lasting - sites.math.washington.edu

12
TODAY Existence of coarse moduli spaces Lasting Quasi coherent sheaves on DM stacks Local structure of DM stacks

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Page 1: Lasting - sites.math.washington.edu

TODAY Existence of coarse modulispaces

LastingQuasi coherentsheaves on DM stacks

Local structure of DM stacks

Page 2: Lasting - sites.math.washington.edu

o.R ew View X as the closest approximationto A which is an adgspace

Tradeoff1 univ property6gFun family

myX moretango

ideally Xiprojective

I.Definitalgstackgatspacestrategy show existence of CMJ

Det A map TTX is a coarse

modulispace Lor cry if Specialcases If Especially

lil fleet Alwyn 5 44 the finiteaus

2 SpeeAke SpecAislamclassesofobjects in Stael Use LocalStructureThm

2 it is universal for mapsto speedfGD Iet Ealgspace ie

St Leoni

ftp yaaYsr

specaax s

glue these in Etaletopology

Page 3: Lasting - sites.math.washington.edu

Rukia 7lb

ftp.Pe we desire stronger µ t.gl

Plxod rLx.dxj

For us if St Sep DM staleX

we will construct X Ptsd Ga Phdto at 40 448

satisfying sci x P X X 0stable under flat base charge

If 0 OxDescortleennyalgstalydgspse

IT is proper Linpartseparated

Let St X be a nap

IT univ homeomorphismIt Xi X Elate cover Lewes Ippt

ISH TIXet Xxxxi 1 Xi ans this

x axis

Page 4: Lasting - sites.math.washington.edu

SozQuotientsbyfinitegroupsy LemmaI Let AG Bringmap

Let G finite group SpecASo G acts on SpecBxoaaAIG eeu.irConsider

SpectB A 1SpeeADefine AG at Ale.EE I7Ig Lawn Law

hoon er A 4Lemma 1 Let R noethring SpalBAF speeB Speth

ykIf A fin genR algebra

then Aa B flat T B J BDµAG

At 1 A Quite AhfingerRay

Ah A integral if a cA In general 4 integralunivhomeo

GILx gac AGED monic poly 1

02 AG A IT Awith a as a root gth

Fenin I naII n9fe FABYE Aa o note.amaDIgIIttR

aH

Conn ayAh finger alR Ng exercise

Page 5: Lasting - sites.math.washington.edu

Gx nad D 4

Ff EA si Ha 1 flax 0

ftp.tagf EAI f'fit flexiSnow Ah 1AAnise Ahfinger R tTLxl4IT

leptin it is a proper univ homeo 1 LT bij or geo pts

it is bij on geom pts since IT is proper ITcontinuous

Consider 4 IT is a homeoSpee A finitegmanat ForATBfinite it I

5 67 Speed SpectB A 1SpaAfgpN honed

Claims it is incomeon gearpts ftp.uwf uhmoYI

Assume R h hi SpacBffAF SpeeB speeth

Let x X CSpecA closedptshomeo

with Gx Gx

Page 6: Lasting - sites.math.washington.edu

Let E be equaliser of XLYPrecisely E FpkX

I o tY rYxY

But I X proper sch domint

ET X4

SlepI IT is univ formaps to chgspace so E TX mono

Espe.AM 1X SpeeAa effpe4 E ixd.gl

fit sa dominated ETXwt.J.iq coil Existence

Raki If 4 is actingthe

plydy Pbl Question is Elate local on

4 Rease Etatedesert urn property

dukes x PlXdid Assume Ah is strictlyHerodian

Lil Uniqueness Suppose Can also assure Y is g compactSt

1 71E uz

Page 7: Lasting - sites.math.washington.edu

Ah stoutlyherselan 3 Reducing to quotientstacksY g compact hers

eat Our strategy44ysfear.gs jAygieyasCspaBKx7

j Ym WEwFWEspaHaIE E

L O f tous forSpecBox Spadix

alt1

t y E Need this is a e'the

ftp.EI.name9niY quinlan

EET th Quesi If f SpecATSpacB

EmperorYAat G egu.ir e'tale when is

e'tale at x 7 ATB Specht specBG e'tale

5 e'tale at tax 4nF 4,34fax pyang

Upshot If G G jwe win

Page 8: Lasting - sites.math.washington.edu

Cory It in additu Cal lb

hold at all closed pts then

SpecAhf CSperBld

D

Sp Aa Etspates

Page 9: Lasting - sites.math.washington.edu

P Question is Etoilelocal aroundTBly

Kos assure Bs shirtybus

SpaA SpecB hers

GpuBk7 L Link'sElspa spaBk

I I hen

SpeeBa SpeSpeeBA

N

Page 10: Lasting - sites.math.washington.edu

54.2K Threeingredientsofproot

Special

AYE.io'Ie

Page 11: Lasting - sites.math.washington.edu

Needto show G holds in a ngbdotw

2 Sep Be a

L ta 74 xxx

Dr Assure 5 Speck active Consider mapofgoopstyQuestion is Zariski local on

IqWEI FEELEY

a sakes to show that to ap D finite1 Awful Awk

closedpt x Clad I openngbd EXzoqeiwqwuefhacaoasemol.gs

Let Speck 12 be a npncfxwlbEW iotalt yw wxawd.ge

Set G GI Efaw 4 open

Local Startin Tbn 1

specHad E Q PillWxxEchIw e W

w x is precisely where Wyat is

61 Authad Auth htt stabbir presetryCan arrange W A stabpres

Page 12: Lasting - sites.math.washington.edu

Cheek R WxW mono

WIR ay space quohut

Use estate closetspecBtf

Q wxawEHWtsputk.IE Ewhy is X sep

1cm o Ion o t Bk x X

Re W SpuAG 1 X sepet

Shue W 12L stab pres so

is KaewTwo square cortesia a

R 4W e'talegapoidof awe shares