The Luroth problem and the Cremona group

Arnaud Beauville

Description

The Luroth problem and the Cremona group Arnaud Beauville Universite de Nice Torino, March 2012 Arnaud Beauville The Luroth problem and the Cremona group

• pn ?99k


• map pn


• rational functions


• riemann surface theory


• surjective rational

Sujets

• Beauville
• Rational function

Informations

• Publié par: pefav
• Nombre de lectures: 34
• Langue: English

Extrait

ehT

Lu¨roth

problemandtheCremona

rAArnaudBeauville

Universite´deNice

Torino,March2012

andueBuaivllehTeuL¨orhtrpboelmnadhterCergpuo

omanrguop
).ciarbeglasifoorps’htoru¨Ltub;yroehtecafrusnnameiRhtiwysaeetiuQ(.lanoitarsievruclanoitarinua:meroehthtoru¨L.VK99∼nPpamlanoitarib∃filanoitarsiV.VK99nPpamlanoitarevitcejrusyllacireneg∃filanoitarinusiVyteiravAsnoitinfieD.ufoeulavenootsdnopserrocCfotnioplarenegatahthcus))u(y,)u(x(→7unoitazehT

iL

ru¨roth

ttheorem

epuorg

mCremona

aeht

rdna

aproblem

pLu¨roth

rehT

eBeauville

hArnaud

tona∃.))t(y,)t(x(→7t:snoitcnuflanoitarybdezirtemarap,evrucciarbegla2C⊂C)5781,htoru¨L(meroehT
pserrocCfotnioplarenegatahthcus))u(y,)u(x(→7unoitazirtemaraprehtona∃.))t(y,)t(x(:snoitcnuflanoitarybdezirtemarap,evrucciarbeglapuorganomerCehtdnamelbor→7t

p2⊂CC

hTheorem(Lu¨roth,1875)

tTheLu¨roththeorem

oru¨LehTellivuaeBduanrA).ciarbeglasifoorps’htoru¨Ltub;yroehtecafrusnnameiRhtiwysaeetiuQ(.lanoitarsievruclanoitarinua:meroehthtoru¨L.VK99∼nPpamlanoitarib∃filanoitarsiV.VK99nPpamlanoitarevitcejrusyllacireneg∃filanoitarinusiVyteiravAsnoitinfieD.ufoeulavenootsdno
puorganomerCehtdnamelborphtoru¨LehTellivuaeBduanrA).ciarbeglasifoorps’htoru¨Ltub;yroehtecafrusnnameiRhtiwysaeetiuQ(.lanoitarsievruclanoitarinua:meroehthtoru¨L.VK99∼nPpamlanoitarib∃filanoitarsiV.VK99nPpamlanoitarevitcejrusyllacireneg∃filanoitarinusiVyteiravAsnoitinfieD.ufoeulavenootsdnopserrocCfotnioplarenegatahthcus))u(y,)u(x(unoitazirtemaraprehtona:snoitcnuflanoitarybdezirtemarap,evrucciarbeglat(x(→7t.))t(y,)

∃

→7

2⊂CC

Theorem(L¨uroth,1875)

TheLu¨roththeorem

puorganomerCehtdnamelborphtoru¨LehTellivuaeBduanrA).ciarbeglasifoorps’htoru¨Ltub;yroehtecafrusnnameiRhtiwysaeetiuQ(.lanoitarsievruclanoitarinua:meroehthtoru¨L.VK9suchthatageneralpointofCcorrespondsto
one
valueofu.

9)t(x(→7t.))t(y,

∼∃

n→7

P2⊂CC

pTheorem(Lu¨roth,1875)

aTheL¨uroththeorem

mlanoitarib∃filanoitarsiV.VK99nPpamlanoitarevitcejrusyllacireneg∃filanoitarinusiVyteiravAsnoitinfieD))u(y,)u(x(unoitazirtemaraprehtona:snoitcnuflanoitarybdezirtemarap,evrucciarbegla
ruclanoitarinua:meroehthtoru¨L.VK99∼nPpamlanoitarib∃filanoitarsiVmap
P
n
99K
V
.

