MATH 249B NOTES: ALTERATIONS 1. Resolution of singularities 2 2. Preparations for the proof of ...

116 Pages·2017·0.63 MB·English

MATH 249B NOTES: ALTERATIONS
LECTURESBYBRIANCONRAD,
NOTESBYTONYFENGANDAARONLANDESMAN
CONTENTS
1. Resolutionofsingularities 2
2. PreparationsfortheproofofdeJong’sTheorem 9
3. Preliminaryreductionsteps 17
4. Constructionofgoodcurvefibrations 21
5. Projection...What’sthepoint? 24
6. Three-pointlemma 37
7. Passagetoaunionofsections 42
8. Stablecurves 43
9. Moduliofstablecurves 51
10. Applicationofmodulistacks 63
11. Semistablecurvesoveradiscretevaluationring 78
12. Resolvingsemistablecurvesoveraregularbase 95
References 116
1
2 BRIANCONRAD,TONYFENG,ANDAARONLANDESMAN
1. RESOLUTION OF SINGULARITIES
1.1. Firstdefinition. Wefirstintroducethenotionofresolutionofsingular-
ities,whichwewilllaterrefineandmodify.
Definition 1.1.1. Let X be a reduced (locally) noetherian scheme. A resolu-
tionofsingularitiesof X isamap f : X(cid:48) → X suchthat
(cid:48)
(1) X isregular,
(2) f isproper,
(3) f is birational: there exists a dense open U(cid:48) ⊂ X(cid:48) and U ⊂ X such
that f restrictstoanisomorphism
F|U(cid:48): U(cid:48) (cid:39) U.
Exercise1.1. Showthat f isbirationalifandonlyifthereexistsadenseopen
U ⊂ Xsuchthat f−1(U) ⊂ X(cid:48) isdenseopenand f−1(U) (cid:39) U. [Hint: shrink
U inthedefinitionbyremoving f(X(cid:48) −U(cid:48)).]
There are stronger notions of resolution of singularities. To motivate
them, let us first mention that for “nice” X (where nice means “excellent”),
theregularsetReg(X) ⊂ X,definedas{x ∈ X: O regular},isZariskiopen
x
in X. If X is finite type over an algebraically closed field, this can be seen
concretelyintermsofJacobians.
For resolutions of singularities, one could further ask for the open sub-
scheme U in (1.1) to be the full regular locus, and (X(cid:48) − f−1(U)) ⊂ X(cid:48) to
red
have nice geometric structure, namely to be a strict normal crossings divisor
(tobedefinedsoon).
Remark1.2. Attheendoftheday,we’llbemostinterestedinvarietiesover
an algebraically closed fields. However, in the middle of arguments one
sometimespassesoutofthisrealm(suchastoworkovercompletionsoflo-
cal rings), so it’s useful to have a more general framework. Also, de Jong’s
method in [deJ] adapts to apply to a relative situation over discrete valua-
tion rings (so, for schemes which are not over a field); this is important for
arithmetic applications and is addressed in the final part of his original pa-
per (and for reasons of time we will say nothing about that in this course,
but we will treat some intermediate steps in wide enough generality to be
applicableintheadaptationtoworkingoverdiscretevaluationrings).
1.2. Strictnormalcrossingsdivisors.
Definition 1.2.1. Let S be a regular scheme. A strict normal crossings divisor
(sncd)in S isaclosedsubscheme D definedbyaninvertibleidealI ⊂ O
D S
(=: effectiveCartierdivisor)suchthat
(1) D isreduced,
(2) the(reduced)irreduciblecomponents {D } of D areregular,
i i∈I
MATH249BNOTES:ALTERATIONS 3
(3) (mostimportant)foranyfinitesubset J ⊂ I,
(cid:92)
D = D
j j
j∈J
isregularofcodimension#J atallpoints;i.e. forall ξ ∈ D
J
dimO = dimO −#J.
