The Cremona group

The Cremona group

Carolina Araujo (IMPA)

The Cremona Group

Bir( P

) :=

n

ϕ : P

− − →

P

birational self-map

o

Setup

Projective spaces:

Pⁿ = Cⁿ⁺¹\ {¯0} .

(x0, . . . ,xn)∼λ(x0, . . . ,xn), λ6= 0 3 (x0:· · ·:xn)

Algebraic sets:

F1, . . . ,Fk ∈ C[x0:· · ·:xn] homogeneous polynomials X =Z(F₁, . . . ,F_k) =n

(a₀:· · ·:a_n)∈Pⁿ

F_i(a₀ :· · ·:a_n) = 0 ∀io Zariski topology: closed sets are the algebraic sets

Projective varieties: irreducible algebraic setsX ⊂P^N

Setup

Projective spaces:

Pⁿ = Cⁿ⁺¹\ {¯0} .

(x0, . . . ,xn)∼λ(x0, . . . ,xn), λ6= 0 3 (x₀:· · ·:x_n)

Algebraic sets:

F1, . . . ,Fk ∈ C[x0:· · ·:xn] homogeneous polynomials X =Z(F₁, . . . ,F_k) =n

(a₀:· · ·:a_n)∈Pⁿ

F_i(a₀ :· · ·:a_n) = 0 ∀io Zariski topology: closed sets are the algebraic sets

Projective varieties: irreducible algebraic setsX ⊂P^N

Setup

Projective spaces:

Pⁿ = Cⁿ⁺¹\ {¯0} .

(x0, . . . ,xn)∼λ(x0, . . . ,xn), λ6= 0 3 (x₀:· · ·:x_n)

Algebraic sets:

F1, . . . ,Fk ∈ C[x0:· · ·:xn] homogeneous polynomials X =Z(F₁, . . . ,F_k) =n

(a₀:· · ·:a_n)∈Pⁿ

F_i(a₀ :· · ·:a_n) = 0 ∀io

Zariski topology: closed sets are the algebraic sets Projective varieties: irreducible algebraic setsX ⊂P^N

Setup

Projective spaces:

Pⁿ = Cⁿ⁺¹\ {¯0} .

(x0, . . . ,xn)∼λ(x0, . . . ,xn), λ6= 0 3 (x₀:· · ·:x_n)

Algebraic sets:

F1, . . . ,Fk ∈ C[x0:· · ·:xn] homogeneous polynomials X =Z(F₁, . . . ,F_k) =n

(a₀:· · ·:a_n)∈Pⁿ

F_i(a₀ :· · ·:a_n) = 0 ∀io Zariski topology: closed sets are the algebraic sets

Projective varieties: irreducible algebraic setsX ⊂P^N

Setup

Projective spaces:

Pⁿ = Cⁿ⁺¹\ {¯0} .

(x0, . . . ,xn)∼λ(x0, . . . ,xn), λ6= 0 3 (x₀:· · ·:x_n)

Algebraic sets:

F1, . . . ,Fk ∈ C[x0:· · ·:xn] homogeneous polynomials X =Z(F₁, . . . ,F_k) =n

(a₀:· · ·:a_n)∈Pⁿ

F_i(a₀ :· · ·:a_n) = 0 ∀io Zariski topology: closed sets are the algebraic sets

Projective varieties: irreducible algebraic setsX ⊂P^N

Setup

Morphisms:

Pⁿ −→ P^m

(x0:· · ·:xn) 7−→ F0(x0, . . . ,xn) :· · ·:Fm(x0, . . . ,xn) F0, . . . ,Fm∈C[x0, . . . ,xn] homogeneous of the same degree withno common zeroes

Rational maps:

Pⁿ − − → P^m

(x₀:· · ·:x_n) 7−→ F₀(x₀, . . . ,x_n) :· · ·:F_m(x₀, . . . ,x_n) F0, . . . ,Fm∈C[x0, . . . ,xn] homogeneous of the same degree

Setup

Morphisms:

Pⁿ −→ P^m

(x0:· · ·:xn) 7−→ F0(x0, . . . ,xn) :· · ·:Fm(x0, . . . ,xn) F0, . . . ,Fm∈C[x0, . . . ,xn] homogeneous of the same degree withno common zeroes

Rational maps:

Pⁿ − − → P^m

(x₀:· · ·:x_n) 7−→ F₀(x₀, . . . ,x_n) :· · ·:F_m(x₀, . . . ,x_n) F0, . . . ,Fm∈C[x0, . . . ,xn] homogeneous of the same degree

Setup

Morphisms:

Pⁿ −→ P^m

(x0:· · ·:xn) 7−→ F0(x0, . . . ,xn) :· · ·:Fm(x0, . . . ,xn) F0, . . . ,Fm∈C[x0, . . . ,xn] homogeneous of the same degree withno common zeroes

Rational maps:

