the lüroth problem - unice.frmath1.unice.fr/~beauvill/conf/colloq.pdf · castelnuovo’s theorem...

The Luroth problem

Arnaud Beauville

Universite de Nice

Ann Arbor, April 2016

Arnaud Beauville The Luroth problem

Luroth 1875

Theorem (Luroth, 1875)

C plane curve, defined by polynomial f px , yq 0, which can beparametrized by rational functions :

t pxptq, yptqq : f pxptq, yptqq 0 .

Then there exists another parametrization u pxpuq, ypuqq suchthat u P C 1:1 px , yq P C , with finitely many exceptions.

In geometric terms :

D rational dominant map C - - C D birational map C 99K C .

Luroth 1875

D rational dominant map C - - C

D birational map C 99K C .

Luroth 1875

An example

-5 -4 -3 -2 -1 0 1 2 3 4 5

-3

-2

-1

1

2

3

Strophoid : parametrized by xptq t1 t3

, yptq t2

1 t3

An example

-5 -4 -3 -2 -1 0 1 2 3 4 5

-3

-2

-1

1

2

3

Strophoid : parametrized by xptq t1 t3

, yptq t2

1 t3

About the proof

Luroth gives a clever, but somewhat mysterious

algebraic proof.

(Modern) proof : Csm C compact Riemann surface.

C - - CsmX XP1 f C

For holomorphic 1-form on C , f holo. 1-form on P1

f 0 0 C P1.

About the proof

algebraic proof.

C - - CsmX XP1 f C

f 0 0 C P1.

About the proof

algebraic proof.

C - - CsmX XP1 f C

f 0 0 C P1.

About the proof

algebraic proof.

C - - CsmX XP1 f C

f 0 0 C P1.

About the proof

algebraic proof.

C - - CsmX XP1 f C

f 0 0 C P1.

About the proof

algebraic proof.

C - - CsmX XP1 f C

f 0

0 C P1.

About the proof

algebraic proof.

C - - CsmX XP1 f C

f 0 0

C P1.

About the proof

algebraic proof.

C - - CsmX XP1 f C

f 0 0 C P1.

Castelnuovo-Enriques

In the decade 1890-1900, Castelnuovo and Enriques build the

theory of algebraic surfaces.

Starting from a rather primitive stage, they obtain in a few years a

rich harvest of results, culminating with an elaborate classification

now called the Enriques classification of surfaces.

now called the Enriques classification of surfaces.Arnaud Beauville The Luroth problem

Castelnuovos theorem

One of the first questions Castelnuovo attacks is the analogue of

the Luroth theorem for surfaces :

Theorem (Castelnuovo, 1893)

S algebraic surface, D C2 - - S D C2 99K S .

or : S unirational S rational .

Castelnuovos first proof was rather technical. Second proof :

S unirational S has no holomorphic 1-form or 2-form

(locally, ppx , yqdx ` qpx , yqdy or rpx , yqdx ^ dy).

[Footnote : This is the modern formulation. Castelnuovo used the

(equivalent) vanishing of the geometric and numerical genera.]

S algebraic surface, D C2 - - S

D C2 99K S .

The Enriques surface

At first Castelnuovo tried to prove that this property characterizes

rational surfaces, but he could not eliminate one particular type of

surfaces.

He asked Enriques, who found a non-rational surface of

this type, now called the Enriques surface :

Guarda un po se fosse tale una superficie del 6o ordine avente

como doppi i 6 spigoli dun tetraedro (se esiste)?

These surfaces play an important role in the Enriques classification.

surfaces. He asked Enriques, who found a non-rational surface of

Enriques surface(s)

Castelnuovos rationality criterion

Then Castelnuovo found the correct characterization (again, in

modern terms) :

Theorem

S rational no (holomorphic) 1-form and quadratic 2-form

(locally f px , yq pdx ^ dyq2 ).

A unirational surface has this property, hence is rational.

