Birational of algebraic varieties

Caucher Birkar

Cambridge University

Rome, 2019 is the study of solutions of systems of polynomial equations and associated geometric structures.

Algebraic geometry is an amazingly complex but beautiful subject.

It is deeply related to many branches of but also to mathematical physics, computer science, etc.

Algebraic geometry and associated geometric structures.

Algebraic geometry is an amazingly complex but beautiful subject.

It is deeply related to many branches of mathematics but also to mathematical physics, computer science, etc.

Algebraic geometry

Algebraic geometry is the study of solutions of systems of polynomial equations Algebraic geometry is an amazingly complex but beautiful subject.

It is deeply related to many branches of mathematics but also to mathematical physics, computer science, etc.

Algebraic geometry

Algebraic geometry is the study of solutions of systems of polynomial equations and associated geometric structures. It is deeply related to many branches of mathematics but also to mathematical physics, computer science, etc.

Algebraic geometry

Algebraic geometry is the study of solutions of systems of polynomial equations and associated geometric structures.

Algebraic geometry is an amazingly complex but beautiful subject. Algebraic geometry

Algebraic geometry is the study of solutions of systems of polynomial equations and associated geometric structures.

Algebraic geometry is an amazingly complex but beautiful subject.

It is deeply related to many branches of mathematics but also to mathematical physics, computer science, etc. Let k = Q, or R, or C.

Let k[t] = polynomials in variable t with coefficients in k.

Given f ∈ k[t], we want to find its solutions. √ Example: t2 − 3t + 1 has no solution in Q but has solutions in R: (3 ± 5)/2. √ Example: t2 + 1 has no solution in Q, not even in R but has solutions in C: ± −1. √ Example: t2 + bt + c has solutions in C: (−b ± b2 − 4c)/2.

Question: Can we find solutions of a general f in terms of its coefficients (by radicals)?

Galois theory says: yes if deg f ≤ 4; not otherwise.

One variable Let k[t] = polynomials in variable t with coefficients in k.

Given f ∈ k[t], we want to find its solutions. √ Example: t2 − 3t + 1 has no solution in Q but has solutions in R: (3 ± 5)/2. √ Example: t2 + 1 has no solution in Q, not even in R but has solutions in C: ± −1. √ Example: t2 + bt + c has solutions in C: (−b ± b2 − 4c)/2.

Question: Can we find solutions of a general f in terms of its coefficients (by radicals)?

Galois theory says: yes if deg f ≤ 4; not otherwise.

One variable

Let k = Q, or R, or C. Given f ∈ k[t], we want to find its solutions. √ Example: t2 − 3t + 1 has no solution in Q but has solutions in R: (3 ± 5)/2. √ Example: t2 + 1 has no solution in Q, not even in R but has solutions in C: ± −1. √ Example: t2 + bt + c has solutions in C: (−b ± b2 − 4c)/2.

Question: Can we find solutions of a general f in terms of its coefficients (by radicals)?

Galois theory says: yes if deg f ≤ 4; not otherwise.

One variable

Let k = Q, or R, or C.

Let k[t] = polynomials in variable t with coefficients in k. √ Example: t2 − 3t + 1 has no solution in Q but has solutions in R: (3 ± 5)/2. √ Example: t2 + 1 has no solution in Q, not even in R but has solutions in C: ± −1. √ Example: t2 + bt + c has solutions in C: (−b ± b2 − 4c)/2.

Question: Can we find solutions of a general f in terms of its coefficients (by radicals)?

Galois theory says: yes if deg f ≤ 4; not otherwise.

One variable

Let k = Q, or R, or C.

Let k[t] = polynomials in variable t with coefficients in k.

Given f ∈ k[t], we want to find its solutions. √ but has solutions in R: (3 ± 5)/2. √ Example: t2 + 1 has no solution in Q, not even in R but has solutions in C: ± −1. √ Example: t2 + bt + c has solutions in C: (−b ± b2 − 4c)/2.

Question: Can we find solutions of a general f in terms of its coefficients (by radicals)?

Galois theory says: yes if deg f ≤ 4; not otherwise.

One variable

Let k = Q, or R, or C.

Let k[t] = polynomials in variable t with coefficients in k.

Given f ∈ k[t], we want to find its solutions.

Example: t2 − 3t + 1 has no solution in Q √ Example: t2 + 1 has no solution in Q, not even in R but has solutions in C: ± −1. √ Example: t2 + bt + c has solutions in C: (−b ± b2 − 4c)/2.

Question: Can we find solutions of a general f in terms of its coefficients (by radicals)?

Galois theory says: yes if deg f ≤ 4; not otherwise.

One variable

Let k = Q, or R, or C.

Let k[t] = polynomials in variable t with coefficients in k.

Given f ∈ k[t], we want to find its solutions. √ Example: t2 − 3t + 1 has no solution in Q but has solutions in R: (3 ± 5)/2. √ but has solutions in C: ± −1. √ Example: t2 + bt + c has solutions in C: (−b ± b2 − 4c)/2.

Question: Can we find solutions of a general f in terms of its coefficients (by radicals)?

Galois theory says: yes if deg f ≤ 4; not otherwise.

One variable

Let k = Q, or R, or C.

Let k[t] = polynomials in variable t with coefficients in k.

Given f ∈ k[t], we want to find its solutions. √ Example: t2 − 3t + 1 has no solution in Q but has solutions in R: (3 ± 5)/2.

Example: t2 + 1 has no solution in Q, not even in R √ Example: t2 + bt + c has solutions in C: (−b ± b2 − 4c)/2.

Question: Can we find solutions of a general f in terms of its coefficients (by radicals)?

Galois theory says: yes if deg f ≤ 4; not otherwise.

One variable

Let k = Q, or R, or C.

Let k[t] = polynomials in variable t with coefficients in k.

Given f ∈ k[t], we want to find its solutions. √ Example: t2 − 3t + 1 has no solution in Q but has solutions in R: (3 ± 5)/2. √ Example: t2 + 1 has no solution in Q, not even in R but has solutions in C: ± −1. Question: Can we find solutions of a general f in terms of its coefficients (by radicals)?

Galois theory says: yes if deg f ≤ 4; not otherwise.