D ,ξ S,ξ
J
Exercise 1.3. Show that the definition is equivalent to saying that for all
ξ ∈ D andtheset {D } of D ’sthrough ξ,wehave
i i∈I i
ξ
I = t O
D,ξ i S,ξ
i
forsuch i with {t } partofaregularsystemofparametersofO .
i i∈I S,ξ
ξ
Oneunfortunatefeatureofthedefinitionsncdisthatitisnot“verylocal”;
i.e., not local for the e´tale or “analytic” topologies. This is due to the basic
butfundamentalfactthatirreducibilityisnote´tale-local(ormoreconcretely,
acompletionofanoetherianlocaldomainneednotbeadomain):
Example 1.4. Let D = {y2 = x2(x+1)} ⊂ A2, for char k (cid:54)= 2. This integral
k
divisor fails to be a sncd because it is non-regular at the origin. However,
“very locally” at the origin (namely, upon passing to the finite e´tale cover
near (0,0) defined by u2 = 1+x) it looks like the intersection of two trans-
versely intersecting regular divisors y = ±ux, which is what an sncd looks
likeatitsnon-regularpoints. Anotherwaytomakethispreciseistopassto
completions: Spec O(cid:98) ⊃ Spec O(cid:98) is a sncd (since the completion con-
√ A2,0 D,0
tains 1+x,byHensel’sLemma).
Definition1.2.2. AreducedCartierdivisor DinaregularschemeSisanor-
malcrossingsdivisor(ncd)ifforallξ ∈ D,thereexistsane´taleneighborhood
S(cid:48) → S of ξ suchthatsuchthat (S(cid:48),D(cid:48) := S(cid:48) × D) isansncd.
S
Remark 1.5. For S excellent, D ⊂ S is an ncd if and only if for all ξ ∈ D the
closed subscheme Spec O(cid:98) ⊂ Spec O(cid:98) is an ncd. This is not obvious; it
D,ξ S,ξ
uses Artin-Popescu approximation (which we will discuss later); the hard
partistoshowthatbeingansncdaftercompletionatapointensurestheex-
istence of an e´tale neighborhood (much less drastic than completion!) over
which D becomes an sncd. This is useful in practice because completions
are often a more convenient framework to perform concrete calculations
(e.g.,thenatureofe´talemapssimplifiesalotafterpassingtocompletions).
To get back to a sncd from a ncd at the end of the day, we will need to
knowtherelationshipbetweenthetwonotions.
4 BRIANCONRAD,TONYFENG,ANDAARONLANDESMAN
Exercise1.6. Showthatanormalcrossingsdivisorisastrictnormalcrossing
divisor precisely when its irreducible components D (given the reduced
i
structure)areregular.
1.3. Hironaka’s Theorem. Hironaka proved that in characteristic 0, vari-
eties have the best imaginable resolutions of singularities. Here is a sample
version of his result (other variants are also available, in analytic settings
andfor“embeddedresolution”,etc.):
Theorem 1.3.1 (Hironaka). Let X be reduced, separated of finite type over
afield k ofcharacteristic0. Thereexistsaresolutionofsingularities X(cid:48) → X
suchthat
(1) f isanisomorphismoverU := Reg(X) = Xsm ⊂ X.
(2) (X(cid:48) − f−1(U)) ⊂ X(cid:48) isncd.
red
Remark 1.7. What Hironaka proved was much stronger. For example, he
showed that f can be built as a composition of blow-ups along smooth
closedsubschemes(wewilldiscussblow-upsindetaillater).
It is a wide open question whether this holds in characteristic p > 0. de
Jong’s alterations provide a substitute notion that works in most applica-
tions,inanycharacteristic(andeveninsomerelativesituations).
1.4. Alterations. For most applications, one can weaken the birationality
aspectofresolutionofsingularitiesasfollows.