This is a major step in the classification of surfaces; even today,

with our powerful modern methods, it is still a difficult theorem.

modern terms) :

Theorem

modern terms) :

Theorem

modern terms) :

Theorem

modern terms) :

Theorem

modern terms) :

Theorem

This is a major step in the classification of surfaces;

even today,

modern terms) :

Theorem

The Luroth problem

Definition

X complex algebraic variety

X rational if D Cn - -X ( CpX q Cpt1, . . . , tnq)

X unirational if D Cn - - X ( CpX q Cpt1, . . . , tnq).

X rational X unirational+3X`

?Luroth problem

In algebraic terms:

C K Cpt1, . . . , tnq K Cpu1, . . . , upq ?

The Luroth problem

Definition

?Luroth problem

In algebraic terms:

The Luroth problem

Definition

X rational if D Cn - -X

( CpX q Cpt1, . . . , tnq)

?Luroth problem

In algebraic terms:

The Luroth problem

Definition

?Luroth problem

In algebraic terms:

The Luroth problem

Definition

X unirational if D Cn - - X

( CpX q Cpt1, . . . , tnq).

?Luroth problem

In algebraic terms:

The Luroth problem

Definition

?Luroth problem

In algebraic terms:

The Luroth problem

Definition

X rational X unirational+3

X`

?Luroth problem

In algebraic terms:

The Luroth problem

Definition

?Luroth problem

In algebraic terms:

The Luroth problem

Definition

?Luroth problem

In algebraic terms:

Higher dimension

In 1912, Enriques claims to give a counter-example: a smooth

complete intersection of a quadric and a cubic V2,3 P5 .Actually he proves that it is unirational, and relies on an earlier

paper of Fano (1908) for the non-rationality.

But Fanos analysis is incomplete. The

geometry in dimension 3 is much morecomplicated than for surfaces; the intuitive

methods of the Italian geometers were

insufficient.

Higher dimension

complete intersection of a quadric and a cubic V2,3 P5 .

Actually he proves that it is unirational, and relies on an earlier

insufficient.

Higher dimension

insufficient.

Higher dimension

But Fanos analysis is incomplete.

The

insufficient.

Higher dimension

geometry in dimension 3 is much morecomplicated than for surfaces;

the intuitive

insufficient.

Higher dimension

insufficient.

Doubts

Fano made various other attempts (1915, 1947); in the last one he

claims that a general cubic hypersurface in 4-space is not rational,

a longstanding conjecture.

But none of these attempts are acceptable by modern standards.

A detailed criticism of Fanos attempts appears

in the 1955 book Algebraic threefolds, with

special regard to problems of rationality by the

British mathematician Leonard Roth, who

concludes that none of these can be considered

as correct.

Doubts

Fano made various other attempts (1915, 1947);

in the last one he

as correct.

Doubts

as correct.

Doubts

as correct.

Doubts

as correct.

More doubts

Roth goes on giving a counter-example of his own, by mimicking in

dimension 3 the construction of Enriques surface.

He shows that it is unirational, and not simply-connected hence

not rational, because a rational (smooth, projective) variety is

simply-connected.

Alas, 4 years later Serre showed that a

unirational variety is simply-connected, so Roth

also was wrong...

More doubts

simply-connected.

also was wrong...

More doubts

simply-connected.

also was wrong...

More doubts

simply-connected.

also was wrong...

The modern era

Starting in the 50s new methods (sheaves, cohomology...)

revolutionized algebraic geometry.

At first the development was rather abstract, and oriented towards

arithmetic aspects, but progressively appeared a trend to use these

methods to revisit classical problems, particularly in the US and

the Soviet Union.

In 1971 appeared almost simultaneously 3 indisputable examples of

unirational, non rational varieties :

The modern era

the Soviet Union.

The modern era

arithmetic aspects,

but progressively appeared a trend to use these

the Soviet Union.

The modern era

the Soviet Union.

The modern era

the Soviet Union.

The 3 counter-examples

Authors Example Method

Clemens-Griffiths V3 P4 JV (Hodge theory)

Iskovskikh-Manin some V4 P4 BirpV q (Fanos idea)

Artin-Mumford specific TorsH3pV ,Zq

Comments

The 3 papers have been very influential: many other examples

have been worked out along the same lines.

