Oh .1& Ch.
2 Affine & Projective Tork Varieties.
Chris Eur
I. Algebraic torus
*h_alg.tv# ( a - variety) Te(E*Y= { ( an . . same And aitovil
.
= Spec EGG, .⇒Cn]x
, ,↳ . .xn= Specified!1! [ 2 " ]
.
Fact alg . map ( E*M→( GYM correspond to A :Im→Zn.
Eg= ( e*p→(e*P ( s ,t)i→ ( s2,
st ,t4.
A=[ 20'
, :].
!1! [ 22 ]← E[ 7<3 ].÷I M= Horn ( T
,E* ) characterIatt= ( 01 . )
N= Horn ( E*,T ) oneparametersubgrplattice Crow )• Pairing :
MxN→Z( E*→T→e* is the )
.
• T= Spec E[ M ] = N @ e*
Deft A toricvariety_ X is variety wl a dense open torus TCX st
T×T→T extends toTx÷I. Affine toric varieties -Construction M€2
"
a lattice . A = { al ,- yar }CM . Then consider
Cnn . &H -8 ) T→€r Top : the ( ya 'Ct ),
Xa2C±1 ,...
, Xa " Ct ) ).
Then take YA = IMI.
[ to ] ,[2o ] CHIH ( s,
sz )
E.ge 14=2,
A= { 1,2 }.
⇒ E*→¢2 th ( t.tt )
YA = V ( y - xzjcc ? !1! * AYA via
t . ( x ,y ) = ( tx, tty )
.
E#14=22
,A = { to],[ ?],[t],[8]}→ twisted cubic.( aff cone of )
( s , -4 H (53,s2t,st2,t3 ) . ✓ ( xz - y2, Ztyw ,
xw - yz )
PropH.8= ACM. YA isafftoricvar wl char .
lattice ZA.
( so dim YA = rktDeft (aff)semigrp_ SCM H.g. ) .
!1! [ S ] 4 !1! [ M ] .
E.ge S= { 0,2/3,4 ,- . } .
E[ 4,2/3,2/4,... ]= !1! [2/2,43] C E[X± ]
.
Spec Q[ S ]=Y=V(y2-x3 )
Propttk If S= INA,
then Spec as ] = YA .
Thml.lt aft . tor.
var .⇐7 YA # prime binomial ideal ( 1-1.9.)-
B- Noradtoric var . ⇐ semigrp of a polyhed .cone
.
Cone o.CN → or cM=N✓ in
Sjfornm,Urtspecctso]
Deft or = { men 1 { u ,m > 20 then }.
⇐ M¥4,
→::* '
or = Cone Gei - ez,
ez ) CNR in ) or = Cone ( ei, eitzez ) C MR
So = IN { [ 'o ]
,[ ! ],[ I } .
Uo= Is,TtstHV( xz - YYCQ?
-pt3.Bt5_ Ni → Nz st a → Jz gives atogrcmpUr
, → Use Tixxi → xi( t±x→Y.
)<
E.ge/IropN# N.
Its ⇒ ( Ur)¢m=Ut where t=rnHm.
( 1.3.16 )
E* ft TG,
above : gives "#%, ← ¥1:
= on Her.
So
↳ { ( tit ,- -
, -4 }
(2.4-21.2)XA = IMTEAEIP " ( closure in Pn instead of An ).
E# A= { [ 8 ],[ to ],[9],[ ', ] }
.
( s ,t)H [ 1 :S :t : st ] .
XA= Vcxw -
yz ) < P?
prop.2.l.tt ( aff .cone of XA)= YA when all treat . is in row
space of Althe matrix of A ).
÷I Normal proj . var . ⇐> polytope .
Pms Liner ) normal fan Ep → Ur for each cone TEEP .
E# normal fan : fy±y→ ¥1.bg#g6rresp.vertamaxadime
facet ← raysQ ← of
P is icyif PAM - V generate
GnecpnmtvThM2.3= P very ample .Then Uq= Spec E[ Gnecpnm - v ) ]
.
W
t.ge/..gzA={[8],[•o]
,[ ! ],[ I ] }
.[ 1 :S : st :st2 ]
x y Z W
' ' YXA=V( yw
-z2)EP3(Projective cone over §€ ) .
U°AP3 = dehomog at x : ¥1U , A1P3 = at
yV ( we
, ,z2) c- Spec [ x. Z ,w ] #€2
U2nP3 = at z V ( yw- i ) yµe→-
I Ia G*×¢
%t-
E# F Conv.
( 0, G. ez
,Gtezt2e3 ) E 1123
.p§t¥#Tf#→€not
very ample .
Xpnm = P3 =/ Xzp
SmoothlsimplicialoImplicitif all rays Formtin
. indep . set .
smooth "can be extended to #oasis .
THMIEKURsmth # T Smith .
Pr# Prod a prod .