hilbert–kunz multiplicity of fibers and bertini theorems 2021 rankeya datta.pdf

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HILBERT–KUNZMULTIPLICITYOFFIBERSANDBERTINITHEOREMS
RANKEYADATTAANDAUSTYNSIMPSON
>⊆Pn
k
isanequidimensionalsubschemewithHilbert–Kunzmultiplicitylessthanλatallpointsx∈X,
thenforageneralhyperplaneH⊆Pn,theHilbert–KunzmultiplicityofX∩Hislessthanλatall
k
pointsx∈X∩-Rojas,Schwede
andTucker,whoseconclusionisthesameasourswhenX⊆,we
k
substantiallygeneralizecertainuniformestimatesonHilbert–Kunzmultiplicitiesofflbersofmaps
obtainedbytheaforementionedauthorsthatshouldbeofindependentinterest.
CONTENTS
....................................................................1
....................................2
...................................................................3
................................3
–Kunzmultiplicityofalocalring.......................................4
–Kunzmultiplicity.............................................4
.....................................5
....................................5
......6
..........................................................7
..............8
.....................................10
–Kunzmultiplicityofflbers............................11
–Kunzandgeometricallyreducedflbers.........12
–Kunzandnon-reducedflbers..................18
–Kunzmultiplicity.....................................22
(A2′)forHilbert–Kunzmultiplicity..........................24
–Kunzmultiplicity.........................29
..............................................................31
References.........................................................................31
arXiv:[]22Mar2020