Except for a new idea of Kollar (1995), they are essentially

the only methods known to prove non-rationality.

The 3 methods are very different, and apply to different kinds

of varieties.

At the time, the three proofs used Hironakas resolution of

singularities, which was far out of reach for Italian geometers.

Only the Artin-Mumford method gives examples in dimension

3; they are of the form V Cn, where V is a quiteparticular threefold. In contrast, the first two methods give

many examples in dimension 3.

Comments

of varieties.

Comments

of varieties.

Comments

of varieties.

Comments

of varieties.

Comments

of varieties.

3; they are of the form V Cn, where V is a quiteparticular threefold.

In contrast, the first two methods give

Comments

of varieties.

Fano complete intersections

For instance, among complete intersections with no holo. 3-forms:

Variety Unirational Rational Method

V3 P4 yes no JV

V4 P4 some no BirpV q

V2,2 P5 yes yes

V2,3 P5 yes no (generic) JV ,BirpV q

V2,2,2 P6 yes no JV

So most of these are unirational, not rational.

V3 P4 yes no JV

V2,2 P5 yes yes

V2,2,2 P6 yes no JV

V3 P4 yes no JV

V2,2 P5 yes yes

V2,2,2 P6 yes no JV

V3 P4 yes no JV

V2,2 P5 yes yes

V2,2,2 P6 yes no JV

V3 P4 yes no JV

V2,2 P5 yes yes

V2,2,2 P6 yes no JV

V3 P4 yes no JV

V2,2 P5 yes yes

V2,2,2 P6 yes no JV

V3 P4 yes no JV

V2,2 P5 yes yes

V2,2,2 P6 yes no JV

V3 P4 yes no JV

V2,2 P5 yes yes

V2,2,2 P6 yes no JV

The intermediate Jacobian approach

Since Riemann we associate to a curve (= compact Riemann

surface) C of genus g its Jacobian pJC ,q.

This is a principally polarized abelian variety (p.p.a.v.), that is,

a complex torus JC Cg{ together with a hermitian form H onCg with certain integrality properties w.r.t. .This form allows to define the Riemann theta function which

vanishes along a hypersurface JC , well-defined up totranslation; conversely determines H.

The Jacobian is a fundamental tool in the study of algebraic

curves.

a complex torus JC Cg{ together with a hermitian form H onCg with certain integrality properties w.r.t. .

This form allows to define the Riemann theta function which

curves.

The Clemens-Griffiths criterion

In the same way, Hodge theory associates to any threefold X with

no holomorphic form a p.p.a.v. pJX ,q, the intermediateJacobian of X .

If X is rational, pJX ,q pJC ,q for some curve C .

This leads to another very classical question, the Schottky

problem: how to distinguish Jacobians among all p.p.a.v.?

p.p.a.v. of dimension g form a variety of dimension 12gpg ` 1q,while Jacobians form a subvariety of dimension 3g 3 .

The most commonly used criterion goes back again to Riemann:

for a Jacobian pJC ,q, dim Singpq g 4.

It is usually quite difficult to control Singpq for an intermediateJacobian. One exception: X is a conic bundle, i.e. D X P2

with fibers P1 or P1 Yp P1. Then pJX ,q is a Prym variety, aparticular type of p.p.a.v. which is more manageable. For instance:

Theorem

X P4 smooth cubic. Then dim JX 5, Singpq tpu, hence Xis not rational.

In the previous list, V3 and V2,2,2 are conic bundles, from which

one deduces that they are not rational. The others are not, but

they admit a mild degeneration to a conic bundle; this allows to

prove that they are generically non-rational.

It is usually quite difficult to control Singpq for an intermediateJacobian.

One exception: X is a conic bundle, i.e. D X P2

Theorem

with fibers P1 or P1 Yp P1.

Then pJX ,q is a Prym variety, aparticular type of p.p.a.v. which is more manageable. For instance:

Theorem

one deduces that they are not rational.