Definition 1.4.1. Suppose X is an integral noetherian scheme. An alteration
of X isamap f : X(cid:48) → X suchthat
(cid:48)
(1) X isintegralandregular,
(2) f isproperanddominant,
(3) [k(X(cid:48)): k(X)] < ∞.
Exercise1.8. Showthatthethirdconditionisequivalenttotheexistenceofa
dense openU ⊂ X such that f−1(U) → U is finite (and flat, after shrinking
again);notethat f−1(U)isautomaticallydensein X(cid:48) sinceitisanon-empty
(cid:48)
open subset (by dominance of f) and X is assumed to be irreducible (even
integral).
deJong’smaintheoremisaweakenedformofHironaka’sTheorem,with
alterationsinplaceofresolutionofsingularities,inwhichonelosesallcon-
trol over the open subscheme U over which the morphism is actually finite
(in particular, we cannot guarantee that it contains any particular closed
pointintheregularlocus):
Theorem 1.4.2. Let X be a variety over a field k; i.e., an integral separated
scheme of finite type over k. For Z ⊂ X any proper closed subset, there
MATH249BNOTES:ALTERATIONS 5
existsanalteration f : X(cid:48) → X andanopenimmersion
J: X(cid:48) (cid:44)→ X(cid:48)
suchthat
(cid:48)
(1) X isregularandprojective,
(2) f−1(Z)∪(X(cid:48) − j(X(cid:48)))) ⊂ X(cid:48) issncd.
red
(cid:124) (cid:123)(cid:122) (cid:125)
complementof f−1(X−Z)
Remark 1.9. In the statement of the theorem, why not simply take Z to
be empty? The answer is that the theorem will be proved by an inductive
means, and we really need the flexibility of a closed subscheme Z for the
inductive step (even if in practice we may often only care about the case
of empty Z). Informally, the role of Z is inspired by Hironaka’s result on
embeddedresolutionofsingularities(whichwedidnotdefine).
Remark 1.10. If k is perfect, the proof will give that we can arrange for the
extension k(X(cid:48))/k(X) to be separable (so f : X(cid:48) → X is finite e´tale over
some dense open in X). The proof doesn’t control the degree of this field
extension,butGabberlatermadeanimprovementarrangingthedegreenot
to be divisible by any desired finite set of primes away from char(k). In
laterwork,deJongprovedanequivariantversionofhisresultfortheaction
by finite groups. But we stress again that the method cannot control U: it
doesn’tevenguaranteefinitenessof f overanyparticularclosedpointof X.
1.5. Applications. We now give some applications of alterations (where
they can be used as a substitute for resolution of singularities), and also
an example where de Jong’s Theorem appears to be insufficient to replace
Hironaka’sTheorem.
1.5.1. Grauert-Remmert Theorem. Let X be a normal C-scheme, locally of fi-
nite type. Given a surjective finite e´tale map E → X, the analytification is a
finite-degreecoveringspace Ean → Xan. Thisdefinesafunctor
Fe´t(X) → Fe´t(Xan)
between finite e´tale X-schemes and analytic spaces finite e´tale over Xan
(which is categorically “the same” as finite-degree proper local homeomor-
phismsto X(C));thesefinitee´talemapsmaynotbesurjective.
The theorem of Grauert and Remmert is that this functor is an equiva-
lence of categories. This is amazing because it holds even when X is not
proper! TheideaoftheproofistouseHironakatoreducetothecasewhere
X is the complement of a normal crossings divisor in a smooth projective
variety. In the projective case, we can use GAGA. The details are carried
over[SGA1,XII].
6 BRIANCONRAD,TONYFENG,ANDAARONLANDESMAN
Exercise 1.11. Adapt Grothendieck’s argument in [SGA1] to use just alter-
ations,ratherthanresolutionofsingularities.
1.5.2. Artin comparison theorem. Let X be a separated, finite type scheme
over C,andF aconstructibleabeliansheafon X ,oraconstructible (cid:96)-adic
e´t
sheaf.
