April 8 Zoom- -

Recall Bloc AI ) = { Cv , Cbs) E Atx Dn- '

IX= vee }up

*fire P=%¥ ,

-

p : X - B" "

is the most important ex .

of a" line bundle

"

called "tautological bundle"-

on Ifn- I

let Z be a variety & p : L- Z a

morphism of varieties s .t.

① HZEZ , p''CZ ) E At fibers are lines

it③ HZEZ .

7- U CZ & Piso . between

ZE FLU ) & Ux INp- '

CU ) -9 Ux IA that commuter with=

Pf,

Lpr, Each Ftz) is a l- dim

v. s . & Q is tin - iso

( same def- I notion for top . Spaces lmomiefoc.ds)fiber bundle #Sms F some variety)Vec. i . At c-At

Exe X smooth curve

TX = tangent bundle of X

= ¥↳ tangent spaceto X at point p

-

EI k= B X = circle

L = ( infinite ) M bins strip

L ¥ X x B - cylinder

¥⑤1 HIM

-

Locally trivial

Ee : check * Bloc EI ) c.AIX Dn"

f p -_pz

Dn- I

is a line bundleover Ipn

- !

Rein There is an operation of tensor

product of line bundles .

L, Lz L

, @ L2

pity & Bf →

p fx x

x

• Fact :Theset

ofline bundles over X

form a group C ! ) called Pic (X) .

Picard groupof X .

Trivial line bundle

Xx At is the identity element .-

Back to blow -up :

( see Eisenbud '' Comm - aly . with a view "

towards alg . geo .p . 148 Sec

. 5-2 Blow up alg . )X affine alg . van. YCX salwar.

Z = Blow upof X along Y .

Ali -- i ar h-aly . gen . for k[X]_Goi - - 19 , gen - of ideal I= Ily ) CkCX)-

R =KCX] aux. var .

-ee

k Cx , . - - Xr , yo . - - its] → Rct]"

⑦Rtii30

Xi 1- Ai

Yj 1- gjt

E : Images of Q is

pIthe alg . : ④ ⑦ It ⑦ Itt @ - - i

-Blow- up alg .

of I

•kernel of Q is homog . in the Yj .

& blowup of X along Y to be

Shuar . of # x Tfs defined by

this ideal .R

- -

Ex. Blow -up alg .

of me My Ckcx ,- - -xn)

R ⑦ m ⑦ MZ @ . . . .

Z = Blycx ) C #r× Ds- t

k ¥:x c

⇒ I R @ I @ I# . . .

G hybrid affine e homog . Coor.

quotient of k[Xi --Xr , Yo -- Ys]

Exercise heftily) ] = RE -0 Izz@ - - -

Defy IC Rc.pets

R I@ IZ@ . . .

- Blowuponly .

or

Rees alg .

Rg -0 ¥2 @- - -

- associated gradedof ( R , I) alg .

( ily ) related to normal nexof Y)

Aside :

Algebraically you can generalize theconstruction to a decreasing sequence of

ideals.

§=(Io J I, D Iz D - - - . ) ideals in R

RCI.) = ⑦Ii ti

i 7,0

gz.ph = ⑦I it .

i > o

-

Geometric meaning of Blye X) :

At p E Y you look atnonmalcoreofY in X at p , projectivize it &

attach it to p .

-singularities .

Rein Blow upis used in resolution of

Hironaka : res . of sing . can be achieved

by a sequence of blow - ups .

Changing gears :

Introduction to" intersection theory

"

-

Problem# -

- , fn given poly - in k[X, . - i Xn]

•Find # of Sol

.

of f , = - - -=fn =D .

. d the solution of the system↳ Compute

- fi 's linear - Gaussian elimination

- fi 's poly - -well- known algo .

& kalg .

closed in Grisbner basis

theory

( see Cox- Little -O'Shea

(very general ,

" Using alg . geo ." )

Pnobtem (of intense . theory)

X variety

Y,o -→Yr C X

Shuar.

Describe Y, c - - - i Yr .

Exaples_ (of results in intense. theory )

Berzout theorem :fcxiy , Z) ho - og . poly ,

gcx, y ,z)of deg .

d , & dz .

XIV (f ) & Y = Tcg) c BZ

IX n y ) = d, da-counted with multiplicityeach intense. point

Defect theorem )N

X C D proj . subvarietyof din d

deg CX ) : -_ IX n Ll

where L is an proj. plane of

de N- d

'' general" (or

'' generic")

( Hiddentheorem : this

number is well- def)

Next time BKK theorem.