One variable

Let k = Q, or R, or C.

Let k[t] = polynomials in variable t with coefficients in k.

Given f ∈ k[t], we want to find its solutions. √ Example: t2 − 3t + 1 has no solution in Q but has solutions in R: (3 ± 5)/2. √ Example: t2 + 1 has no solution in Q, not even in R but has solutions in C: ± −1. √ Example: t2 + bt + c has solutions in C: (−b ± b2 − 4c)/2. Galois theory says: yes if deg f ≤ 4; not otherwise.

One variable

Let k = Q, or R, or C.

Let k[t] = polynomials in variable t with coefficients in k.

Given f ∈ k[t], we want to find its solutions. √ Example: t2 − 3t + 1 has no solution in Q but has solutions in R: (3 ± 5)/2. √ Example: t2 + 1 has no solution in Q, not even in R but has solutions in C: ± −1. √ Example: t2 + bt + c has solutions in C: (−b ± b2 − 4c)/2.

Question: Can we find solutions of a general f in terms of its coefficients (by radicals)? One variable

Let k = Q, or R, or C.

Let k[t] = polynomials in variable t with coefficients in k.

Given f ∈ k[t], we want to find its solutions. √ Example: t2 − 3t + 1 has no solution in Q but has solutions in R: (3 ± 5)/2. √ Example: t2 + 1 has no solution in Q, not even in R but has solutions in C: ± −1. √ Example: t2 + bt + c has solutions in C: (−b ± b2 − 4c)/2.

Question: Can we find solutions of a general f in terms of its coefficients (by radicals)?

Galois theory says: yes if deg f ≤ 4; not otherwise. Examples: 2 2 t1 + t2 + 1 ∈ Q[t1, t2] has no solutions over Q but has solutions over C and F3.

2 2 t1 + t2 − 1 ∈ R[t1, t2] has lots of solutions in R.

m m t1 + t2 − 1, for m ≥ 3, has only trivial solutions in Q: when t1 = 0 or t2 = 0. This is Fermat’s last theorem.

2 t2 − t1 and t3 − t1t2 have common solutions in C: 2 3 t1 = a, t2 = a , t3 = a , a ∈ C.

In general it is hard to see if a system of equations has a solution. But it is relatively easy when working with C.

Question: can we visualise the set of solutions?

Multi-variable 2 2 t1 + t2 + 1 ∈ Q[t1, t2] has no solutions over Q but has solutions over C and F3.

2 2 t1 + t2 − 1 ∈ R[t1, t2] has lots of solutions in R.

m m t1 + t2 − 1, for m ≥ 3, has only trivial solutions in Q: when t1 = 0 or t2 = 0. This is Fermat’s last theorem.

2 t2 − t1 and t3 − t1t2 have common solutions in C: 2 3 t1 = a, t2 = a , t3 = a , a ∈ C.

In general it is hard to see if a system of equations has a solution. But it is relatively easy when working with C.

Question: can we visualise the set of solutions?

Multi-variable

Examples: 2 2 t1 + t2 − 1 ∈ R[t1, t2] has lots of solutions in R.

m m t1 + t2 − 1, for m ≥ 3, has only trivial solutions in Q: when t1 = 0 or t2 = 0. This is Fermat’s last theorem.

2 t2 − t1 and t3 − t1t2 have common solutions in C: 2 3 t1 = a, t2 = a , t3 = a , a ∈ C.

In general it is hard to see if a system of equations has a solution. But it is relatively easy when working with C.

Question: can we visualise the set of solutions?

Multi-variable

Examples: 2 2 t1 + t2 + 1 ∈ Q[t1, t2] has no solutions over Q but has solutions over C and F3. m m t1 + t2 − 1, for m ≥ 3, has only trivial solutions in Q: when t1 = 0 or t2 = 0. This is Fermat’s last theorem.

2 t2 − t1 and t3 − t1t2 have common solutions in C: 2 3 t1 = a, t2 = a , t3 = a , a ∈ C.

In general it is hard to see if a system of equations has a solution. But it is relatively easy when working with C.

Question: can we visualise the set of solutions?

Multi-variable

Examples: 2 2 t1 + t2 + 1 ∈ Q[t1, t2] has no solutions over Q but has solutions over C and F3.

2 2 t1 + t2 − 1 ∈ R[t1, t2] has lots of solutions in R. This is Fermat’s last theorem.

2 t2 − t1 and t3 − t1t2 have common solutions in C: 2 3 t1 = a, t2 = a , t3 = a , a ∈ C.

In general it is hard to see if a system of equations has a solution. But it is relatively easy when working with C.

Question: can we visualise the set of solutions?

Multi-variable

Examples: 2 2 t1 + t2 + 1 ∈ Q[t1, t2] has no solutions over Q but has solutions over C and F3.

2 2 t1 + t2 − 1 ∈ R[t1, t2] has lots of solutions in R.

m m t1 + t2 − 1, for m ≥ 3, has only trivial solutions in Q: when t1 = 0 or t2 = 0. 2 t2 − t1 and t3 − t1t2 have common solutions in C: 2 3 t1 = a, t2 = a , t3 = a , a ∈ C.

In general it is hard to see if a system of equations has a solution. But it is relatively easy when working with C.

Question: can we visualise the set of solutions?

Multi-variable

Examples: 2 2 t1 + t2 + 1 ∈ Q[t1, t2] has no solutions over Q but has solutions over C and F3.

2 2 t1 + t2 − 1 ∈ R[t1, t2] has lots of solutions in R.

m m t1 + t2 − 1, for m ≥ 3, has only trivial solutions in Q: when t1 = 0 or t2 = 0. This is Fermat’s last theorem. In general it is hard to see if a system of equations has a solution. But it is relatively easy when working with C.

Question: can we visualise the set of solutions?

Multi-variable

Examples: 2 2 t1 + t2 + 1 ∈ Q[t1, t2] has no solutions over Q but has solutions over C and F3.

2 2 t1 + t2 − 1 ∈ R[t1, t2] has lots of solutions in R.

m m t1 + t2 − 1, for m ≥ 3, has only trivial solutions in Q: when t1 = 0 or t2 = 0. This is Fermat’s last theorem.