Theorem1.5.1. Thenaturalpullbackmap
(1.1) H∗(X ,F) −→∼ H∗(Xan,Fan)
e´t
isanisomorphism.
Remark 1.12. This is also true, and much easier to prove, for cohomology
withcompactsupports. Whyisthatcaseeasiertoprove? Thereasonisthat
cohomologywithcompactsupportsismuchbetterbehavedwithrespectto
stratification by locally closed sets due to the excision sequence involving
∗
H ’s throughout; in contrast, the excision sequence for ordinary cohomol-
c
ogy involves an anomalous term H∗(X,·) for cohomology with supports
Z
along a closed set, and that messes up the induction. The upshot of the
∗
more convenient excision sequence for H is that it permits a reasonably
c
rapidreductiontothecaseofcurves.
The proof of (1.1) uses resolution of singularities in dimension ≤ dimX.
For an analogue with non-Archimedean e´tale cohomology allowing posi-
tive characteristic, an entirely different proof was developed by Berkovich
that adapts almost verbatim to the complex-analytic setting above for con-
F
structible abelian . So for this much, resolution of singularities is not
needed. However, it is needed to bootstrap to the (cid:96)-adic case (that, oddly
enough, does not seem to be documented in the literature); this is handled
by a beautiful argument of Deligne that can also be carried out using alter-
ations.
1.5.3. Deligne’stheoryofmixedHodgestructures. Thereisanotionof“proper
hypercovering”,whichisanabstractionofCechcoveringsandallowsother
types of maps than “open embeddings” to be used in the study of suffi-
ciently topological cohomological investigations. In particular, due to the
proper base change theorem in e´tale cohomology, it is fruitful to work with
hypercoverings whose constituents are proper smooth varieties with suitable
“sncd boundary”, and such hypercovers can be built by systematic use of
deJong’stheoreminsteadofHironaka’sTheorem.
1.5.4. A non-application. Here is an example in which de Jong’s alterations
do not appear to suffice in place of Hironaka’s Theorem. Let X be a smooth
MATH249BNOTES:ALTERATIONS 7
scheme over C, separated of finite type. We can form the algebraic de Rham
complex
Ω• = (O −→d Ω1 −→d Ω2 −→d ...)
XC X X/C X/C
Grothendieck defined the algebraic de Rham cohomology to be the hyper-
cohomologyofthiscomplex:
Hi (X/C) = Hi(Ω• ).
dR X/C
Ω•
There’s a natural “analytification” (being careful about the fact that
Xan/C
O
the d-mapsarenot -linear!),soonehasamap
X
θi : Hi (X/C) = Hi(Ω• ) → Hi(Ω• ).
X dR X/C Xan/C
By the ∂-Poincare´ Lemma, the latter is isomorphic to Hi(X(C),C). In the
proper case, GAGA combined with a spectral sequence for hypercohomol-
ogyimpliesthat θi isanisomorphism.
X
GrothendieckprovedinIHES29thatθi isanisomorphismevenwithout
X
properness. AstrikingconsequenceisthatifXisaffine,thenHi(X(C),C) (cid:39)
Hi(Ω• ) = Hi(Ω• ) (the first isomorphism by Grothendieck’s result,
X/C A/C
and the second by acyclicity of coherent sheaves on affines). That is, topo-
logicalC-valuedcohomologyofX(C)forsmoothaffineX canbecomputed
entirelyintermsofalgebraicdifferentialforms!
