The Luroth problem and the Cremona group
122 pages
English

The Luroth problem and the Cremona group

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122 pages
English
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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

Informations

Publié par
Nombre de lectures 34
Langue English

Exrait

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

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