2 t2 − t1 and t3 − t1t2 have common solutions in C: 2 3 t1 = a, t2 = a , t3 = a , a ∈ C. But it is relatively easy when working with C.

Question: can we visualise the set of solutions?

Multi-variable

Examples: 2 2 t1 + t2 + 1 ∈ Q[t1, t2] has no solutions over Q but has solutions over C and F3.

2 2 t1 + t2 − 1 ∈ R[t1, t2] has lots of solutions in R.

m m t1 + t2 − 1, for m ≥ 3, has only trivial solutions in Q: when t1 = 0 or t2 = 0. This is Fermat’s last theorem.

2 t2 − t1 and t3 − t1t2 have common solutions in C: 2 3 t1 = a, t2 = a , t3 = a , a ∈ C.

In general it is hard to see if a system of equations has a solution. Question: can we visualise the set of solutions?

Multi-variable

Examples: 2 2 t1 + t2 + 1 ∈ Q[t1, t2] has no solutions over Q but has solutions over C and F3.

2 2 t1 + t2 − 1 ∈ R[t1, t2] has lots of solutions in R.

m m t1 + t2 − 1, for m ≥ 3, has only trivial solutions in Q: when t1 = 0 or t2 = 0. This is Fermat’s last theorem.

2 t2 − t1 and t3 − t1t2 have common solutions in C: 2 3 t1 = a, t2 = a , t3 = a , a ∈ C.

In general it is hard to see if a system of equations has a solution. But it is relatively easy when working with C. Multi-variable

Examples: 2 2 t1 + t2 + 1 ∈ Q[t1, t2] has no solutions over Q but has solutions over C and F3.

2 2 t1 + t2 − 1 ∈ R[t1, t2] has lots of solutions in R.

m m t1 + t2 − 1, for m ≥ 3, has only trivial solutions in Q: when t1 = 0 or t2 = 0. This is Fermat’s last theorem.

2 t2 − t1 and t3 − t1t2 have common solutions in C: 2 3 t1 = a, t2 = a , t3 = a , a ∈ C.

In general it is hard to see if a system of equations has a solution. But it is relatively easy when working with C.

Question: can we visualise the set of solutions? Multi-variable

Examples: 2 2 t1 + t2 + 1 ∈ Q[t1, t2] has no solutions over Q but has solutions over C and F3.

2 2 t1 + t2 − 1 ∈ R[t1, t2] has lots of solutions in R.

m m t1 + t2 − 1, for m ≥ 3, has only trivial solutions in Q: when t1 = 0 or t2 = 0. This is Fermat’s last theorem.

2 t2 − t1 and t3 − t1t2 have common solutions in C: 2 3 t1 = a, t2 = a , t3 = a , a ∈ C.

In general it is hard to see if a system of equations has a solution. But it is relatively easy when working with C.

Question: can we visualise the set of solutions? Line: we can represent points on a line with real numbers.

Plane: we can represent points on a plane with two numbers.

Spaces – numbers Plane: we can represent points on a plane with two numbers.

Spaces – numbers

Line: we can represent points on a line with real numbers. Plane: we can represent points on a plane with two numbers.

Spaces – numbers

Line: we can represent points on a line with real numbers. Spaces – numbers

Line: we can represent points on a line with real numbers.

Plane: we can represent points on a plane with two numbers. Spaces – numbers

Line: we can represent points on a line with real numbers.

Plane: we can represent points on a plane with two numbers. Likewise we can define the n-dimensional space n k = {(a1,..., an) | ai ∈ k}.

Spaces – numbers

Space: we can represent points in 3-dimensional space with three numbers. Likewise we can define the n-dimensional space n k = {(a1,..., an) | ai ∈ k}.

Spaces – numbers

Space: we can represent points in 3-dimensional space with three numbers. Spaces – numbers

Space: we can represent points in 3-dimensional space with three numbers.

Likewise we can define the n-dimensional space n k = {(a1,..., an) | ai ∈ k}. n Given an equation f ∈ k[t1,..., tn] we can think of its solutions as points in k .

2 2 2 Example: shape of t1 + t2 − 1 in k :

Shape of equations 2 2 2 Example: shape of t1 + t2 − 1 in k :

Shape of equations

n Given an equation f ∈ k[t1,..., tn] we can think of its solutions as points in k . Shape of equations

n Given an equation f ∈ k[t1,..., tn] we can think of its solutions as points in k .

2 2 2 Example: shape of t1 + t2 − 1 in k : Shape of equations

n Given an equation f ∈ k[t1,..., tn] we can think of its solutions as points in k .

2 2 2 Example: shape of t1 + t2 − 1 in k : Shape of equations

n Given an equation f ∈ k[t1,..., tn] we can think of its solutions as points in k .

2 2 2 Example: shape of t1 + t2 − 1 in k : Shape of equations

n Given an equation f ∈ k[t1,..., tn] we can think of its solutions as points in k .

2 2 2 Example: shape of t1 + t2 − 1 in k : Shape of equations

2 3 2 3 2 2 Example: shape of t2 − t1 and t2 − t1 − t1 in R : Shape of equations

2 3 2 3 2 2 Example: shape of t2 − t1 and t2 − t1 − t1 in R : Let C[t1,..., tn] = polynomials in variables ti over C.

An (affine) variety is the solution set n X = {(a1,..., an) ∈ C | fj (a1,..., an) = 0, ∀j}

for some f1,..., fr ∈ C[t1,..., tn].

Everything about X is encoded in the

k[t1,..., tn]/hf1,..., fr i.

Goal: understand the local and global shape of X.

Question: how can we put such X in different groups according to their geometry?

Algebraic varieties An (affine) variety is the solution set n X = {(a1,..., an) ∈ C | fj (a1,..., an) = 0, ∀j}

for some f1,..., fr ∈ C[t1,..., tn].

Everything about X is encoded in the ring

k[t1,..., tn]/hf1,..., fr i.

Goal: understand the local and global shape of X.

Question: how can we put such X in different groups according to their geometry?

Algebraic varieties

Let C[t1,..., tn] = polynomials in variables ti over C. Everything about X is encoded in the ring

k[t1,..., tn]/hf1,..., fr i.