RecallthattheHilbert–KunzmultiplicityofaNoetherianlocalring(R,m)ofprimecharacteristic
p>0,denotedeHK(R),isthelimit
[pe]
ℓR(R/m)
eHK(R):=limedimR.
e−→∞p
AnaturalprimecharacteristicanalogueoftheHilbert–Samuelmultiplicity,eHK(R)hasbeenfre-
quentlyusedtostudythesingularitiesofRsinceitsproofofexistencein[Mon83].Thegeneral
sloganisthattheclosereHK(R)istoone,the“better”,undermild
ThesecondauthorwassupportedbyNSFRTGgrantDMS-1246844.
1:.
2RANKEYADATTAANDAUSTYNSIMPSON
assumptions,eHK(R)=1preciselywhenRisregular[WY00],andifeHK(R)issufflcientlyclose
to1thenRisF-regularandGorenstein[BE04,AE08].
ThegoalofthispaperistoproveaBertinitypetheoremfortheHilbert–
classicalBertinitheoremstatesthatifXisasmoothsubschemeofPnoveranalgebraicallyclosed
k
fleldk,thenageneralhyperplanesectionofXisalsosmooth[Har77,ChapterII,].
Inspiredbythisclassicalresult,oneexpectsthesingularitiesofgeneralhyperplanesectionsofXto
–
KunzmultiplicityandanswersaconjectureofCarvajal-Rojas,SchwedeandTucker[CRST17,Re-
],whoobtainedasimilarresultwithnormalityhypotheses[CRST17,]:
MainTheorem().Letkbeanalgebraicallyclosedfleldofcharacteristicp>0,and
letX⊆≥(O)<λforall
kHKX,x
x∈X,thenforageneralhyperplaneH⊆Pnandforallx∈X∩H,e(O)<λ.
kHKX∩H,x
ThetheoremisinspiredbythefactthatitsanalogueholdsfortheHilbert–Samuelmultiplicity
ofirreduciblesubvarietiesofPnincharacteristic0by[dFEM03,],aresultusually
k
,withoutirreducibilityornormalityhypotheses,theMainTheorem
requiressubstantiallymoreefforttoprove.
Theprimarytoolweemployisawell-knownframeworkdevelopedin[CGM86]toestablish
-
workhasbeensuccessfullyusedtoestablishBertinitheoremsforpropertiessuchasweaknor-
malityincharacteristic0[CGM86],F-purityandstrongF-regularity[SZ13,],the
F-signature[CRST17,],-awarethattheaxiomatic
framework,whichwenowsummarize,allowsonetoproveBertinitheoremsformoregenerallinear
–Kunz
multiplicity().However,wehavechosentoemphasizethemostinterestingcaseof
.
,GrecoandManaresishowedthatif
PisalocalpropertyofNoetherianschemesthatsatisflesthefollowingtwoaxioms,andXisa
subschemeofPnsatisfyingP,thenageneralhyperplanesectionofXalsosatisflesP[CGM86,
k
Theorem1]:
(AX1)Wheneverϕ:Y→ZisaflatmorphismwithregularflbersandZisPthenYisP.
(AX2)Letϕ:Y→SbeaflnitetypemorphismwhereYisexcellentandSisintegralwithgeneric
,thenthereexistsanopenneighborhood
η∈U⊆SsuchthatYsisPforeachs∈U.
Thus,onewaytoprovetheMainTheoremistoestablish(AX1)and(AX2)forthefollowinglocal
propertyofalocallyNoetherianschemeX:
PHK,λ:=eHK(OX,x)<λ,forallx∈X,andaflxedrealnumberλ≥1.
ThatPHK,λsatisfles(AX1)hasbeenknownsincethe1970sbytheworkofKunz(seeTheorem
).ThemaincontentofourpaperisthatPHK,λsatisfles(AX2)withoutnormalityhypotheses.
Thestatementof(AX2)suggeststhatitsveracitywilldependonwhetherPHK,λbehavesuniformly
onthenearbyflbersofaflnitetypemap,sothatwecanspreadoutPHK,λfromthegenerictoa
,thisturnsouttobethecase,andweshowthatafairlygeneralclass
offlnitetyperinghomomorphismsϕ:A→R()satisflesa
uniformconvergenceresultonthegeneralflbersofϕ().
Thestudyofuniformbehaviorisarecurringthemeincommutativealgebraandalgebraicgeome-
try,,Ein,LazarsfeldandSmithusedthe:.
HILBERT–KUNZMULTIPLICITYOFFIBERSANDBERTINITHEOREMS3
theoryofmultiplieridealstoprovesurprisinguniformestimatesonsymbolicpowerandAbhyankar
valuationideals[ELS01,ELS03],andtheirtechniqueshavefoundwide-rangingapplicationsinthe
studyofsingularitiesinequalcharacteristic0,primecharacteristicp>0,andmorerecently,even
mixedcharacteristic(see[HH02,Har05,Tak06,LM09,JM12,Cut14,Li17,Dat17,Blu18,MS18]
forsomeapplications).Moreover,certainuniformHilbert–Kunzestimateswereattheheartof
Tucker’sproofoftheexistenceofF-signature[Tuc12],whichisanimportantprimecharacteristic
invariantthatbehaves,insomeaspects,likethemirrorimageoftheHilbert–-
ilarglobaluniformestimateswerealsousedbySmirnovtoprovetheuppersemi–continuityofthe
Hilbert–Kunzmultiplicity[Smi16],(2).
Thus,wefeelthatourstudyoftheuniformbehaviorofHilbert–Kunzmultiplicityoftheflbersofa
flnitetypemapisinterestinginitsownright,andnotjustforitsrelevanceto(AX2).
NilpotentelementsmustbehandleddelicatelyinthestudyoftheHilbert–
example,[Mon83]and[Tuc12]flrstanalyzeeHK(M)forflnitelygeneratedmodulesoverRred;
moregeneralstatementsthenfollowbyviewingM=Fe0RasamoduleoverR,fore≫0.
∗red0
,butwiththeaddeddifflcultyofuniformly
controllingtheHilbert–KunzmultiplicitiesofthegeneralflbersofϕusingtherelativeFrobenius
:
(1)ShowauniformconvergenceresultonmodulesovergeneralflbersofflnitetypemapsA→
Rwithequidimensionalandgeometricallyreducedgenericflbers();
(2),foraflnitetypemapA→Rwithequidimensional
genericflbers,picke0sufflcientlylargesothatthegenericflbersof
1/pe0
A→(RA1/pe0)red
aregeometricallyreducedandequidimensional,andsuchthatFe0Risamoduleover
∗
(RA1/pe0)red();
(3),showauniformconvergenceresultongeneralflbersofarbitrary
flnitetypemapsA→Rwithequidimensionalgenericflbers().

