course outlinejarod/math582c/slides/...course outline site shears stacks iii t.ee eo as we are here...
TRANSCRIPT
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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
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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
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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
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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
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Summaryofimportantresutscont Assume Noetherian
Today
Idk
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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
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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
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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
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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
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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
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THI R Te U e'tale respsmooth groupoid
7 ulRf is DM stackresp algstack