Goal: understand the local and global shape of X.

Question: how can we put such X in different groups according to their geometry?

Algebraic varieties

Let C[t1,..., tn] = polynomials in variables ti over C.

An (affine) variety is the solution set n X = {(a1,..., an) ∈ C | fj (a1,..., an) = 0, ∀j}

for some f1,..., fr ∈ C[t1,..., tn]. Goal: understand the local and global shape of X.

Question: how can we put such X in different groups according to their geometry?

Algebraic varieties

Let C[t1,..., tn] = polynomials in variables ti over C.

An (affine) variety is the solution set n X = {(a1,..., an) ∈ C | fj (a1,..., an) = 0, ∀j}

for some f1,..., fr ∈ C[t1,..., tn].

Everything about X is encoded in the ring

k[t1,..., tn]/hf1,..., fr i. Question: how can we put such X in different groups according to their geometry?

Algebraic varieties

Let C[t1,..., tn] = polynomials in variables ti over C.

An (affine) variety is the solution set n X = {(a1,..., an) ∈ C | fj (a1,..., an) = 0, ∀j}

for some f1,..., fr ∈ C[t1,..., tn].

Everything about X is encoded in the ring

k[t1,..., tn]/hf1,..., fr i.

Goal: understand the local and global shape of X. Algebraic varieties

Let C[t1,..., tn] = polynomials in variables ti over C.

An (affine) variety is the solution set n X = {(a1,..., an) ∈ C | fj (a1,..., an) = 0, ∀j}

for some f1,..., fr ∈ C[t1,..., tn].

Everything about X is encoded in the ring

k[t1,..., tn]/hf1,..., fr i.

Goal: understand the local and global shape of X.

Question: how can we put such X in different groups according to their geometry? Algebraic varieties

Let C[t1,..., tn] = polynomials in variables ti over C.

An (affine) variety is the solution set n X = {(a1,..., an) ∈ C | fj (a1,..., an) = 0, ∀j}

for some f1,..., fr ∈ C[t1,..., tn].

Everything about X is encoded in the ring

k[t1,..., tn]/hf1,..., fr i.

Goal: understand the local and global shape of X.

Question: how can we put such X in different groups according to their geometry? Examples: 2 3 f = t2 − t1 ∈ C[t1, t2] defines a singularity at (0, 0). 2 3 2 f = t2 − t1 − t1 ∈ C[t1, t2] also defines a singularity at (0, 0).

Singular points x ∈ X are found by solving ∂f (x) = ∂f (x) = 0. ∂t1 ∂t2

Singular points 2 3 2 f = t2 − t1 − t1 ∈ C[t1, t2] also defines a singularity at (0, 0).

Singular points x ∈ X are found by solving ∂f (x) = ∂f (x) = 0. ∂t1 ∂t2

Singular points

Examples: 2 3 f = t2 − t1 ∈ C[t1, t2] defines a singularity at (0, 0). Singular points x ∈ X are found by solving ∂f (x) = ∂f (x) = 0. ∂t1 ∂t2

Singular points

Examples: 2 3 f = t2 − t1 ∈ C[t1, t2] defines a singularity at (0, 0). 2 3 2 f = t2 − t1 − t1 ∈ C[t1, t2] also defines a singularity at (0, 0). Singular points x ∈ X are found by solving ∂f (x) = ∂f (x) = 0. ∂t1 ∂t2

Singular points

Examples: 2 3 f = t2 − t1 ∈ C[t1, t2] defines a singularity at (0, 0). 2 3 2 f = t2 − t1 − t1 ∈ C[t1, t2] also defines a singularity at (0, 0). Singular points

Examples: 2 3 f = t2 − t1 ∈ C[t1, t2] defines a singularity at (0, 0). 2 3 2 f = t2 − t1 − t1 ∈ C[t1, t2] also defines a singularity at (0, 0).

Singular points x ∈ X are found by solving ∂f (x) = ∂f (x) = 0. ∂t1 ∂t2 The n-dimensional is n P = {(a0 : ··· : an) | ai ∈ C, aj 6= 0 for some j} subject to: (a0 : ··· : an) = (ba0 : ··· : ban), ∀ b ∈ C \{0}.

Pn = k n ∪ Pn−1 is a "compactification" of k n.

A in Pn is the solution set of homogeneous polynomials

F1,..., Fr ∈ k[s0,..., sn].

Examples: 2 1 s0 − s1 ∈ C[s0, s1, s2] defines a "line" in P which is a sphere ' P . 2 3 2 2 s0s2 − s1 + s0s1 ∈ C[s0, s1, s2] defines an elliptic curve in P .

Projective varieties subject to: (a0 : ··· : an) = (ba0 : ··· : ban), ∀ b ∈ C \{0}.

Pn = k n ∪ Pn−1 is a "compactification" of k n.

A projective variety in Pn is the solution set of homogeneous polynomials

F1,..., Fr ∈ k[s0,..., sn].

Examples: 2 1 s0 − s1 ∈ C[s0, s1, s2] defines a "line" in P which is a sphere ' P . 2 3 2 2 s0s2 − s1 + s0s1 ∈ C[s0, s1, s2] defines an elliptic curve in P .

Projective varieties

The n-dimensional projective space is n P = {(a0 : ··· : an) | ai ∈ C, aj 6= 0 for some j} Pn = k n ∪ Pn−1 is a "compactification" of k n.

A projective variety in Pn is the solution set of homogeneous polynomials

F1,..., Fr ∈ k[s0,..., sn].

Examples: 2 1 s0 − s1 ∈ C[s0, s1, s2] defines a "line" in P which is a sphere ' P . 2 3 2 2 s0s2 − s1 + s0s1 ∈ C[s0, s1, s2] defines an elliptic curve in P .

Projective varieties

The n-dimensional projective space is n P = {(a0 : ··· : an) | ai ∈ C, aj 6= 0 for some j} subject to: (a0 : ··· : an) = (ba0 : ··· : ban), ∀ b ∈ C \{0}. A projective variety in Pn is the solution set of homogeneous polynomials

F1,..., Fr ∈ k[s0,..., sn].

