Table Of ContentLecture Notes in Mathematics 2270
Vincent Cossart
Uwe Jannsen
Shuji Saito
Desingularization:
Invariants and
Strategy
Application to Dimension 2
With Contributions by Bernd Schober
Lecture Notes in Mathematics
Volume 2270
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Vincent Cossart (cid:129) Uwe Jannsen (cid:129) Shuji Saito
Desingularization: Invariants
and Strategy
Application to Dimension 2
With Contributions by Bernd Schober
VincentCossart UweJannsen
UniversitéParis-Saclay,UVSQ Fakulta¨tfu¨rMathematik
LMV(UMR8100)CNRS Universita¨tRegensburg
VersaillesCedex,France Regensburg,Bayern,Germany
ShujiSaito
GraduateSchoolofMathematicalSciences
UniversityofTokyo
Meguro-ku,Tokyo,Japan
ISSN0075-8434 ISSN1617-9692 (electronic)
LectureNotesinMathematics
ISBN978-3-030-52639-9 ISBN978-3-030-52640-5 (eBook)
https://doi.org/10.1007/978-3-030-52640-5
MathematicsSubjectClassification:14-02,14E15
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Abstract
This first part is a course, almost self-contained on Hironaka’s methods providing
rigorousproofsoftheoremsoffunctorial,canonicalembeddedandnon-embedded
resolution of singularities for excellent two-dimensional schemes. The major part
(Chaps.2–9) is written for schemes of arbitrary dimension, in the hope that this
might be useful for further investigations. Chapters 2 and 3 give the classical
notions.InChaps.4and5,weinvestigatethecasewhereanormalcrossingdivisor
hastoberespectedduringtheprocess.InChaps.6–9,weexplainhowtohandlethe
computations.Chapters11–16aredevotedtothespecialcaseofdimension2.
v
Contents
1 Introduction................................................................. 1
1.1 WhatIsDesingularization?.......................................... 1
1.2 VeryShortHistoryofDesingularization............................ 2
1.3 HowDidweStart?................................................... 3
1.4 Summary.............................................................. 4
1.5 ConventionsandConcludingRemarks.............................. 13
2 BasicInvariantsforSingularities......................................... 15
2.1 InvariantsofGradedRingsandHomogeneousIdealsin
PolynomialRings .................................................... 15
2.2 InvariantsforLocalRings ........................................... 21
2.3 InvariantsforSchemes............................................... 25
3 PermissibleBlow-Ups...................................................... 37
4 B-PermissibleBlow-Ups:TheEmbeddedCase......................... 49
5 B-PermissibleBlow-Ups:TheNon-embeddedCase.................... 65
6 MainTheoremsandStrategyforTheirProofs.......................... 79
7 (u)-standardBases ......................................................... 105
8 CharacteristicPolyhedraofJ ⊂R....................................... 117
9 TransformationofStandardBasesUnderBlow-Ups................... 133
10 TerminationoftheFundamentalSequencesofB-Permissible
Blow-Ups,andtheCaseex(X)=1....................................... 145
11 AdditionalInvariantsintheCaseex(X)=2............................ 155
12 ProofintheCaseex(X)=esx(X)=2,I:SomeKeyLemmas ........ 161
13 ProofintheCaseex(X) = ex(X) = 2,II:SeparableResidue
Extensions ................................................................... 167
vii
viii Contents
14 ProofintheCaseex(X)=ex(X)=2,III:InseparableResidue
Extensions ................................................................... 175
15 Non-existenceofMaximalContactinDimension2..................... 191
16 AnAlternativeProofofTheorem6.17 ................................... 201
17 Functoriality,LocallyNoetherianSchemes,AlgebraicSpaces
andStacks................................................................... 205
18 Appendix by B. Schober: Hironaka’s Characteristic
Polyhedron.NotesforNovices ............................................ 211
18.1 Introduction........................................................... 211
18.2 TheNewtonPolyhedronofanIdeal................................. 216
18.3 TheProjectedPolyhedronandItsRelationtotheNewton
Polyhedron............................................................ 221
18.4 TheDirectrixandItsRole:Choosing(u)........................... 228
18.5 DeterminingtheCharacteristicPolyhedron:Optimizingthe
Choiceof(f;y) ..................................................... 233
18.6 InvariantsfromthePolyhedronandtheEffectofBlowingUp .... 238
References......................................................................... 249
Index............................................................................... 255
Chapter 1
Introduction
1.1 WhatIsDesingularization?
LetXbeanirreducibleandreduced excellentnoetherianscheme.
Whatisaregularpointx ∈X?