Avariety
V
is
unirational
if
∃
genericallysurjectiverational

Definitions

suchthatageneralpointofCcorrespondsto
one
valueofu.

t
7→
(
x
(
t
)
,
y
(
t
))
.
∃
anotherparametrizationu
7→
(
x
(
u
)
,
y
(
u
))

C
⊂
C
2
algebraiccurve,parametrizedbyrationalfunctions:

Theorem(Lu¨roth,1875)

TheLu¨roththeorem

puorganomerCehtdnamelborphtoru¨LehTellivuaeBduanrA).ciarbeglasifoorps’htoru¨Ltub;yroehtecafrusnnameiRhtiwysaeetiuQ(.lanoitarsiev
puorganomerCehtdnamelborphtoru¨LehTellivuaeBduanrA).ciarbeglasifoorps’htoru¨Ltub;yroehtecafrusnnameiRhtiwysaeetiuQ(.lanoitarsievruclanoitarinua:meroehthtoru¨LV
is
rational
if
∃
birationalmap
P
n
9
∼
9K
V
.

map
P
n
99K
V
.

Avariety
V
is
unirational
if
∃
genericallysurjectiverational

Definitions

suchthatageneralpointofCcorrespondsto
one
valueofu.

t
7→
(
x
(
t
)
,
y
(
t
))
.
∃
anotherparametrizationu
7→
(
x
(
u
)
,
y
(
u
))

C
⊂
C
2
algebraiccurve,parametrizedbyrationalfunctions:

Theorem(Lu¨roth,1875)

TheLu¨roththeorem

TheLu¨roththeorem

Theorem(Lu¨roth,1875)

C
⊂
C
2
algebraiccurve,parametrizedbyrationalfunctions:

t
7→
(
x
(
t
)
,
y
(
t
))
.
∃
anotherparametrizationu
7→
(
x
(
u
)
,
y
(
u
))

suchthatageneralpointofCcorrespondsto
one
valueofu.

Definitions

Avariety
V
is
unirational
if
∃
genericallysurjectiverational

map
P
n
99K
V
.

V
is
rational
if
∃
birationalmap
P
n
99
∼
K
V
.

Lu¨roththeorem:aunirationalcurveisrational.

rAandueBuaivllehTeuL¨orhtrpboelmnadhterCmenoarguopQ(laugietbeareia.cs)yiwhtiRmenanusfrcaehtoeyr;ubtuL¨orhts’rpofosi
TheLu¨roththeorem

Theorem(Lu¨roth,1875)

C
⊂
C
2
algebraiccurve,parametrizedbyrationalfunctions:

t
7→
(
x
(
t
)
,
y
(
t
))
.
∃
anotherparametrizationu
7→
(
x
(
u
)
,
y
(
u
))

suchthatageneralpointofCcorrespondsto
one
valueofu.

Definitions

Avariety
V
is
unirational
if
∃
genericallysurjectiverational

map
P
n
99K
V
.

V
is
rational
if
∃
birationalmap
P
n
99
∼
K
V
.

Lu¨roththeorem:aunirationalcurveisrational.

(QuiteeasywithRiemannsurfacetheory;butLu¨roth’sproofis
algebraic.)

rAandueBuaivllehTeuL¨orhtrpboelmnadhterCmenoarguop
:derational.

ris

asurface

eunirational

pa

p(1894):

aCastelnuovo

sHigherdimension

epuorg

lTheLu¨rothproblemandtheCremona

pArnaudBeauville

maxe-retnuoc”nredom“eerht1791dnuorA.sdradnatsnredomybelbatpeccatontub,)7491,5191(stpmettarehtrufedamonaF.etelpmocnisisisylanas’onaFtuB.ytilanoitar-nonehtrof)8091(onaFforepapreilraenanoseilerdna,ytilanoitarinusevorpseuqirnEyllautcA.5P⊂3,2V:elpmaxe-retnuocdesoporp:)2191(seuqirnE