Examples: 2 1 s0 − s1 ∈ C[s0, s1, s2] defines a "line" in P which is a sphere ' P . 2 3 2 2 s0s2 − s1 + s0s1 ∈ C[s0, s1, s2] defines an elliptic curve in P .

Projective varieties

The n-dimensional projective space is n P = {(a0 : ··· : an) | ai ∈ C, aj 6= 0 for some j} subject to: (a0 : ··· : an) = (ba0 : ··· : ban), ∀ b ∈ C \{0}.

Pn = k n ∪ Pn−1 is a "compactification" of k n. Examples: 2 1 s0 − s1 ∈ C[s0, s1, s2] defines a "line" in P which is a sphere ' P . 2 3 2 2 s0s2 − s1 + s0s1 ∈ C[s0, s1, s2] defines an elliptic curve in P .

Projective varieties

The n-dimensional projective space is n P = {(a0 : ··· : an) | ai ∈ C, aj 6= 0 for some j} subject to: (a0 : ··· : an) = (ba0 : ··· : ban), ∀ b ∈ C \{0}.

Pn = k n ∪ Pn−1 is a "compactification" of k n.

A projective variety in Pn is the solution set of homogeneous polynomials

F1,..., Fr ∈ k[s0,..., sn]. 2 3 2 2 s0s2 − s1 + s0s1 ∈ C[s0, s1, s2] defines an elliptic curve in P .

Projective varieties

The n-dimensional projective space is n P = {(a0 : ··· : an) | ai ∈ C, aj 6= 0 for some j} subject to: (a0 : ··· : an) = (ba0 : ··· : ban), ∀ b ∈ C \{0}.

Pn = k n ∪ Pn−1 is a "compactification" of k n.

A projective variety in Pn is the solution set of homogeneous polynomials

F1,..., Fr ∈ k[s0,..., sn].

Examples: 2 1 s0 − s1 ∈ C[s0, s1, s2] defines a "line" in P which is a sphere ' P . Projective varieties

The n-dimensional projective space is n P = {(a0 : ··· : an) | ai ∈ C, aj 6= 0 for some j} subject to: (a0 : ··· : an) = (ba0 : ··· : ban), ∀ b ∈ C \{0}.

Pn = k n ∪ Pn−1 is a "compactification" of k n.

A projective variety in Pn is the solution set of homogeneous polynomials

F1,..., Fr ∈ k[s0,..., sn].

Examples: 2 1 s0 − s1 ∈ C[s0, s1, s2] defines a "line" in P which is a sphere ' P . 2 3 2 2 s0s2 − s1 + s0s1 ∈ C[s0, s1, s2] defines an elliptic curve in P . Projective varieties

The n-dimensional projective space is n P = {(a0 : ··· : an) | ai ∈ C, aj 6= 0 for some j} subject to: (a0 : ··· : an) = (ba0 : ··· : ban), ∀ b ∈ C \{0}.

Pn = k n ∪ Pn−1 is a "compactification" of k n.

A projective variety in Pn is the solution set of homogeneous polynomials

F1,..., Fr ∈ k[s0,..., sn].

Examples: 2 1 s0 − s1 ∈ C[s0, s1, s2] defines a "line" in P which is a sphere ' P . 2 3 2 2 s0s2 − s1 + s0s1 ∈ C[s0, s1, s2] defines an elliptic curve in P . Example: If F ∈ C[s0, s1, s2] is homog of degree d, then its set X is a donut with g holes where 1 g = genus = (d − 1)(d − 2) 2 (assuming no singular points):

Projective varieties Projective varieties

Example: If F ∈ C[s0, s1, s2] is homog of degree d, then its set X is a donut with g holes where 1 g = genus = (d − 1)(d − 2) 2 (assuming no singular points): Projective varieties

Example: If F ∈ C[s0, s1, s2] is homog of degree d, then its set X is a donut with g holes where 1 g = genus = (d − 1)(d − 2) 2 (assuming no singular points): An important example is the blowup: it is a morphism φ: Y → Cn which collapses a copy of Pn−1 to the origin x = (0,..., 0).

φ is birational, that is, it modifies only a “small" part of the variety.

Blowups can be used to remove singularities.

Theorem (Hironaka, 1964) For each variety X, there is a (projective) birational morphism W → X where W is smooth.

W can be obtained by a sequence of blowups.

For fields of positive characteristic, resolution of singularities is a major conjecture.

Singularities

A morphism Y → X between varieties is a map defined by polynomials. φ is birational, that is, it modifies only a “small" part of the variety.

Blowups can be used to remove singularities.

Theorem (Hironaka, 1964) For each variety X, there is a (projective) birational morphism W → X where W is smooth.

W can be obtained by a sequence of blowups.

For fields of positive characteristic, resolution of singularities is a major conjecture.

Singularities

A morphism Y → X between varieties is a map defined by polynomials.

An important example is the blowup: it is a morphism φ: Y → Cn which collapses a copy of Pn−1 to the origin x = (0,..., 0). that is, it modifies only a “small" part of the variety.

Blowups can be used to remove singularities.

Theorem (Hironaka, 1964) For each variety X, there is a (projective) birational morphism W → X where W is smooth.

W can be obtained by a sequence of blowups.

For fields of positive characteristic, resolution of singularities is a major conjecture.

Singularities

A morphism Y → X between varieties is a map defined by polynomials.

An important example is the blowup: it is a morphism φ: Y → Cn which collapses a copy of Pn−1 to the origin x = (0,..., 0).

φ is birational, Blowups can be used to remove singularities.

Theorem (Hironaka, 1964) For each variety X, there is a (projective) birational morphism W → X where W is smooth.

W can be obtained by a sequence of blowups.

For fields of positive characteristic, resolution of singularities is a major conjecture.

Singularities

A morphism Y → X between varieties is a map defined by polynomials.

An important example is the blowup: it is a morphism φ: Y → Cn which collapses a copy of Pn−1 to the origin x = (0,..., 0).

φ is birational, that is, it modifies only a “small" part of the variety. Theorem (Hironaka, 1964) For each variety X, there is a (projective) birational morphism W → X where W is smooth.

W can be obtained by a sequence of blowups.

For fields of positive characteristic, resolution of singularities is a major conjecture.