The others are not, but

Theorem

Symmetries

A much easier approach, but which applies only to some particular

varieties, is based on automorphism groups.

For instance, the Klein cubic threefold X :

iPZ{5X 2i Xi`1 0

has an action of PSL2pF11q.

If JX JC , the Torelli theorem for curves implies

PSL2pF11q AutpC q .

But #PSL2pF11q 660, and #AutpC q 84pgpC q 1q 336.

The same method applies to the V2,3

Xi

X 2i

X 3i 0 in P6 .

This has implications on the finite subgroups of the group BirpP3q.

Symmetries

iPZ{5X 2i Xi`1 0

PSL2pF11q AutpC q .

Xi

X 2i

X 3i 0 in P6 .

Symmetries

iPZ{5X 2i Xi`1 0

PSL2pF11q AutpC q .

Xi

X 2i

X 3i 0 in P6 .

Symmetries

iPZ{5X 2i Xi`1 0

PSL2pF11q AutpC q .

Xi

X 2i

X 3i 0 in P6 .

Symmetries

iPZ{5X 2i Xi`1 0

PSL2pF11q AutpC q .

But #PSL2pF11q 660,

and #AutpC q 84pgpC q 1q 336.

Xi

X 2i

X 3i 0 in P6 .

Symmetries

iPZ{5X 2i Xi`1 0

PSL2pF11q AutpC q .

Xi

X 2i

X 3i 0 in P6 .

Symmetries

iPZ{5X 2i Xi`1 0

PSL2pF11q AutpC q .

Xi

X 2i

X 3i 0 in P6 .

Symmetries

iPZ{5X 2i Xi`1 0

PSL2pF11q AutpC q .

Xi

X 2i

X 3i 0 in P6 .

A few words about birational rigidity

The method of Iskovskikh-Manin (inspired by Fanos ideas) proves:

Theorem

Let V4 P4 be a smooth quartic threefold. Any birational mapV4 99K X onto a Mori fiber space X is an isomorphism.

Mori fiber spaces form a large class, including any threefold in

our list, conic bundles, etc.

In particular we see that BirpV4q AutpV4q; this is a finite group,while BirpP3q is gigantic hence V4 is not rational.

Since then birational rigidity has been proved for a number of

varieties, in particular any smooth Vn Pn (de Fernex, 2013).However, in dimension 4, they are not known to be unirational.

Theorem

The stable Luroth problem

The abundance of unirational, non rational threefolds leads to look

for an intermediate notion :

X stably rational if X Pm rational for some m (Zariski, 1949)

CpX qpt1, . . . , tmq purely transcendental.

rational +3

#+

unirational

stably rational

3;

dl

MU

? ?1 2X X

1 was asked by Zariski (1949); counter-example by Colliot-

Thelene, Sansuc, Swinnerton-Dyer, AB (1985), using JX .

rational +3

#+

unirational

stably rational

3;

dl

MU

? ?1 2X X

rational +3

#+

unirational

stably rational

3;

dl

MU

? ?1 2X X

rational +3

#+

unirational

stably rational

3;

dl

MU

? ?1 2X X

rational +3

#+

unirational

stably rational

3;

dl

MU

? ?1 2X X

rational +3

#+

unirational

stably rational

3;

dl

MU

? ?1 2

X X1 was asked by Zariski (1949); counter-example by Colliot-

rational +3

#+

unirational

stably rational

3;

dl

MU

? ?

1 2X

X

rational +3

#+

unirational

stably rational

3;

dl

MU

? ?

1 2X X1 was asked by Zariski (1949); counter-example by Colliot-

The Artin-Mumford example

2 : unirational stably rational? No by Artin-Mumford.

They prove: X stably rational TorsH3pX ,Zq 0 ,

and construct a unirational threefold X with H3pX ,Zq Z{2.

Their example is a double covering of P3 branched along aparticular quartic surface with 10 nodes (quartic symmetroid):

u2 px , y , z , tq with detpLijq ,

pLijq 4 4-symmetric matrix of linear forms on P3.