InthissectionwerecallthebasicfactsaboutHilbert–Kunzmultiplicitythatwillbeusedinthe
article;additionally,wedevelopmachineryonflnitetyperinghomomorphismsthatwillbeintegral
–Kunz
theorywerecommendthesurveyarticlebyHuneke[Hun13].

∈SpecR,wedenotebyκ(p),
κ(p)=Rp/pRp.
IfRhasprimecharacteristicp>0,thee-thFrobeniusendomorphismFe:R→Risdeflned
pee
byr7→-module,F∗MdenotestheR-modulewhichagreeswithMasanabelian
groupbutwhoseR-,ifr∈Rand
eepeee
m∈M,r·F∗(m)=F∗(rm)whereF∗(m)istheelementofF∗
thatRisF-flniteifFeisaflnitemapforsome(equivalently,forall)e>0.
WhenRisreduced(inparticular,adomain),itisconvenienttoidentifytheR-algebraFeRwith
e∗
R1/p,
formoreonournotationalconventions.
ForaflnitelygeneratedR-moduleM,weuseµR(M)todenotetheminimalnumberofgenerators
(R,m,k)islocal,thenrecallthatNakayama’slemmaimpliesthatµR(M)=dimk(k⊗R:.
4RANKEYADATTAANDAUSTYNSIMPSON
M).Inparticular,ifM,NareflnitelygeneratedmodulesoveralocalringR,thenµR(M⊕N)=
µR(M)+µR(N)becausevectorspacedimensionisadditiveoverdirectsums.
–,assumethat(R,m,k)isaNoe-
therianlocalringofprimecharacteristicp>(M)todenotethelengthofaflnitely
generatedArtinianR-∈N,Fe(−)isexactasitisjustrestrictionof
∗

e1/pe
ℓR(F∗M)=[k:k]ℓR(M).()
[pe]pe
IfI⊆Risanideal,denotebyI=hr|r∈-module
M,
e[pe]∼e
F∗(M/IM)=R/I⊗RF∗M.()
Deflnition-.[Mon83,]Let(R,m,k)bead-dimensionallocalNoether-
ianringofcharacteristicp>-module,andIisanm-primary

[pe]ede(d−1)
ℓR(M/IM)=eHK(I,M)p+O(p).
Thelimite(I,M)=limeHK(I,M)existsandiscalledtheHilbert–KunzmultiplicityofMwith
HKe−→∞ped
respecttoI.
IfI=mweuseeHK(M)insteadofthemorecumbersomenotationeHK(m,M).
TheHilbert–KunzmultiplicityisknowntosatisfytheanalogueofLech’sconjectureforthe
Hilbert–SamuelmultiplicitybyHanes’sthesis.
.[Han99,]Let(R,m)→(S,n)beaflatlocalhomomorphismof
(R)≤eHK(S).
–,wedeflne
γ(R):=max{log[κ(q)1/p:κ(q)]|q∈min(R)},()
p
whichfeaturesinthecomputationoflocalHilbert–Kunzmultiplicityinthefollowingmanner:
(R,m,k)beanF-flniteNoetherianlocalringofprimecharacteristicp>0,and
letMbeaflnitelygeneratedR->0,wehave
[pe]e
ℓR(M/mM)µR(F∗M)
edim(R)=eγ(R).
pp
Inparticular,
µ(FeM)
R∗
eHK(M)=limeγ(R).
e−→∞p
e1/pe[pe]
µR(F∗M)=[k:k]ℓR(M/mM)whichisa
eγ(R)1/peedim(R)
consequenceof(),andtheidentityp=[k:k]-
(1)appliedtoaminimalprimeofRandthedeflnitionofγ(R).
-,FeMwillnotbe
∗
aflnitelygeneratedR-moduleevenifMisaflnitelygeneratedR-module.
DeStefani,PolstraandYao’sinsightisthatthepreviouslemmaglobalizes,yieldingarobust
notionofHilbert–Kunzmultiplicityfornon-localF-:.
HILBERT–KUNZMULTIPLICITYOFFIBERSANDBERTINITHEOREMS5
Deflnition-.[DSPY19,]IfRisanF-flni