Singularities

A morphism Y → X between varieties is a map defined by polynomials.

An important example is the blowup: it is a morphism φ: Y → Cn which collapses a copy of Pn−1 to the origin x = (0,..., 0).

φ is birational, that is, it modifies only a “small" part of the variety.

Blowups can be used to remove singularities. W can be obtained by a sequence of blowups.

For fields of positive characteristic, resolution of singularities is a major conjecture.

Singularities

A morphism Y → X between varieties is a map defined by polynomials.

An important example is the blowup: it is a morphism φ: Y → Cn which collapses a copy of Pn−1 to the origin x = (0,..., 0).

φ is birational, that is, it modifies only a “small" part of the variety.

Blowups can be used to remove singularities.

Theorem (Hironaka, 1964) For each variety X, there is a (projective) birational morphism W → X where W is smooth. For fields of positive characteristic, resolution of singularities is a major conjecture.

Singularities

A morphism Y → X between varieties is a map defined by polynomials.

An important example is the blowup: it is a morphism φ: Y → Cn which collapses a copy of Pn−1 to the origin x = (0,..., 0).

φ is birational, that is, it modifies only a “small" part of the variety.

Blowups can be used to remove singularities.

Theorem (Hironaka, 1964) For each variety X, there is a (projective) birational morphism W → X where W is smooth.

W can be obtained by a sequence of blowups. Singularities

A morphism Y → X between varieties is a map defined by polynomials.

An important example is the blowup: it is a morphism φ: Y → Cn which collapses a copy of Pn−1 to the origin x = (0,..., 0).

φ is birational, that is, it modifies only a “small" part of the variety.

Blowups can be used to remove singularities.

Theorem (Hironaka, 1964) For each variety X, there is a (projective) birational morphism W → X where W is smooth.

W can be obtained by a sequence of blowups.

For fields of positive characteristic, resolution of singularities is a major conjecture. The blowup of C2 at x induces a morphism W → X which resolves the singularity.

Singularities

2 2 3 2 Example: recall X ⊂ C defined by t2 − t1 − t1 is singular at x = (0, 0). Singularities

2 2 3 2 Example: recall X ⊂ C defined by t2 − t1 − t1 is singular at x = (0, 0).

The blowup of C2 at x induces a morphism W → X which resolves the singularity. Singularities

2 2 3 2 Example: recall X ⊂ C defined by t2 − t1 − t1 is singular at x = (0, 0).

The blowup of C2 at x induces a morphism W → X which resolves the singularity. The blowup of C3 at x induces a morphism ψ : W → X which resolves the singularity.

E := ψ−1{x} = P1.

E 2 = degree of normal bundle = −2.

Singularities

3 2 2 2 Example: X ⊂ C defined by t1 + t2 + t3 = 0 is singular at x = (0, 0, 0). E := ψ−1{x} = P1.

E 2 = degree of normal bundle = −2.

Singularities

3 2 2 2 Example: X ⊂ C defined by t1 + t2 + t3 = 0 is singular at x = (0, 0, 0).

The blowup of C3 at x induces a morphism ψ : W → X which resolves the singularity. E := ψ−1{x} = P1.

E 2 = degree of normal bundle = −2.

Singularities

3 2 2 2 Example: X ⊂ C defined by t1 + t2 + t3 = 0 is singular at x = (0, 0, 0).

The blowup of C3 at x induces a morphism ψ : W → X which resolves the singularity. E 2 = degree of normal bundle = −2.

Singularities

3 2 2 2 Example: X ⊂ C defined by t1 + t2 + t3 = 0 is singular at x = (0, 0, 0).

The blowup of C3 at x induces a morphism ψ : W → X which resolves the singularity.

E := ψ−1{x} = P1. Singularities

3 2 2 2 Example: X ⊂ C defined by t1 + t2 + t3 = 0 is singular at x = (0, 0, 0).

The blowup of C3 at x induces a morphism ψ : W → X which resolves the singularity.

E := ψ−1{x} = P1.

E 2 = degree of normal bundle = −2. is singular at x = (0, 0, 0).

There is a resolution of singularities ψ : W → X.

We can choose ψ such that −1 S 1 ψ {x} = Ei is a chain of n − 1 copies of P , 2 Ei = −2.

In some sense ψ is a “minimal" resolution.

Singularities

3 2 2 n Example: X ⊂ C defined by t1 + t2 + t3 = 0 for n ≥ 2 There is a resolution of singularities ψ : W → X.

We can choose ψ such that −1 S 1 ψ {x} = Ei is a chain of n − 1 copies of P , 2 Ei = −2.

In some sense ψ is a “minimal" resolution.

Singularities

3 2 2 n Example: X ⊂ C defined by t1 + t2 + t3 = 0 for n ≥ 2 is singular at x = (0, 0, 0). We can choose ψ such that −1 S 1 ψ {x} = Ei is a chain of n − 1 copies of P , 2 Ei = −2.

In some sense ψ is a “minimal" resolution.

Singularities

3 2 2 n Example: X ⊂ C defined by t1 + t2 + t3 = 0 for n ≥ 2 is singular at x = (0, 0, 0). There is a resolution of singularities ψ : W → X. 2 Ei = −2.

In some sense ψ is a “minimal" resolution.

Singularities

3 2 2 n Example: X ⊂ C defined by t1 + t2 + t3 = 0 for n ≥ 2 is singular at x = (0, 0, 0). There is a resolution of singularities ψ : W → X.

We can choose ψ such that −1 S 1 ψ {x} = Ei is a chain of n − 1 copies of P , In some sense ψ is a “minimal" resolution.

Singularities

3 2 2 n Example: X ⊂ C defined by t1 + t2 + t3 = 0 for n ≥ 2 is singular at x = (0, 0, 0). There is a resolution of singularities ψ : W → X.

We can choose ψ such that −1 S 1 ψ {x} = Ei is a chain of n − 1 copies of P , 2 Ei = −2. In some sense ψ is a “minimal" resolution.

Singularities

3 2 2 n Example: X ⊂ C defined by t1 + t2 + t3 = 0 for n ≥ 2 is singular at x = (0, 0, 0). There is a resolution of singularities ψ : W → X.