For a long time this has been essentially the only example of a

unirational, non stably rational projective variety together with a

later example by Dolgachev-Gross with TorsH3pX ,Zq Z{3.

2 : unirational stably rational?

No by Artin-Mumford.

New results

The situation has changed spectacularly with a

new idea of Claire Voisin:

Theorem (Voisin, 2015)

A double covering of P3 branched along a general quartic surface isnot stably rational.

general := outside a countable union of strict subvarieties of

the moduli space

Known to be unirational, not rational (AB 77, Voisin 86)

New results

the moduli space

New results

the moduli space

New results

the moduli space

New results

the moduli space

Other results

Elaborations of Voisins idea give the non-stable rationality of the

general (in chronological order):

1 Quartic threefold (Colliot-Thelene-Pirutka).

2 Sextic double solid (AB).

3 Hypersurface of degree 2R

n ` 23

V

in Pn`1 (Totaro); e.g.

V4 P4 or P5, V6 P6,P7,P8, etc.4 Most conic bundles (Hassett-Kresch-Tschinkel).

5 V2,3, V2,2,2 and many others (Hassett-Tschinkel).

Other results

n ` 23

V

Other results

n ` 23

V

V4 P4 or P5, V6 P6,P7,P8, etc.

4 Most conic bundles (Hassett-Kresch-Tschinkel).

Other results

n ` 23

V

Other results

n ` 23

V

V4 P4 or P5, V6 P6,P7,P8, etc.

4 Most conic bundles (Hassett-Kresch-Tschinkel).

Other results

n ` 23

V

Other results

n ` 23

V

The degeneration argument

Idea : degenerate general quartic into symmetroid.

B : space of quartic surfaces in P3 ; family pXbqbPB , with

Xb : double cover of P3 branched along the surface b.

o P B quartic symmetroid. Xo has 10 ordinary double points;X : desingularization of Xo has TorsH3pX ,Zq 0.

TorsH3pXb,Zq 0 for general b, so this gives nothing. But:

Proposition (Voisin)

pXbqbPB family of projective varieties, o P B smooth. Assume:piq Xo has only ordinary double points;piiq A desingularization X of Xo satisfies TorsH3pX ,Zq 0 .

Then Xb is not stably rational for general b.

pXbqbPB family of projective varieties, o P B smooth. Assume:

piq Xo has only ordinary double points;piiq A desingularization X of Xo satisfies TorsH3pX ,Zq 0 .

pXbqbPB family of projective varieties, o P B smooth. Assume:piq Xo has only ordinary double points;

piiq A desingularization X of Xo satisfies TorsH3pX ,Zq 0 .

Decomposition of the diagonal

Key ingredient : for V smooth projective variety, find property

DDV (decomposition of the diagonal) such that :

1 V stably rational DDV TorsH3pV ,Zq 0;

2 In a family pXbqbPB as above, DDX does not hold DDXb does not hold for general b.

Then: TorsH3pX ,Zq 0 DDX

does not hold

DDXb does not hold for general b Xb not stably rational.

Definition

A decomposition of the diagonal is an equality in HpV V ,Zq:

V V tptu `

niZi . (DDV )

with V V V diagonal, and pr1pZi q V .

1 V stably rational DDV TorsH3pV ,Zq 0;2 In a family pXbqbPB as above, DDX does not hold

DDXb does not hold for general b.

does not hold

Definition

V V tptu `

niZi . (DDV )

1 V stably rational DDV TorsH3pV ,Zq 0;

2 In a family pXbqbPB as above, DDX does not hold DDXb does not hold for general b.

does not hold

Definition

V V tptu `

niZi . (DDV )

does not hold

Definition

V V tptu `

niZi . (DDV )

does not hold

Definition

V V tptu `

niZi . (DDV )

Decomposition of the diago

the lüroth problem - unice.frmath1.unice.fr/~beauvill/conf/colloq.pdf · castelnuovo’s theorem...

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