How did Grothendieck achieve this? The idea is to bootstrap from the
proper case by first forming a compactification X (cid:44)→ X and then using a
strong enough form of Hironaka’s theorem (applied to X) to find a reso-
lution of singularities X(cid:48) → X such that if X(cid:48) is the pre-image of X then
not only is X(cid:48) → X an isomorphism but also X(cid:48) −X(cid:48) is a normal crossings
divisor. Inotherwords,wehaveaCartesiandiagram
X(cid:48)(cid:31)(cid:127) (cid:47)(cid:47) X(cid:48)
(cid:15)(cid:15) (cid:15)(cid:15)
(cid:31)(cid:127) (cid:47)(cid:47)
X X
whose left side is an isomorphism that thereby identifies X as the comple-
(cid:48)
ment of a normal crossings divisor in the smooth proper X . Grothendieck
relatedthedeRhamcohomologyof X = X(cid:48) tothehypercohomologyofthe
an
deRham complex on X(cid:48) modified to permit controlled poles along that
normalcrossingsdivisor.
The key point is that one knows that X is untouched by the resolution of
singularities, being in the smooth locus of X. This is what breaks down for
alterations. IndeJong’sversion,onecannotcontrolwherethemaprestricts
8 BRIANCONRAD,TONYFENG,ANDAARONLANDESMAN
to a finite morphism, let alone an e´tale map. This is bad because formation
of differential forms interacts badly with respect to a map that is not e´tale,
Ω• Ω•
soonecan’trelate to inanyusefulway.
X/C X(cid:48)/C
Remark1.13. Bhattprovedaresultthatgetsaroundthisissueinthelcicase,
establishing a version of Grothendieck’s isomorphism beyond the proper
casewithlcisingularitiesbyworkingwiththeso-calledinfinitesimalsite(in
lieuofthedeRhamcomplex,towhichitiscloselyrelatedonlyinthesmooth
case)andusingmoreadvancedhomologicaltoolsthanwereavailableinthe
1960’s.
1.6. Regular vs smooth. We want to emphasize the technical distinction
between regular and smooth. Let S be a locally finite type scheme over a
field k.
O
Definition 1.6.1. The scheme S is regular if all the local rings are reg-
S,s
ular. (It is sufficient to verify this at closed points s ∈ S, in view of Serre’s
theorem that the localization at a prime ideal for any regular local ring is
againregular.)
TheschemeSissmoothoverkifS isregular. Thereareseveralequivalent
k
formulationsofthis“geometric”condition:
• Sk(cid:48) isregularforallfiniteextensions k(cid:48)/k,
• Sk(cid:48) isregularforallextensions k(cid:48)/k,
• Sk(cid:48) isregularforoneperfectextension k(cid:48)/k,
• S → Spec k satisfiestheinfinitesimalcriterionforsmoothness.
Smoothness over a field k always implies regularity, but the converse is
false (if k is not perfect). For instance, smoothness is preserved by ground
fieldextension,butregularityisnot:
Exercise 1.14. Let k be an imperfect field of characteristic p > 0, and a ∈
k − kp. Pick m > 1 such that p (cid:45) m (e.g., m = 2 when p is odd). Check
that {ym = xp −a} ⊂ A2 is Dedekind, but obviously C = {ym = zp} (for
k k
z := x−a1/p)isnotregularat (0,0).
Example1.15. Let K/k(x) beafiniteextension. Considerthediagram
K ⊃ A
finite integralclosure
K(x) ⊃ k[x]
Then Spec A is dense open in a unique projective k-curve C with function
field K. It can happen that C is geometrically irreducible over k yet C is
k
MATH249BNOTES:ALTERATIONS 9
nowhere reduced, even if k ⊂ K is relatively algebraically closed. (We de-
mand this final condition on how k sits inside K to avoid the lame example
inwhich k admitsanon-trivialpurelyinseparableextensioninside K.)
Exercise 1.16 (MacLane). Take k = F (s,t), Spec A = {sxp +typ = 1} ⊂
p
A2. Show that A is Dedekind, and k ⊂ Frac(A) is relatively algebraically
closed. Evidently, after adjoining the pth roots of s and t the curve Spec A
becomeseverywherenon-reduced.