We can choose ψ such that −1 S 1 ψ {x} = Ei is a chain of n − 1 copies of P , 2 Ei = −2. Singularities

3 2 2 n Example: X ⊂ C defined by t1 + t2 + t3 = 0 for n ≥ 2 is singular at x = (0, 0, 0). There is a resolution of singularities ψ : W → X.

We can choose ψ such that −1 S 1 ψ {x} = Ei is a chain of n − 1 copies of P , 2 Ei = −2.

In some sense ψ is a “minimal" resolution. 4 Example: X ⊂ C defined by t1t2 − t3t4 = 0 is singular at x = (0, 0, 0, 0).

Blowing up x ∈ C4 induces a resolution of singularities ψ : W → X.

−1 3 1 1 E := ψ {x} is defined by t1t2 − t3t4 = 0 as a subset of P . Thus E ' P × P .

1 The two projections E → P induce birational morphisms W → Y1 and W → Y2.

Singularities x ∈ C4 induces a resolution of singularities ψ : W → X.

−1 3 1 1 E := ψ {x} is defined by t1t2 − t3t4 = 0 as a subset of P . Thus E ' P × P .

1 The two projections E → P induce birational morphisms W → Y1 and W → Y2.

Singularities

4 Example: X ⊂ C defined by t1t2 − t3t4 = 0 is singular at x = (0, 0, 0, 0). −1 3 1 1 E := ψ {x} is defined by t1t2 − t3t4 = 0 as a subset of P . Thus E ' P × P .

1 The two projections E → P induce birational morphisms W → Y1 and W → Y2.

Singularities

4 Example: X ⊂ C defined by t1t2 − t3t4 = 0 is singular at x = (0, 0, 0, 0).

Blowing up x ∈ C4 induces a resolution of singularities ψ : W → X. Thus E ' P1 × P1.

1 The two projections E → P induce birational morphisms W → Y1 and W → Y2.

Singularities

4 Example: X ⊂ C defined by t1t2 − t3t4 = 0 is singular at x = (0, 0, 0, 0).

Blowing up x ∈ C4 induces a resolution of singularities ψ : W → X.

−1 3 E := ψ {x} is defined by t1t2 − t3t4 = 0 as a subset of P . 1 The two projections E → P induce birational morphisms W → Y1 and W → Y2.

Singularities

4 Example: X ⊂ C defined by t1t2 − t3t4 = 0 is singular at x = (0, 0, 0, 0).

Blowing up x ∈ C4 induces a resolution of singularities ψ : W → X.

−1 3 1 1 E := ψ {x} is defined by t1t2 − t3t4 = 0 as a subset of P . Thus E ' P × P . Singularities

4 Example: X ⊂ C defined by t1t2 − t3t4 = 0 is singular at x = (0, 0, 0, 0).

Blowing up x ∈ C4 induces a resolution of singularities ψ : W → X.

−1 3 1 1 E := ψ {x} is defined by t1t2 − t3t4 = 0 as a subset of P . Thus E ' P × P .

1 The two projections E → P induce birational morphisms W → Y1 and W → Y2. Singularities

4 Example: X ⊂ C defined by t1t2 − t3t4 = 0 is singular at x = (0, 0, 0, 0).

Blowing up x ∈ C4 induces a resolution of singularities ψ : W → X.

−1 3 1 1 E := ψ {x} is defined by t1t2 − t3t4 = 0 as a subset of P . Thus E ' P × P .

1 The two projections E → P induce birational morphisms W → Y1 and W → Y2. Both have the right to claim to be a minimal resolution.

The diagram Y1 99K Y2

! } X is the Atiyah flop, the simplest example of a flop.

Flops and flips are very important in algebraic geometry. They are needed to run the so-called in higher dimension.

Singularities

Both Y1 and Y2 are smooth, so both Y1 → X and Y2 → X are resolutions. The diagram Y1 99K Y2

! } X is the Atiyah flop, the simplest example of a flop.

Flops and flips are very important in algebraic geometry. They are needed to run the so-called minimal model program in higher dimension.

Singularities

Both Y1 and Y2 are smooth, so both Y1 → X and Y2 → X are resolutions. Both have the right to claim to be a minimal resolution. Flops and flips are very important in algebraic geometry. They are needed to run the so-called minimal model program in higher dimension.

Singularities

Both Y1 and Y2 are smooth, so both Y1 → X and Y2 → X are resolutions. Both have the right to claim to be a minimal resolution.

The diagram Y1 99K Y2

! } X is the Atiyah flop, the simplest example of a flop. They are needed to run the so-called minimal model program in higher dimension.

Singularities

Both Y1 and Y2 are smooth, so both Y1 → X and Y2 → X are resolutions. Both have the right to claim to be a minimal resolution.

The diagram Y1 99K Y2

! } X is the Atiyah flop, the simplest example of a flop.

Flops and flips are very important in algebraic geometry. Singularities

Both Y1 and Y2 are smooth, so both Y1 → X and Y2 → X are resolutions. Both have the right to claim to be a minimal resolution.

The diagram Y1 99K Y2

! } X is the Atiyah flop, the simplest example of a flop.

Flops and flips are very important in algebraic geometry. They are needed to run the so-called minimal model program in higher dimension. On each non-singular (or mildy singular) variety X we can define "curvature".

We say X is   Fano if the curvature is everywhere positive, Calabi-Yau if the curvature is everywhere zero,  canonically polarised if the curvature is everywhere negative.

These are of great importance in algebraic/differential/arithmetic geometry, mirror symmetry, mathematical physics.

Example: X ⊂ Pn defined by one polynomial of degree r is   Fano if r < n + 1, Calabi-Yau if r = n + 1,  canonically polarised if r > n + 1.

The minimal model program transforms each variety birationally into one which is canonically polarised or admits a Fano or Calabi-Yau fibration.

Global geometry We say X is   Fano if the curvature is everywhere positive, Calabi-Yau if the curvature is everywhere zero,  canonically polarised if the curvature is everywhere negative.

These are of great importance in algebraic/differential/arithmetic geometry, mirror symmetry, mathematical physics.

