LECTURE 6 First properties of alg spaces and stacks

where are weCourse outlineSite shears stacks

iii t.EE Eo aswe are here Moduli of stable curves

QRe Keydets

Let F G be a mapAn algspace is a sheaf X on

of presheavesprestadesover Schet Schey set I scheme

U and

We say F Gis repnbyster U X repnby

schemes e'tale say

tf tf S G from a schemeS DeligneMunford

Fxas is ascheme A DM.SI is a stack It on

we say F Gis repe Schey

set I scheme U andestate sus

tf tf S G from a schemeS U FL representable

Fxas is an algspare

Dictan FES sAnalg.SI is a stack a o

I 1 Scheyset I scheme U and

F G U K representablesmoothlesu

We can discuss properties of maps

repn mpnby salons

EXAMPLES ExI Hironaka Fsmoothproper3ftwith a free 042 actionLast timesa F orbit not contained in any

Egg algebraic aEn9efeiniex1kyExeI.Showthe stackBuy

of week bolls n C of ranker legaltagspaceludtsche

Ex 4 treeis algebraic cT zNXE 6 742 MAT thatGED nonsepacciasmooth Tonk r y X G is algspace

linenot scheme

qayayDMSKY.LITwo reasonscoasennelopsy

singular birationalThfntfeqghsanfanefve.ita SpeckCx7x7ilT The diagonal YtYxY is1cone over a quack not Loc closed inn

EE GmMA in 15 5 7 agx xxx4A'TUNED naffffmet e y 1 4

Summary of importantresuts TO BE

BROKEN

Propertiesofthediagonalf

Keyp Diagonal encodesstickiness

In part the stabilise Gx isan

Recall that Va.be'T IX algspace Infactitisssate

Isaf laid 1TOffer we will impose further condites

T on St xxT p Icab

I xxx Exactive finite

s T n Isomesfftaf b

None of the axis ofbaby stuck

peck dd

define the stabilizer asa 7Speck FIN

Lexx

at Xxx

Summaryofimportantresutscont Assume Noetherian

Today

Idk

So1Properbesofmophismse't

e'tEe Etoile surjective Xxyy'tX v

L Lyl 1Y gonsonekaget

i If P is smoothlocal car desk

Ex almost everythingpropertyPot a Y ifalg spades

except projectivity

Same defn for smooth10cal

on same target

s i EE

a iewww.dim

2 Properties of stacks Topological properties

Def We say a DMstackfias P

V Etalepres U St U Steakleyhas P Speck µ

equiv F speakSame for smoothlocal

Ex Loc ninth regular redued Eye G 4z MIA t.x Xare smooth6cal at GAYA X 1

Upset Make sure of algstak

Wires k9

EE Gm MA t Lxytfxity ICAHN4A

pl cattle.is EEGmMAlCIAYendogyp.enIYed11MUSD

Jefe An algstackIt is quasiempat

conned or irreducible if104 is

A morphism Y is quasicompactif 104 1 ly l is

A morphism yis ftype if

Loc f type quasicompact

Exe Show St g compact

Fsm pres specdA

Exert If Ixxx g compact

1174 sobertop spae

everyirred closed

a ge.ptsubset has

3 Equiv relations groupoids Think of R as ascheme ofrelates

relation rre R M SCH trIn U compositorCutty o xIw lu

w

identity u y

inner Wy m 14

R Te U equiv relation

Fat most one retakesbetween ar two points

of 4

Samefor smooth

Ext G somegthgroup I fiddle Det Let Rst U bea smoothgap

U h scheme w Gactor Define WpfPreas prostate

R a u Usmooth set airports RGB

C pEtaleif

gea u Aguadf.se Define CURT as gndpalguots

equiv retake girlyCree

stackikcalian

Cie axational Exes F cart diagram

E Let be DMstack R 5 µ

Let U 5A Etoilepas tf o LP

R Ux U zU U cakeEtalegropoid

R 4x4

iv relater F Stay spaceI 0 Lpxp

ego qypy icukd.suR7

THI R Te U e'tale respsmooth groupoid

7 ulRf is DM stackresp algstack