This means that the local ring (O ,m ) is regular, i.e., that dim mx =
X,x x k(x) m2
dimO , where k(x) = O /m is the residue field. This is equivalent to:xthe
X,x X,x x
gradedringgrm (OX,x)isapolynomialringovertheresiduefieldk(x)=OX,x/mx
x
[Se,(Ch.IVD)Th.9].Whenx ∈Xisnotregular,itiscalledsingular.
(cid:3)
ThegoalofdesingularizationistofindamodelX ofX,whichsharesalotofthe
informationwithX,butwhichisregular,i.e.,whichdoesnothavesingularpoints.
Afirstdefinitionofdesingularizationcouldbe:
Definition1.1 Let X be a reduced, noetherian and excellent scheme. A desingu-
(cid:3)
larization of X is an everywhere regular scheme X and a surjective projective
morphism π : X(cid:3) −→ X such that π induces an isomorphism π−1(U) −∼→ U,
where ∅ (cid:6)= U ⊂ X is a (dense) open subset. The last condition means that π is
birational.
In the case where X is an irreducible and reduced variety over k = C, the
definition above could be rephrased as “there is a global parametrization of X”.
This has been the first motivation to solve this problem [Pu]. Another point of
viewistheclassificationofsingularitiesbytheirdesingularization.Thenoneneeds
a minimal resolution (valid only in dimension (cid:2)2 [Hart, Theorems V.5.7, V.5.8])
or, a canonical procedure of desingularization. Another motivation formulated by
S.S.Abhyankar[Ab4]IntroductionandA.Grothendieck[EGAIV](7.9.6)and[Gr]
inthe1960sistheimportanceforstudyinghomologicalandhomotopicalproperties
ofschemes.
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2 1 Introduction
1.2 VeryShort HistoryofDesingularization
The interested reader might read the picturesque historical article of our late
colleagueH.Reitberger[R]and[Ko]sections1and2.
ResolutionofsingularitiesofcurveswasachievedinthenineteenthCenturyby
three different methods. Indeed, a possibly singular germ of irreducible curve can
beviewedalternatelyas:
– acoveringofaregulargerm(PuiseuxandRiemann),
– anintegraldomainDofdimensionone,essentiallyoffinitetypeovertheground
field(Dedekind),
– orageometricobjectCdefinedbyvariablesandequationsvanishingatacertain
orderatthesingularpoint(M.Noether).
Correspondingapproachestothestudyofthesingularityrespectivelyconsistin:
studyingthelocalfundamentalgroupofthepointedline,thenormalizationof D,
or the effect of a quadratic transform on the order of the equations. While the last
twoapproachesgiveaproof fortheexistenceofaresolutionwhichischaracteristic
free, the first one does not, due to the failure of the Puiseux theorem in positive
characteristic([CP2]Introduction).
Resolution of singularities of surfaces appeared to be extremely difficult. Over
C,J.Walker’sproof[W](1935)isconsideredasthefirstcompleteone. Moreover,
we must quote Albanese’s contribution [Al] (1924) who, by a sequence of stere-
ographic projections (for a projective variety over an algebraically closed field of
characteristic0orp > 0),reachesthecaseofmultiplicity(cid:2) 2whichisnowadays
easytosolve.
ThiswasfollowedbyZariski’stremendouscontributionforsurfaces[Za1,Za2]
and then for three-folds [Za3]. In 1939, Zariski [Za1] proved the existence of
a desingularization for irreducible surfaces over algebraically closed fields of
characteristic zero (i.e., Theorem 1.2 without canonicity or functoriality). Five
years later, in [Za3], he proved the existence of desingularization for surfaces
over fields of characteristic zero which are embedded in a regular threefold (i.e.,
Corollary 1.5, again without canonicity or functoriality). In 1966, in his book
[Ab4],Abhyankarextendedthislastresulttoallalgebraicallyclosedfields,making
heavyuseofhispapers[Ab3]and[Ab5],usingvaluationtheoryandGaloistheory.
Aroundthesametime,Hironaka[H6]sketchedashorterproofofthesameresult,
over all algebraically closed ground fields, using his characteristic polyhedron
[H3], which will also play a crucial role in the present book. Recently a shorter
accountofAbhyankar’s resultswasgivenbyCutkosky[Cu2]andashortproofof
desingularizationoftwo-dimensionalhypersurfacesoveralgebraicallyclosedfields
in[Cu1,Theorems1.2and5.6].
For all excellent schemes of characteristic zero, i.e., whose residue fields all
have characteristic zero, and of arbitrary dimension, Theorems 1.2 and 1.4 were
∗
provedbyHironakainhisfamous1964paper[H1](MainTheorem1 ,p.138,and
Corollary 3, p. 146), so Theorem 1.3 holds in arbitrary dimension as well, except