Pⁿ − − → P^m

(x₀:· · ·:x_n) 7−→ F₀(x₀, . . . ,x_n) :· · ·:F_m(x₀, . . . ,x_n) F0, . . . ,Fm∈C[x0, . . . ,xn] homogeneous of the same degree

Setup

Morphisms:

Pⁿ −→ P^m

(x0:· · ·:xn) 7−→ F0(x0, . . . ,xn) :· · ·:Fm(x0, . . . ,xn) F0, . . . ,Fm∈C[x0, . . . ,xn] homogeneous of the same degree withno common zeroes

Rational maps:

Pⁿ − − → P^m

(x₀:· · ·:x_n) 7−→ F₀(x₀, . . . ,x_n) :· · ·:F_m(x₀, . . . ,x_n) F₀, . . . ,F_m∈C[x₀, . . . ,x_n] homogeneous of the same degree

Setup

Morphisms:

f :X → Y

Rational maps:

ϕ:X 99K Y

Function fields: C(X) =

n

meromorphic functions on X o

C(Pⁿ) ∼= C(x1, . . . ,xn)

Setup

Morphisms:

f :X → Y

Rational maps:

ϕ:X 99K Y

Function fields: C(X) =

n

meromorphic functions on X o

C(Pⁿ) ∼= C(x1, . . . ,xn)

Setup

Morphisms:

f :X → Y

Rational maps:

ϕ:X 99K Y

Function fields: C(X) =

n

meromorphic functions on X o

C(Pⁿ) ∼= C(x1, . . . ,xn)

Setup

Morphisms:

f :X → Y

Rational maps:

ϕ:X 99K Y

Function fields: C(X) =

n

meromorphic functions on X o

C(Pⁿ) ∼= C(x1, . . . ,xn)

Setup

Morphisms:

f :X → Y

Rational maps:

ϕ:X 99K Y

Function fields:

C(X) = n

meromorphic functions on X o

C(Pⁿ) ∼= C(x1, . . . ,xn)

Setup

Morphisms:

f :X → Y

Rational maps:

ϕ:X 99K Y

Function fields:

C(X) = n

meromorphic functions on X o

C(Pⁿ) ∼= C(x1, . . . ,xn)

Automorphisms of projective varieties

X ⊂Pⁿ complex projective variety

Aut(X) = n

f :X →X automorphism o

Example (X =Pⁿ) Aut(Pⁿ) = PGLn+1(C)

X ⊂Pⁿ complex projective variety Aut(X) = n

f :X →X automorphism o

Example (X =Pⁿ) Aut(Pⁿ) = PGLn+1(C)

X ⊂Pⁿ complex projective variety Aut(X) = n

f :X →X automorphism o

Example (X =Pⁿ) Aut(Pⁿ) = PGLn+1(C)

X ⊂Pⁿ complex projective variety Aut(X) Lie group

Aut⁰(X)⊂Aut(X) connected component ofIX (C-algebraic group)

0 → Aut⁰(X)

| {z }

C-algebraic group

→ Aut(X) → Aut(X)/Aut⁰(X)

| {z }

countable discrete group

→ 0

X ⊂Pⁿ complex projective variety Aut(X) Lie group

Aut⁰(X)⊂Aut(X) connected component ofIX (C-algebraic group)

0 → Aut⁰(X)

| {z }

C-algebraic group

→ Aut(X) → Aut(X)/Aut⁰(X)

| {z }

countable discrete group

→ 0

X ⊂Pⁿ complex projective variety 0 → Aut⁰(X)

| {z }

C-algebraic group

→ Aut(X) → Aut(X)/Aut⁰(X)

| {z }

countable discrete group

→ 0

Example (X smooth projective curve of genus g )

&% '$

g = 0 g = 1

· · ·

g ≥2 g = 0 Aut⁰(P¹) = Aut(P¹) = PGL2(C)

g = 1 Aut⁰(X) ∼= X g ≥2 Aut(X) is finite

X ⊂Pⁿ complex projective variety 0 → Aut⁰(X)

| {z }

C-algebraic group

→ Aut(X) → Aut(X)/Aut⁰(X)

| {z }

countable discrete group

→ 0

Example (X smooth projective curve of genus g )

&%

'$

g = 0 g = 1

· · ·

g ≥2

g = 0 Aut⁰(P¹) = Aut(P¹) = PGL2(C) g = 1 Aut⁰(X) ∼= X

g ≥2 Aut(X) is finite

X ⊂Pⁿ complex projective variety 0 → Aut⁰(X)

| {z }

C-algebraic group

→ Aut(X) → Aut(X)/Aut⁰(X)

| {z }

countable discrete group

→ 0

Example (X smooth projective curve of genus g )

&%

'$

g = 0 g = 1

· · ·

g ≥2 g = 0 Aut⁰(P¹) = Aut(P¹) = PGL2(C)

g = 1 Aut⁰(X) ∼= X g ≥2 Aut(X) is finite

X ⊂Pⁿ complex projective variety 0 → Aut⁰(X)

| {z }

C-algebraic group

→ Aut(X) → Aut(X)/Aut⁰(X)

| {z }

countable discrete group

→ 0

Example (X smooth projective curve of genus g )

