494 |

Equiresolvable stratifications | 498 |

Simultaneous resolution of quasiordinary singularities | 501 |

References | 503 |

Resolution of weighted homogeneous surface singularities | 507 |

F | 518 |

H Reitberger | 533 |

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### Common terms and phrases

Abhyankar Algebraic Geometry algebraically closed algorithm analytic assume basic object birational blowup Cartier divisor chart closed point codimension cone construction coordinates courbe curves of genus defined Definition denote desingularization differential equations dimension embedded equivariant example exceptional divisor Exercise exists extension fiber foliation follows function field functor henselian Hilbert scheme Hilbert-Samuel Hironaka holomorphic hypersurface ideal induction integral intersection invariant irreducible component isomorphism Lemma maximal Maxt Maxw-ord moduli scheme moduli space monomial morphism multz Newton polygon nonsingular normal crossings normal crossings divisor pointed curves polynomial projective proof prove regular Remark resolution of singularities Section Show Sing(J singular locus singular point singularité smooth solution stable curves stable n-pointed curve stable pointed strict transform subscheme subset surface Theorem théorème theory toric variety toroidal vector w-ord X C W Zariski