By the Nullstellensatz we have fi x y. If 2 I. If X o y o or 7 0. p Pz Ps. R yz XZ x. If X Z l o 7 x co or 2

I ^{R yz}

we ^use

the Nullshelleusatz and

2 I

I

X2 yz

If ^{X Z l o 7 x co or 2}

If

If

_y

By

Nullstellensatz

have

fI

^y

²

^{l y x2}

p Pz Ps

of ^Al At 2 bereft

k

KID

xp t

foe XZ

y

y ^s

t4

2 _t

f ^y

^{to I} Ily

FE XI E _{f fof fz EI}

Conversely

let FEI

Kay

kg

deg ^x 3 elegy_4

deg^z 5

fo f fz

of

^{8,9 10}

Modulo

J

may

any fEk xy 27

f ^{limit AIN}

^BHI _y

plugin

I

y Atty

Blt

t4 at Its

z

t5 T T T

0 mod

3 _I

₃

A _B C

f

J ^ZI

I ^cannot be

by fewer ^elements

there

elements of ^degrees

⁷

F _x

yw

L

How

G

Let c ² 1

xy

3 YI

Claim

7

G C

L

lis ^L

Z

F G

ICC

C

Let PE

EG p

p

^u3 f

^p

I

f

why

y

2

Xy x3

p

x3

C

Note yw

ay

y2

0

Alternation ^{P'xp IP embedding}

doily

44

_Niko

4 1

_yw uiuo lu.yku.ro

Vo Uf ^ro

_y

a

Krull's

b

c

Ily

^of 1 II ^I

^and

Y ^{2 F} G ^F

^Xz

G _Xz

XE Xz

_{Xz Xix}

VEG

IX _{Xz xET X F iz G}

Xoxz

Kal X ^F ko G

Feat ^I

I ^is _prime

a

Let _p

CP2 _{why PE}

AT

ng

p

17 12

y

Xy

i

Oc.pt ^kEyI7

^kEMcx.a

Ocp C y ^tix

^D

b _wlog p Il

0cgp k 4YwI ^KEV

^normal

Uavw

QQz _p Hurry

2

kEuw

or

k _Luiu.us ELE ^{w3 ronal}

c

bix _{y HEHE not normal}

te KE

but t

XZ ^t2

d

A ^AIX

A

A

Ap

^up

Am ^und

Ox

^nonal

X nonal

e See

the

normal noethian

al

_Hmi R

p ^{R KIRI}

R

htp ^P

C

we

want to

that f

^to

_{regular function}

p

n

may

Y

_affine

Let A ACY Being normal

local

after _replacing

Y with

assume Let

p

^{p Mp}

Y

Pick

gemp ^p Agc ^Ap

Y

U ^D

^f

^Dcg

hemp_{D g}

T

f

^O

Ag

here

f

AP

Hence

f ^EA _by

f

^nbh of ^p

I ^{is regular}

^Alt

^but

is no extension to

Atl

Moy X ^{and Y}

^affine

Let

^K YI

_Ox

YI Hu

Oy q 4th

mg Given

of

_mg5 Kai HI

Define

f ^{U Y}

7

by f ^{Hi int}

199m74 xn

This

of _p

_cely

clearly

g _{up my}

_f ^Cpl _of

If _{y p}

g ^V

^X glyi ^ym7 _luckily

which

defied

of _g

we

have fog

^V

g of

^U

See the ^{notes on} birational

maps

J

4

^to

2 4 ²

E

90 only y ^Coco

Suiyuanpoint

Themultiplicity is ² at p ^seria

tem

The

rat

y _y

A _B C D

f Hey

_Xy

y

y 6y5

_{x6ty6 Xy xy}

_y

4xy

61 ⁴

₁₆

we have _x ^o y ^{o so} 190 _{is the only singular}

point

f invariant under

y

_y

f y2xx4ty4 ^x3

J ₄

4y3

TI XT

37

II y

yao

y't ^y4

4 ^{0 x x3 o}

IF

11 ³4 and _y I

t.i

0,07 is the only

super

This ^is the_cusp

X3 y2

⁴⁴³⁰

no negative values

for

f X4ty4 Ry

I

y y2

4

Ry

Macaulay 2

TI

This ^{has to be} the triple pout by ^elevation It is clear

that _p

f x4ty4 Ry

degreey ^degree3

If

thin

flat

^{It't f It't}

A ^B

C

an F Xy 72 J ⁴² 2xy ^ZZ

i suiylxl

ZLT.tl

line y ^{Z o}

also X ^is

a fortyof

parabolas

b F

Rey

singlx ^a

2Gt

A

of

c F

x3ty3

su

ylxt

Z ^does not _appear air F

T

minyanim

ay ^yep

141

f axtby

g gadgeteer

F

brfCo7

_gem

p

b

a Hp ^{Z 1710,07 become} of ^{the XZ}

b

Yip

xp

c

Yp

^l

ya

d Yp

^at p ¹⁹⁰⁷ My yt

Y ^t

^Y

by ^{Krull's PID}

dim

Cf.gg ^0440g artinian k

Y Z

p dimk

YiP1g1Lx.b

Wl _{og p}

_ol

AT

f f

fat

^YI

G ^Cfb type

R kGiy

µ hiy

y zip

_g

^µatb

ftp.matbykoszul

121µA X

RI ^{yb R µatb} Mf _g ^µatb

UN ^{1 1}

f

_g

dim Rdf

g^{µatby 7 dinis}

Keat

µb

atb

att

I

C wlog p ¹⁰¹⁰⁷

f Eaijx

^{fat fat}

f

Zly

^YplYl

Hasan ^Hi.Y

KEN

f

For ^most aek

fda

dimk

fdxaxltf.ee dim ^k

1

c xd¹

d L Ya

E limp

wlog ^Y

Hy

flxiyl

e e ^l

x

tag f

p matp ^Cxio7

zeroof^multm

L.y

ELL

PE^LnY

The point at infinity p ^Coilo

Y ^is _given by ²

dgl

^et

4 _copO d e _n L Y d e te d

r

p ^{char k}

If

pt

ydt ^zd

J

^{dyd dzd}

If pld _Flay

yzd

2

yd

xd

yd

J

^xd

zd

_yd

_za

Can ^check that ₂ J ^{n Y} I ^set 7 ¹ etc

a Noles

b Why the ^{Turgut lines are} 11 ⁰ any y ^O

flag _{_Xy thxy} ^h high^degreeterns

Blow up ^{U X} Uoy ^o 40 ¹ chat _y ^U X

Total tmfm

f

X4u

F

ailersects E 2^CX at ^the _point ^{u x o}

one morepoint

the ^other chart 4 ^{1 X} up^{y at}

fluoy _y y2uothluoy.gg ^{y2fuo 0 4 0}

UT is nonsrnynlw along E nY amice it ^is _given

by

and ^there the 252 do not ^vanish