&%

'$

g = 0 g = 1

· · ·

g ≥2 g = 0 Aut⁰(P¹) = Aut(P¹) = PGL2(C)

g = 1 Aut⁰(X) ∼= X

g ≥2 Aut(X) is finite

X ⊂Pⁿ complex projective variety 0 → Aut⁰(X)

| {z }

C-algebraic group

→ Aut(X) → Aut(X)/Aut⁰(X)

| {z }

countable discrete group

→ 0

Example (X smooth projective curve of genus g )

&%

'$

g = 0 g = 1

· · ·

g ≥2 g = 0 Aut⁰(P¹) = Aut(P¹) = PGL2(C)

g = 1 Aut⁰(X) ∼= X g ≥2 Aut(X) is finite

X ⊂Pⁿ complex projective variety

1 → Aut⁰(X)

| {z }

C-algebraic group

→ Aut(X) → Aut(X)/Aut⁰(X)

| {z }

countable discrete group

→ 1

1 → Lin Aut⁰(X)

| {z }

linear algebraic group

→ Aut⁰(X) → Ab

|{z}

projective

→ 1

X ⊂Pⁿ complex projective variety

1 → Aut⁰(X)

| {z }

C-algebraic group

→ Aut(X) → Aut(X)/Aut⁰(X)

| {z }

countable discrete group

→ 1

1 → Lin Aut⁰(X)

| {z }

linear algebraic group

→ Aut⁰(X) → Ab

|{z}

projective

→ 1

Birational Geometry

The classification problem in Algebraic Geometry

In dimension 1 :

&%

'$

g = 0 g = 1

· · ·

g ≥2

M_g moduli space of curves of genus g

In higher dimensions : too many isomorphism classes

The classification problem in Algebraic Geometry

In dimension 1 :

&%

'$

g = 0 g = 1

· · ·

g ≥2

M_g moduli space of curves of genus g

In higher dimensions : too many isomorphism classes

The classification problem in Algebraic Geometry

In dimension 1 :

&%

'$

g = 0 g = 1

· · ·

g ≥2

M_g moduli space of curves of genus g

In higher dimensions : too many isomorphism classes

Example (The Blowup of P²)

P = (0 : 0 : 1)∈P²

P²×P¹

∪ X˜ = Γ

P² P¹

(x:y :z) (x :y)

πP

π₂ π₁

π₁: ˜X\E −−−→^∼= P²\ {P} E = π₁⁻¹(P) ∼=P¹

Example (The Blowup of P²)

P = (0 : 0 : 1)∈P²

P²×P¹

∪ X˜ = Γ

P² P¹

(x:y :z) (x :y)

πP

π₂ π₁

π₁: ˜X\E −−−→^∼= P²\ {P} E = π₁⁻¹(P) ∼=P¹

Example (The Blowup of P²)

P = (0 : 0 : 1)∈P²

P²×P¹

∪ X˜ = Γ

P² P¹

(x:y :z) (x :y)

πP

π₂ π₁

π₁: ˜X\E −−−→^∼= P²\ {P} E = π₁⁻¹(P) ∼=P¹

The Blowup ofP² at a pointP

The Blowup ofX at a pointP

The Blowup ofX along a proper subvariety Z X˜\E −−−→^∼= X\Z

The Blowup ofP² at a pointP

The Blowup ofX at a pointP

The Blowup ofX along a proper subvariety Z X˜\E −−−→^∼= X\Z

The Blowup ofP² at a pointP

The Blowup ofX at a pointP

The Blowup ofX along a proper subvariety Z

X˜\E −−−→^∼= X\Z

The Blowup ofP² at a pointP

The Blowup ofX at a pointP

The Blowup ofX along a proper subvariety Z X˜\E −−−→^∼= X\Z

Definition

X andY are birational equivalentif ∃ dense open subsets U ⊂X and V ⊂Y and isomorphism

X ⊃ U −−−→^∼= V ⊂ Y X − − →^∼ Y

Equivalently:

C(X) ∼=_C C(Y)

The problem of birational classification :

Given a projective variety X, to find asimplest representativein its birational class - minimal model ofX

Constructmoduli spaces ofminimal models (with fixed discrete invariants)

Definition

X andY are birational equivalentif ∃ dense open subsets U ⊂X and V ⊂Y and isomorphism

X ⊃ U −−−→^∼= V ⊂ Y X − − →^∼ Y

Equivalently:

C(X) ∼=_C C(Y)

The problem of birational classification :

Given a projective variety X, to find asimplest representativein its birational class - minimal model ofX

Constructmoduli spaces ofminimal models (with fixed discrete invariants)

Definition

X andY are birational equivalentif ∃ dense open subsets U ⊂X and V ⊂Y and isomorphism