In de Jong’s theorem with imperfect k one cannot expect to arrange that
the k-variety X has an alteration X(cid:48) → X with X(cid:48) generically smooth over k.
To see what is an obstruction to this, suppose a smooth alteration exist, so
wehavebetweendenseopensafiniteflatsurjection
U(cid:48) ⊃ X
finiteflat
(cid:15)(cid:15)
U ⊃ X(cid:48)
(cid:48)
Regularitydescendsacrossfaithfullyflatmaps[Mat2,23.7(i)]. SoifU were
(cid:48) (cid:48)
k-smooth (as we could arrange if X were generically k-smooth) then U
k
were regular, so U would be regular, so X would be generically smooth;
k
equivalently, k(X)/k would be separable in the sense of field theory (i.e.,
k(X)⊗ k is reduced for all finite extensions k(cid:48)/k). But this latter property
k
canfailevenifk ⊂ k(X)isrelativelyalgebraicallyclosed,asillustratedwith
MacLane’s example above. Thus, regularity without generic smoothness is
thebestwecanhopeforwhen k isimperfect.
Remark1.17. TheregularitywillbeachievedindeJong’sproofbybuilding
(cid:48)
analteration X thatissmoothoverafiniteextension(typicallyinseparable)
of k. Thatis,evenif k isalgebraicallyclosedin k(X),wedonotarrangethat
k isalsoalgebraicallyclosedin k(X(cid:48)).
2. PREPARATIONS FOR THE PROOF OF DE JONG’S THEOREM
2.1. Basic outline. Our object of interest is a variety X over k. By a trivial
application of Chow’s Lemma, we may assume that X is quasi-projective.
Also,wewillseethatitsufficestoreplacekwithk;eventuallywewillchase
coefficientsin k tomakethedesiredconstructionover k fromoneover k.
The method involves induction on d = dimX. Using normalization, the
cases d ≤ 1 are easy. In general, with d ≥ 2, we’ll blow up X appropriately
10 BRIANCONRAD,TONYFENG,ANDAARONLANDESMAN
sothatitis“fiberedincurves”,meaningthatthereisasurjectivemap
X
f
(cid:15)(cid:15)
Pd−1
k
Withallgeometricfibersconnectedofdimension1(butpossiblywithnasty
singularities or non-reduced), and such that the generic fiber X is smooth.
η
M
Next we will use properties of the moduli stack and the Semistable
g,n
Reduction Theorem for curves (which will be discussed shortly), to pro-
duce an alteration Y → Pd−1 and also an alteration X (cid:16) X fitting into a
k
commutativediagram
X alteration (cid:47)(cid:47)
X
f(cid:48) f
(cid:15)(cid:15) (cid:15)(cid:15)
Y (cid:47)(cid:47) Pd−1
alteration k
(cid:48)
where f isflatwithsemistablegeometricallyconnectedfibersofdimension
1 (and smooth generic fiber). By induction, we can find a regular alteration
of Y(cid:48) → Y, so X(cid:48) := X × Y(cid:48) (which remains integral) is an alteration of X
Y
(cid:48)
thatissemistableoveraregularY (withsmoothgenericfiber). Then,inthis
(cid:48)
nicesituation,weresolve X byhand(usingcarefully-chosenblow-ups).
Remark 2.1. Non-trivial extensions of function fields arise in the method
(i.e., alterations rather than birational maps) due to the role of field exten-
sionsintheSemistableReductionTheoremasdiscussedbelow.
2.2. Semistable curves. We first discuss the notion of semistable curves
over a field k; the version over more general base schemes will be crucial
lateron,atwhichtimewewillrevisitthetopicfromawiderpointofview.
Theorem 2.2.1. Let X be a finite type scheme over k of pure dimension 1.
Forclosedpoints x ∈ X,thefollowingareequivalent:
(1) As k-algebras, O(cid:98) is isomorphic to k[[t]] or k[[u,v]]/(uv) for some
X ,x
k
k-point x over x,