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

Ut-VCxZ-y4xeCfxII.Projectivetovicvicexjnei0YntY@4P.n

↳ { ( 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 .