X ⊃ U −−−→^∼= V ⊂ Y X − − →^∼ Y

Equivalently:

C(X) ∼=_C C(Y)

The problem of birational classification :

Given a projective variety X, to find asimplest representativein its birational class - minimal model ofX

Constructmoduli spaces ofminimal models (with fixed discrete invariants)

Definition

X andY are birational equivalentif ∃ dense open subsets U ⊂X and V ⊂Y and isomorphism

X ⊃ U −−−→^∼= V ⊂ Y X − − →^∼ Y

Equivalently:

C(X) ∼=_C C(Y)

The problem of birational classification :

Given a projective variety X, to find asimplest representativein its birational class - minimal model ofX

Constructmoduli spaces ofminimal models (with fixed discrete invariants)

The Minimal Model Program (MMP)

Given a projective variety X, to find asimplest representativein its birational class - minimal model of X

Dimension 1: Riemann (19^th century)

Dimension 2: Italian school (early 20^th century) Dimension 3: Mori (1988)

Dimension ≥4: Birkar-Cascini-Hacon-McKernan (2010)

Definition

X is rationalif it is birationally equivalent to Pⁿ

Equivalently:

C(X) ∼=_C C(x₁, . . . ,x_n)

Problem

Which algebraic varieties are rational?

Problem

For which d andn, is the generic hypersurfaceX_d ⊂Pⁿ⁺¹ rational?

Definition

X is rationalif it is birationally equivalent to Pⁿ Equivalently:

C(X) ∼=_C C(x₁, . . . ,x_n)

Problem

Which algebraic varieties are rational?

Problem

For which d andn, is the generic hypersurfaceX_d ⊂Pⁿ⁺¹ rational?

Definition

X is rationalif it is birationally equivalent to Pⁿ Equivalently:

C(X) ∼=_C C(x₁, . . . ,x_n)

Problem

Which algebraic varieties are rational?

Problem

For which d andn, is the generic hypersurfaceX_d ⊂Pⁿ⁺¹ rational?

Definition

X is rationalif it is birationally equivalent to Pⁿ Equivalently:

C(X) ∼=_C C(x₁, . . . ,x_n)

Problem

Which algebraic varieties are rational?

Problem

For which d andn, is the generic hypersurfaceX_d ⊂Pⁿ⁺¹ rational?

Problem

For which d andn, is the generic hypersurfaceX_d ⊂Pⁿ⁺¹ rational?

Necessary Condition

d ≤n+ 1

Proof.

pg(X) =h⁰(X, ω_X) is a birational invariant for smooth projective varieties

ω_Pⁿ ∼=O_Pⁿ(−n−1) =⇒ p_g(Pⁿ) = 0 ω_Xd ∼=O_Pⁿ(−n−2 +d)_|Xd

pg(Xd) = 0 ⇐⇒ d ≤n+ 1

Problem

For which d andn, is the generic hypersurfaceX_d ⊂Pⁿ⁺¹ rational?

Necessary Condition

d ≤n+ 1

Proof.

pg(X) =h⁰(X, ω_X) is a birational invariant for smooth projective varieties

ω_Pⁿ ∼=O_Pⁿ(−n−1) =⇒ p_g(Pⁿ) = 0 ω_Xd ∼=O_Pⁿ(−n−2 +d)_|Xd

pg(Xd) = 0 ⇐⇒ d ≤n+ 1

Problem

For which d andn, is the generic hypersurfaceX_d ⊂Pⁿ⁺¹ rational?

Necessary Condition

d ≤n+ 1

Proof.

p_g(X) =h⁰(X, ω_X) is a birational invariant for smooth projective varieties

ω_Pⁿ ∼=O_Pⁿ(−n−1) =⇒ p_g(Pⁿ) = 0 ω_Xd ∼=O_Pⁿ(−n−2 +d)_|Xd

pg(Xd) = 0 ⇐⇒ d ≤n+ 1

Problem

For which d andn, is the generic hypersurfaceX_d ⊂Pⁿ⁺¹ rational?

Necessary Condition

d ≤n+ 1

Proof.

p_g(X) =h⁰(X, ω_X) is a birational invariant for smooth projective varieties

ω_Pⁿ ∼=O_Pⁿ(−n−1) =⇒ p_g(Pⁿ) = 0

ω_Xd ∼=O_Pⁿ(−n−2 +d)_|Xd pg(Xd) = 0 ⇐⇒ d ≤n+ 1

Problem

For which d andn, is the generic hypersurfaceX_d ⊂Pⁿ⁺¹ rational?

Necessary Condition

d ≤n+ 1

Proof.

p_g(X) =h⁰(X, ω_X) is a birational invariant for smooth projective varieties

ω_Pⁿ ∼=O_Pⁿ(−n−1) =⇒ p_g(Pⁿ) = 0 ω_Xd ∼=O_Pⁿ(−n−2 +d)_|Xd

pg(Xd) = 0 ⇐⇒ d ≤n+ 1

Problem

For which d andn, is the generic hypersurfaceX_d ⊂Pⁿ⁺¹ rational?

