is Bf EXEIPE in X r plane An REE field extasio Grcr n Wendi've dim - - PDF document

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is Bf EXEIPE in X r plane An REE field extasio Grcr n Wendi've dim - - PDF document

There 160,6504 17 160,839417 are 3 7 dim't planes in cubic a hypersurface joint with Tom Bachmann R field fi fjE Rf Xo Xn homogeneous X CD fi 03 intersection complete is Bf EXEIPE in X r plane An REE field extasio Grcr n


slide-1
SLIDE 1

There

are

160,839417

160,6504 17

3

planes

in

a

7 dim't

cubic hypersurface

joint

with

Tom Bachmann

R field

fi

fjE Rf Xo

Xn

homogeneous

X

fi 03

CD

complete

intersection An

r plane

in X

is Bf EXEIPE

REE

fieldextasio

Grcr n

Wendi've

Vectorspace

dimW

S

Gr

rin

tautological bundle

Sang

W

Symds

any

Polynomials of

degree d

  • n

W

slide-2
SLIDE 2

g

di

deg fi

Fi

determines

Tf

e P symdiS

by

Te CW

fit w

ra

c PCV

V

i.IO symdis

Fr

X

Scheme of

r planes in X

5

because

PWC

x

film

Hi Debarre Manivel

For general fi

Fry

is

smooth

  • f

expected

dim 8

slide-3
SLIDE 3

OI

when

8

how many

r planes

are

in X

A

R

Euler class

ECU

eCV

r

Eros

degpr

  • ff

PEGr

are isolated

s.tn TCP

zeros

E

y

r planes

  • ff

in X

are simple

Pear

it

preserves

TCP

  • rientation

R

R

when

V

is

  • rientable
slide-4
SLIDE 4

e CV

sign

Ja

Pe GrCIR

M

  • J

E

r planes

READ

  • ver IR

Jae F detloff

in

weighted count A

homotopy theory

Morel Voevodsky 98J

Euler

class

Barge

Morel

Morel

Kass W CFasel

DEglise Jin

Khan

Hopkins

M Levine

Raksit

Serre

slide-5
SLIDE 5

when

din X

rank V

and V

is

relatively

  • riented

eCV

c

Gwen

GW

ring

group

  • f

formal differences

  • f

symmetric

non degenerate

bilinear

forms

presentation

  • ver

a field

i

generators La

aeRYµ p

Las Kirk

K

City taxy

relations La

Lb

Lab

a

Sb L atb

1 LabcatD

In

L

th D

La

tea

EI

GW CR

2417

022 13

slide-6
SLIDE 6

Et

GWCQ 2

EI

GWAR

EZEK

EI

GWCFp

E

2x Fp Fpt

2

Crank disc

transfers

k

1

finite

separable

field extension

TrL1kiGWCLJ

3GWCKJ

xv BsDi

sCVxVEL59grefiT.Y.Lam

An

Introduction to

slide-7
SLIDE 7

Backtocountingr

planese

Kass

W

eCSym3s

Gr 1,317

15217

121 17

R field

M Levine

e

Sym'd s

GrCt dtd

2d D

si ee

cand D

241747

d l

Gd 1 Cad 3

al

k

field

char k t 2 lad h

Ee

is

ecvca

S Mckean

ec

  • cnj

Ph

T Cistus

S Pauli

e

Symds

Gru 3 via

dynamic

D

3 or

intersections

Charkt

2d

slide-8
SLIDE 8

Thin

Bachmann

W

Let 12 be

a ring

with HER

Let

symdis

Germy

be

relatively oriented

with

dinV

dimbrcr.nl

ice

drain dir

c nel

EO mod 2

rel

Cn

r

Ee l din

Then

ecvj

ectzek zpteeeke.ly

where

ee

e

Vca

en

e VCR

Cory

X

is

a generic

complete int

I

Trrepyn LJactCp

r planes P in X

T

slide-9
SLIDE 9

f

eezeir zptee

ER.GR

These

are not CDs andL D's

EI

Finashin

Kharlamov

3

planes

  • n

a 7

dime

cubic surface

eco

321,489

ER

189

e Sym S

Grc3,81

160,83947 1160,650C D

slide-10
SLIDE 10

Proven

with

integrality result

Thin3 Let

be

sm and

proper over 2C'T

da

Lef V be a

rel

  • rientee Vectorbundle onX with

rkV

dimX

Then

e

V

e 2K 17,67 Xd JC GWCk

If d 2

then

either

eerie

n

a

natzke

no

hit s i

t

217

Alternative perspective

X

vector

bundle

Dr

5

03

14kV

xx

Eon L

I

1T

How

slide-11
SLIDE 11

Gwck

I

degit

TirpLJacT7

q ecus

Morel

Il

joint with

1547 122 17

Kass M Levine

J Solomon

EI

Bayer Fluckiger

Serre

IT

from

lines

  • n

a

cubic surface

Trps

A7X thx

h

t

4 17 427 Cz

cx

h

1

747 2 27

depends

  • n

cubic

surface