Whichvarieties are birational to Ph rational Difficult problem k x k - - PowerPoint PPT Presentation

Irrationality of quotient

varieties

Urban Jezernik

University of the Basque Country

Fundedby EUHorizon2020 under MScgrant 748129

Difficult problem

Whichvarieties are birational to Ph

rational k x

k ti

Tn

Problem

Given a nice

smoothprojective variety X rk

is X birational to some Ph

Difficult problem

Whichvarieties are birational to Ph

rational k x

k ti

Tn

Problem

Given a nice

smoothprojective variety X rk

is X birational to some Ph

Liiroth problem

nice

unirational

g Ip teen

X

kCX

kfta

th iiroth problem has a positiveanswer over k fi

when dimX E Z

Noether problem

nice

quotient variety

IG

Spec KEV

G

f

vectorspace

finitegroup

representation of G

G invariant regular functions

Example

Cz A V

IAI

x y to C x

y

VIC

Spec k x2 ay

y

E Speck U v wJ

Cuw

v2

k Vce

E kCee v

Vez

P2

Motivation

If VG is birational to Ph over k

Q then one

can

strongly solve the inverseGabi'sproblem for G

• G E Gall EIQ

for some E

Noether Hilbert

Motivation

If VG is birational to IP

• ver

k

Q then one

can

strongly solve the inverseGabi'sproblem for G

• G E Gall EIQ

for some E

Noether Hilbert

Unfortunately Noetherproblem doesnot have a positive answer

in general

smallest counterexample is Cg

Swan 1969

Lenstra 1974 Plans 2018

Motivation

If VG is birational to IP

• ver

k

Q then one

can

strongly solve the inverseGabi'sproblem for G

• G E Gall EIQ

for some E

Noether Hilbert

Unfortunately Noetherproblem doesnot have a positive answer

in general

smallest counterexample is Cg

Swan 1969

Lenstra 1974 Plans 2018

Assume from now on that k

E

Nonamelemma Thequestion of whether or not WG is

stably rationaldoes notdepend on the representation

YG

P

is rational for some h

CMiyata1971

Nonamelemma Thequestion of whether or not WG is

stably rationaldoes notdepend on the representation

YG

P

is rational for some h

CMiyata1971

Example

G abelian

KG rational

Fischer 1915

Nonamelemma Thequestion of whether or not WG is

stably rationaldoes notdepend on the representation

YG

P

is rational for some h

CMiyata1971

Example

G abelian

KG rational

Fischer 1915

G

Sym n

KG rational

V G

symmetricpolynomials

Nonamelemma Thequestion of whether or not WG is

stably rationaldoes notdepend on the representation

YG

P

is rational for some h

CMiyata1971

Example

G abelian

KG rational

Fischer 1915

G

Sym n

KG rational

V G

symmetricpolynomials

G 2 group

1Gt I 32

VG rational

Chu HuiKang Prokhorov2008

To prove that a variety X

is not

stably rational

• ne uses

a birational invariant

I

varieties

abeliangroups

with

I X

f 0

I Ph

O

To prove that a variety X

is not

stably rational

• ne uses

a birational invariant

I

varieties

abeliangroups

with

I X

f 0

I Ph

O

TorsionHfing

X

E

Z

Artin

Mumford 1971

To prove that a variety X

is not

stably rational

• ne uses

a birational invariant

I

varieties

abeliangroups

with

I X

f 0

I Ph

O

TorsionHfing

X

E

Z

Artin

Mumford 1971

Brnr X

N

image Br

R

Br

Ec x

R DVR

KCR

Ek

Saltman 1984 for X

VIG

unramified Brauergroup

To prove that a variety X

is not

stably rational

• ne uses

a birational invariant

I

varieties

abeliangroups

with

I X

f 0

I Ph

O

TorsionHfing

X

E

Z

Artin

Mumford 1971

Brnr X

N

image Br

R

Br

Ec x

R DVR

KCR

Ek

Saltman 1984 for X

VIG

unramified Brauergroup

For X smooth projective unirational

Brw X

Br X

Torsion H3sing X E Z

Bogomolov 1989

This invariant is computable for X VIG

Brar VIG

A

kernel H4G

HEA ex

AEG

A abelian

Bogomolov 1989

Bogomolov multiplier

BoCG

This invariant is computable for X VIG

Brar VIG

A

kernel H4G

HEA ex

AEG Bogomolov 1989

th NG

A abelian

Bogomolov multiplier

BoCG

Higher unramified cohomology

ColliotThe leine

Ojanguren 1989

id

Him X

EE't

in Leray spectral sequence Xan

Xzar

Only partial extensions of Bogomolov's formula for X VIG

Peyre2008

Hoshi Kanga

Yamasaki 2018 computer calculations

Example

Bo abeliangroup

O

Example

Bo abeliangroup

O

Bo finite simplegroup

Kunyavski 2010

Example

Bo abeliangroup

O

Bo finite simplegroup

Kunyavski 2010

Bo exponent p class 2 quotientof Isl2g

for g 2 2

O

g

ITExi y

1 Saltman 1984

i

n

Example

Bo abeliangroup

O

Bo finite simplegroup

Kunyavski 2010

Bo exponent p class 2 quotientof Isl2g

for g 22

O

g

ITExi y

1 Saltman 1984

i

n

B

somegroups of order 64

Chul

Hu Kang Kunyarski 2012

Bo some groups of orderp5

1 0

Hoshi Kang KunyavskT 2012

Quite surprisingly Bo is often non trivial

log

G

IGI ph

lBoc

G I 2 M

him

1

AM

a log

G

IGI

p

with Sanchez 2018 It follows that Noetherproblem has a strongly negative answer

Bo is

an attractive object

Bo is

an attractive object

it

reappears in

K theory

B G

SKI Epa

kernel Ky 2pG

Kr Qp

G

Oliver 1980

Bo is

an attractive object

it

reappears in

K theory

B G

SKI Epa

kernel Ky 2pG

Kr QpG

Oliver 1980

Representationtheory

1

1

I G abl

p

1 IB G 1

11

11

Lin VC

FpG

Lin I pg with Garcia Rodriguez

Jai

Kin Zapirain 2017

Mathematical physics

soft tensorbraidedauto

equivalences

Outage G

X Bo G

• f Drinfeldcente.ro

category of compatible Davydov 2014

G gradedG vector

spaces

Mathematical physics

soft tensorbraidedauto

equivalences

Outage G

X Bo G

• f DrinfeldcenterofCI

category of compatible Davydov 2014

G gradedG vector

spaces

Moduli spaces

Hurwitzspace

• f connected curves Gcovering D

aepapihantngmesu.FII e

aymmongdromieses

B

Ellenberg Venkatesh Westerland2012