Necessary Condition

d ≤n+ 1

Proof.

p_g(X) =h⁰(X, ω_X) is a birational invariant for smooth projective varieties

ω_Pⁿ ∼=O_Pⁿ(−n−1) =⇒ p_g(Pⁿ) = 0 ω_Xd ∼=O_Pⁿ(−n−2 +d)_|Xd

pg(Xd) = 0 ⇐⇒ d ≤n+ 1

Problem

For which d andn, is the generic hypersurfaceXd ⊂Pⁿ⁺¹ rational?

Example (Quadric hypersurface X2 ⊂Pⁿ⁺¹)

Stereographic projection defines a birational map π_P : X₂ − − →^∼ Pⁿ

Problem

For which d andn, is the generic hypersurfaceXd ⊂Pⁿ⁺¹ rational?

Example (Quadric hypersurface X2 ⊂Pⁿ⁺¹)

Stereographic projection defines a birational map π_P : X₂ − − →^∼ Pⁿ

Problem

For which d andn, is the generic hypersurfaceXd ⊂Pⁿ⁺¹ rational?

Necessary Condition

d ≤n+ 1

Problem

Is the generic cubic hypersurface X3 ⊂Pⁿ⁺¹ rational (n >3)?

Problem

For which d andn, is the generic hypersurfaceXd ⊂Pⁿ⁺¹ rational?

Necessary Condition

d ≤n+ 1

Problem

Is the generic cubic hypersurface X3⊂Pⁿ⁺¹ rational (n >3)?

Problem

Which properties are invariant under birational equivalence?

Aut(X) is nota birational invariant

X X

Y Y

∼=

o o

∼

Definition (The Birational Group) Bir(X) :=

ϕ:X − − →^∼ X birational self-map

Example (Iskovskikh-Manin 1971) Any smooth quartic 3-fold X₄⊂P⁴ is irrational

Problem

Which properties are invariant under birational equivalence?

Aut(X) is nota birational invariant

X X

Y Y

∼=

o o

∼

Definition (The Birational Group) Bir(X) :=

ϕ:X − − →^∼ X birational self-map

Example (Iskovskikh-Manin 1971) Any smooth quartic 3-fold X₄⊂P⁴ is irrational

Problem

Which properties are invariant under birational equivalence?

Aut(X) is nota birational invariant

X X

Y Y

∼=

o o

∼

Definition (The Birational Group) Bir(X) :=

ϕ:X − − →^∼ X birational self-map

Example (Iskovskikh-Manin 1971) Any smooth quartic 3-fold X₄⊂P⁴ is irrational

Problem

Which properties are invariant under birational equivalence?

Aut(X) is nota birational invariant

X X

Y Y

∼=

o o

∼

Definition (The Birational Group) Bir(X) :=

ϕ:X − − →^∼ X birational self-map

Example (Iskovskikh-Manin 1971) Any smooth quartic 3-fold X₄⊂P⁴ is irrational

The Cremona Group

Bir( P

) :=

n

ϕ : P

− − →

P

birational self-map

o

The Cremona Group of the plane

Example (The standard quadratic transformation)

τ : P² − − →^∼ P²

(x :y :z) 7−→ _x¹ : ¹_y : ¹_z

= (yz :xz:xy)

Theorem (Noether-Castelnuovo 1870-1901) Bir(P²) =

Aut(P²) , τ

The Cremona Group of the plane

Example (The standard quadratic transformation)

τ : P² − − →^∼ P²

(x :y :z) 7−→ _x¹ :_y¹ : ¹_z

= (yz :xz:xy)

Theorem (Noether-Castelnuovo 1870-1901) Bir(P²) =

Aut(P²) , τ

The Cremona Group of the plane

Example (The standard quadratic transformation)

τ : P² − − →^∼ P²

(x :y :z) 7−→ _x¹ :_y¹ : ¹_z

= (yz :xz:xy)

Theorem (Noether-Castelnuovo 1870-1901) Bir(P²) =

Aut(P²) , τ

The Cremona Group of the plane

Example (The standard quadratic transformation)

τ : P² − − →^∼ P²

(x :y :z) 7−→ _x¹ :_y¹ : ¹_z

= (yz :xz:xy)

Theorem (Noether-Castelnuovo 1870-1901) Bir(P²) =

Aut(P²) , τ

The Cremona Group of the plane

Theorem (Cantat-Lamy 2013) Bir(P²) is not a simple group.

Proof exploits the action of Bir(P²) in an infinite dimensional hyperbolic space.

Theorem holds over any base field (Lonjou 2016). Any quotient of Bir(P²) is infinite and non-abelian.

Theorem (Bertini 1877, · · · , Dolgachev-Iskovskikh 2009) Classification of finite subgroups of Bir(P²).

The Cremona Group of the plane

Theorem (Cantat-Lamy 2013) Bir(P²) is not a simple group.

Proof exploits the action of Bir(P²) in an infinite dimensional hyperbolic space.

Theorem holds over any base field (Lonjou 2016). Any quotient of Bir(P²) is infinite and non-abelian.

Theorem (Bertini 1877, · · · , Dolgachev-Iskovskikh 2009) Classification of finite subgroups of Bir(P²).

The Cremona Group of the plane

Theorem (Cantat-Lamy 2013) Bir(P²) is not a simple group.

Proof exploits the action of Bir(P²) in an infinite dimensional hyperbolic space.

Theorem holds over any base field (Lonjou 2016).

Any quotient of Bir(P²) is infinite and non-abelian.

Theorem (Bertini 1877, · · · , Dolgachev-Iskovskikh 2009) Classification of finite subgroups of Bir(P²).

The Cremona Group of the plane

Theorem (Cantat-Lamy 2013) Bir(P²) is not a simple group.

Proof exploits the action of Bir(P²) in an infinite dimensional hyperbolic space.

Theorem holds over any base field (Lonjou 2016).

Any quotient of Bir(P²) is infinite and non-abelian.

Theorem (Bertini 1877, · · · , Dolgachev-Iskovskikh 2009) Classification of finite subgroups of Bir(P²).

The Cremona Group of the plane

Theorem (Cantat-Lamy 2013) Bir(P²) is not a simple group.

Proof exploits the action of Bir(P²) in an infinite dimensional hyperbolic space.

Theorem holds over any base field (Lonjou 2016).

Any quotient of Bir(P²) is infinite and non-abelian.

Theorem (Bertini 1877, · · · , Dolgachev-Iskovskikh 2009) Classification of finite subgroups of Bir(P²).

The Cremona Group in higher dimensions

Theorem (Noether-Castelnuovo 1870-1901) Bir(P²) =

Aut(P²) , τ

Theorem (Hilda Hudson 1927)

For n≥3, Bir(Pⁿ) cannot be generated by elements of bounded degree.

Theorem (Blanc-Lamy-Zimmermann 2019) For n≥3:

Bir(Pⁿ) Z/2Z.

The Cremona Group in higher dimensions

Theorem (Noether-Castelnuovo 1870-1901) Bir(P²) =

Aut(P²) , τ

Theorem (Hilda Hudson 1927)

For n≥3, Bir(Pⁿ) cannot be generated by elements of bounded degree.

Theorem (Blanc-Lamy-Zimmermann 2019) For n≥3:

Bir(Pⁿ) Z/2Z.

The Cremona Group in higher dimensions

Theorem (Noether-Castelnuovo 1870-1901) Bir(P²) =

Aut(P²) , τ

Theorem (Hilda Hudson 1927)

For n≥3, Bir(Pⁿ) cannot be generated by elements of bounded degree.

Theorem (Blanc-Lamy-Zimmermann 2019) For n≥3:

Bir(Pⁿ) Z/2Z.

The Cremona Group in higher dimensions

Theorem (Noether-Castelnuovo 1870-1901) Bir(P²) =

Aut(P²) , τ

Theorem (Hilda Hudson 1927)

For n≥3, Bir(Pⁿ) cannot be generated by elements of bounded degree.

Theorem (Blanc-Lamy-Zimmermann 2019) For n≥3:

Bir(Pⁿ) Z/2Z.

Factorizing birational maps ψ : P

99K P

The Sarkisov program (Corti 1995, Hacon-McKernan 2013):

Pⁿ=X0 X1 · · · Xn−1 Xn=Pⁿ

ψ1 ψ2 ψn−1 ψn

ψ

The ψ_i’s areelementary links

The Xi’s areMori fiber spaces(possible outcomes of the MMP)

Factorizing birational maps ψ : P

99K P

The Sarkisov program (Corti 1995, Hacon-McKernan 2013):

Pⁿ=X₀ X₁ · · · Xn−1 X_n=Pⁿ

ψ1 ψ2 ψn−1 ψn

ψ

The ψ_i’s areelementary links

The Xi’s areMori fiber spaces(possible outcomes of the MMP)

Factorizing birational maps ψ : P

99K P

The Sarkisov program (Corti 1995, Hacon-McKernan 2013):

Pⁿ=X₀ X₁ · · · Xn−1 X_n=Pⁿ

ψ1 ψ2 ψn−1 ψn

ψ

The ψ_i’s areelementary links

The Xi’s areMori fiber spaces(possible outcomes of the MMP)

Factorizing birational maps ψ : P

99K P

The Sarkisov program (Corti 1995, Hacon-McKernan 2013):

Pⁿ=X0 X1 · · · Xn−1 Xn=Pⁿ

pt Y1 Yn−1 pt

ψ1 ψ2 ψn−1 ψn

ψ

The ψ_i’s areelementary links

The X_i’s areMori fiber spaces(possible outcomes of the MMP)

The surface case

The Mori fiber spaces are: P² →pt

Fm →P¹ (P¹-bundle) F0 ∼=P¹×P¹

F1 ∼=BlPP²

F2 is the blowup a quadric coneQ ⊂P³

The surface case

The Mori fiber spaces are:

P² →pt

Fm →P¹ (P¹-bundle) F0 ∼=P¹×P¹

F1 ∼=BlPP²

F2 is the blowup a quadric coneQ ⊂P³

The surface case

The Mori fiber spaces are:

P² →pt

Fm →P¹ (P¹-bundle) F0 ∼=P¹×P¹

F1 ∼=BlPP²

F2 is the blowup a quadric coneQ ⊂P³

The surface case

The Mori fiber spaces are:

P² →pt

Fm →P¹ (P¹-bundle) F0 ∼=P¹×P¹

F1 ∼=BlPP²

F2 is the blowup a quadric coneQ ⊂P³

The surface case

The Mori fiber spaces are:

P² →pt

Fm →P¹ (P¹-bundle) F0 ∼=P¹×P¹

F1 ∼=BlPP²

F2 is the blowup a quadric coneQ ⊂P³

The surface case

The Mori fiber spaces are:

P² →pt

Fm →P¹ (P¹-bundle)

The elementary linksare

Type 1 P² F1

pt P¹

Bl_P

Type 2 Z

Fm Fm±1

P¹ P¹

Type 3

F1 P²

P¹ pt

Bl_P

Type 4 F0 F0

P¹ P¹

The surface case

The Mori fiber spaces are:

P² →pt

Fm →P¹ (P¹-bundle)

The elementary linksare

Type 1 P² F1

pt P¹

Bl_P

Type 2 Z

Fm Fm±1

P¹ P¹

Type 3

F1 P²

P¹ pt

Bl_P

Type 4 F0 F0

P¹ P¹

The surface case

The Mori fiber spaces are:

P² →pt

Fm →P¹ (P¹-bundle)

The elementary linksare

Type 1 P² F1

pt P¹

Bl_P

Type 2 Z

Fm Fm±1

P¹ P¹

Type 3

F1 P²

P¹ pt

Bl_P

Type 4 F0 F0

P¹ P¹

The surface case

The Mori fiber spaces are:

P² →pt

Fm →P¹ (P¹-bundle)

The elementary linksare

Type 1 P² F1

pt P¹

Bl_P

Type 2 Z

Fm Fm±1

P¹ P¹

Type 3

F1 P²

P¹ pt

Bl_P

Type 4 F0 F0

P¹ P¹

The surface case

The Mori fiber spaces are:

P² →pt

Fm →P¹ (P¹-bundle)

The elementary linksare

Type 1 P² F1

pt P¹

Bl_P

Type 2 Z

Fm Fm±1

P¹ P¹

Type 3

F1 P²

P¹ pt

BlP

Type 4 F0 F0

P¹ P¹

The surface case

The Mori fiber spaces are:

P² →pt

Fm →P¹ (P¹-bundle)

The elementary linksare

Type 1 P² F1

pt P¹

Bl_P

Type 2 Z

Fm Fm±1

P¹ P¹

Type 3

F1 P²

P¹ pt

BlP

Type 4 F0 F0

P¹ P¹

Theorem (Blanc-Lamy-Zimmermann 2019) For n≥3:

Bir(Pⁿ) Z/2Z.

The Sarkisov program

Pⁿ=X₀ X₁ · · · Xn−1 X_n=Pⁿ

pt Y₁ Yn−1 pt

ψ1 ψ2 ψn−1 ψn

ψ

The ψi’s areelementary links The X_i’s areMori fiber spaces

Theorem (Blanc-Lamy-Zimmermann 2019) For n≥3:

Bir(Pⁿ) Z/2Z.

The Sarkisov program

Pⁿ=X₀ X₁ · · · Xn−1 X_n=Pⁿ

pt Y₁ Yn−1 pt

ψ1 ψ2 ψn−1 ψn

ψ

The ψi’s areelementary links The X_i’s areMori fiber spaces

Automorphisms of Smooth Hypersurfaces

D =D_d ⊂Pⁿ⁺¹ smooth hypersurface of degreed

Theorem (Matsumura-Monsky 1964) If (n,d)6= (1,3),(2,4), then

Aut(Pⁿ⁺¹,D) Aut(D).

C =D3 ⊂P² elliptic curve S =D₄ ⊂P³ smooth K3 surface

In both cases, the image of Aut(Pⁿ⁺¹,D)→Aut(D) is finite

Automorphisms of Smooth Hypersurfaces

D =D_d ⊂Pⁿ⁺¹ smooth hypersurface of degreed

Theorem (Matsumura-Monsky 1964) If (n,d)6= (1,3),(2,4), then

Aut(Pⁿ⁺¹,D) Aut(D).

C =D3 ⊂P² elliptic curve S =D₄ ⊂P³ smooth K3 surface

In both cases, the image of Aut(Pⁿ⁺¹,D)→Aut(D) is finite

Automorphisms of Smooth Hypersurfaces

D =D_d ⊂Pⁿ⁺¹ smooth hypersurface of degreed

Theorem (Matsumura-Monsky 1964) If (n,d)6= (1,3),(2,4), then

Aut(Pⁿ⁺¹,D) Aut(D).

C =D3 ⊂P² elliptic curve S =D₄ ⊂P³ smooth K3 surface

In both cases, the image of Aut(Pⁿ⁺¹,D)→Aut(D) is finite

Automorphisms of Smooth Hypersurfaces

D =D_d ⊂Pⁿ⁺¹ smooth hypersurface of degreed

Theorem (Matsumura-Monsky 1964) If (n,d)6= (1,3),(2,4), then

Aut(Pⁿ⁺¹,D) Aut(D).

C =D3 ⊂P² elliptic curve

S =D₄ ⊂P³ smooth K3 surface

In both cases, the image of Aut(Pⁿ⁺¹,D)→Aut(D) is finite

Automorphisms of Smooth Hypersurfaces

D =D_d ⊂Pⁿ⁺¹ smooth hypersurface of degreed

Theorem (Matsumura-Monsky 1964) If (n,d)6= (1,3),(2,4), then

Aut(Pⁿ⁺¹,D) Aut(D).

C =D3 ⊂P² elliptic curve S =D₄ ⊂P³ smooth K3 surface

In both cases, the image of Aut(Pⁿ⁺¹,D)→Aut(D) is finite

Automorphisms of Smooth Hypersurfaces

D =D_d ⊂Pⁿ⁺¹ smooth hypersurface of degreed

Theorem (Matsumura-Monsky 1964) If (n,d)6= (1,3),(2,4), then

Aut(Pⁿ⁺¹,D) Aut(D).

C =D3 ⊂P² elliptic curve S =D₄ ⊂P³ smooth K3 surface

In both cases, the image of Aut(Pⁿ⁺¹,D)→Aut(D) is finite

C =D₃ ⊂P² elliptic curve

Theorem

Every automorphism ofC is induced by a birational self-map of the ambient space P².

S =D₄⊂P³ smooth K3 surface

Question (Gizatullin-Oguiso 2013)

Is every automorphism of S induced by a birational self-map of the ambient space P³?

Problem (w/ Alessio Corti and Alex Massarenti) Understand the group Bir P³,S

, and the group homomorphism Bir P³,S

→ Aut(S)

C =D₃ ⊂P² elliptic curve Theorem

Every automorphism ofC is induced by a birational self-map of the ambient space P².

S =D₄⊂P³ smooth K3 surface

Question (Gizatullin-Oguiso 2013)

Is every automorphism of S induced by a birational self-map of the ambient space P³?

Problem (w/ Alessio Corti and Alex Massarenti) Understand the group Bir P³,S

, and the group homomorphism Bir P³,S

→ Aut(S)

C =D₃ ⊂P² elliptic curve Theorem

Every automorphism ofC is induced by a birational self-map of the ambient space P².

S =D₄⊂P³ smooth K3 surface

Question (Gizatullin-Oguiso 2013)

Is every automorphism of S induced by a birational self-map of the ambient space P³?

Problem (w/ Alessio Corti and Alex Massarenti) Understand the group Bir P³,S

, and the group homomorphism Bir P³,S

→ Aut(S)

C =D₃ ⊂P² elliptic curve Theorem

Every automorphism ofC is induced by a birational self-map of the ambient space P².

S =D₄⊂P³ smooth K3 surface

Question (Gizatullin-Oguiso 2013)

Is every automorphism of S induced by a birational self-map of the ambient space P³?

Problem (w/ Alessio Corti and Alex Massarenti) Understand the group Bir P³,S

, and the group homomorphism Bir P³,S

→ Aut(S)

C =D₃ ⊂P² elliptic curve Theorem

Every automorphism ofC is induced by a birational self-map of the ambient space P².

S =D₄⊂P³ smooth K3 surface

Question (Gizatullin-Oguiso 2013)

Is every automorphism of S induced by a birational self-map of the ambient space P³?

Problem (w/ Alessio Corti and Alex Massarenti) Understand the group Bir P³,S

, and the group homomorphism Bir P³,S

→ Aut(S)

C =D₃ ⊂P² elliptic curve Theorem

Every automorphism ofC is induced by a birational self-map of the ambient space P².

S =D₄⊂P³ smooth K3 surface

Question (Gizatullin-Oguiso 2013)

Is every automorphism of S induced by a birational self-map of the ambient space P³?

Problem (w/ Alessio Corti and Alex Massarenti) Understand the group Bir P³,S

, and the group homomorphism Bir P³,S

→ Aut(S)

Thank you!

1Thanks to Santiago Arango, Daniela Paiva, Charles Staats and Wikipedia for the nice pictures