Example: X ⊂ Pn defined by one polynomial of degree r is   Fano if r < n + 1, Calabi-Yau if r = n + 1,  canonically polarised if r > n + 1.

The minimal model program transforms each variety birationally into one which is canonically polarised or admits a Fano or Calabi-Yau fibration.

Global geometry

On each non-singular (or mildy singular) variety X we can define "curvature". These are of great importance in algebraic/differential/arithmetic geometry, mirror symmetry, mathematical physics.

Example: X ⊂ Pn defined by one polynomial of degree r is   Fano if r < n + 1, Calabi-Yau if r = n + 1,  canonically polarised if r > n + 1.

The minimal model program transforms each variety birationally into one which is canonically polarised or admits a Fano or Calabi-Yau fibration.

Global geometry

On each non-singular (or mildy singular) variety X we can define "curvature".

We say X is   Fano if the curvature is everywhere positive, Calabi-Yau if the curvature is everywhere zero,  canonically polarised if the curvature is everywhere negative. Example: X ⊂ Pn defined by one polynomial of degree r is   Fano if r < n + 1, Calabi-Yau if r = n + 1,  canonically polarised if r > n + 1.

The minimal model program transforms each variety birationally into one which is canonically polarised or admits a Fano or Calabi-Yau fibration.

Global geometry

On each non-singular (or mildy singular) variety X we can define "curvature".

We say X is   Fano if the curvature is everywhere positive, Calabi-Yau if the curvature is everywhere zero,  canonically polarised if the curvature is everywhere negative.

These are of great importance in algebraic/differential/arithmetic geometry, mirror symmetry, mathematical physics. The minimal model program transforms each variety birationally into one which is canonically polarised or admits a Fano or Calabi-Yau fibration.

Global geometry

On each non-singular (or mildy singular) variety X we can define "curvature".

We say X is   Fano if the curvature is everywhere positive, Calabi-Yau if the curvature is everywhere zero,  canonically polarised if the curvature is everywhere negative.

These are of great importance in algebraic/differential/arithmetic geometry, mirror symmetry, mathematical physics.

Example: X ⊂ Pn defined by one polynomial of degree r is   Fano if r < n + 1, Calabi-Yau if r = n + 1,  canonically polarised if r > n + 1. Global geometry

On each non-singular (or mildy singular) variety X we can define "curvature".

We say X is   Fano if the curvature is everywhere positive, Calabi-Yau if the curvature is everywhere zero,  canonically polarised if the curvature is everywhere negative.

These are of great importance in algebraic/differential/arithmetic geometry, mirror symmetry, mathematical physics.

Example: X ⊂ Pn defined by one polynomial of degree r is   Fano if r < n + 1, Calabi-Yau if r = n + 1,  canonically polarised if r > n + 1.

The minimal model program transforms each variety birationally into one which is canonically polarised or admits a Fano or Calabi-Yau fibration. Li and Lj intersect at one point for i 6= j.

Blowup an intersection point. We will have the Li and a new curve E. Blowup an intersection point of E and some Li , and continue.

We get W → P2. Collapsing some of the curves gives W → X.

Often X is a .

Fano’s

2 Example: Let L1, L2, L3 ⊂ P be defined by s0, s1, s2 respectively. Blowup an intersection point. We will have the Li and a new curve E. Blowup an intersection point of E and some Li , and continue.

We get W → P2. Collapsing some of the curves gives W → X.

Often X is a Fano variety.

Fano’s

2 Example: Let L1, L2, L3 ⊂ P be defined by s0, s1, s2 respectively. Li and Lj intersect at one point for i 6= j. We get W → P2. Collapsing some of the curves gives W → X.

Often X is a Fano variety.

Fano’s

2 Example: Let L1, L2, L3 ⊂ P be defined by s0, s1, s2 respectively. Li and Lj intersect at one point for i 6= j.

Blowup an intersection point. We will have the Li and a new curve E. Blowup an intersection point of E and some Li , and continue. We get W → P2. Collapsing some of the curves gives W → X.

Often X is a Fano variety.

Fano’s

2 Example: Let L1, L2, L3 ⊂ P be defined by s0, s1, s2 respectively. Li and Lj intersect at one point for i 6= j.

Blowup an intersection point. We will have the Li and a new curve E. Blowup an intersection point of E and some Li , and continue. Collapsing some of the curves gives W → X.

Often X is a Fano variety.

Fano’s

2 Example: Let L1, L2, L3 ⊂ P be defined by s0, s1, s2 respectively. Li and Lj intersect at one point for i 6= j.

Blowup an intersection point. We will have the Li and a new curve E. Blowup an intersection point of E and some Li , and continue.

We get W → P2. Often X is a Fano variety.

Fano’s

2 Example: Let L1, L2, L3 ⊂ P be defined by s0, s1, s2 respectively. Li and Lj intersect at one point for i 6= j.

Blowup an intersection point. We will have the Li and a new curve E. Blowup an intersection point of E and some Li , and continue.

We get W → P2. Collapsing some of the curves gives W → X. Fano’s

2 Example: Let L1, L2, L3 ⊂ P be defined by s0, s1, s2 respectively. Li and Lj intersect at one point for i 6= j.

Blowup an intersection point. We will have the Li and a new curve E. Blowup an intersection point of E and some Li , and continue.

We get W → P2. Collapsing some of the curves gives W → X.

Often X is a Fano variety. Fano’s

2 Example: Let L1, L2, L3 ⊂ P be defined by s0, s1, s2 respectively. Li and Lj intersect at one point for i 6= j.

Blowup an intersection point. We will have the Li and a new curve E. Blowup an intersection point of E and some Li , and continue.

We get W → P2. Collapsing some of the curves gives W → X.

Often X is a Fano variety. Because their singularities get deeper and deeper.

Theorem (B, 2016)

For each d ∈ N the set {X | X is Fano of dimension d with "good" singularities}

is a bounded family.

All such varieties can then be embedded in some Pn with equations of bounded degree.

Fano’s

This simple recipe produces too many Fano’s that they cannot be parametrised by finitely many parameters. Theorem (B, 2016)

For each d ∈ N the set {X | X is Fano of dimension d with "good" singularities}

is a bounded family.

All such varieties can then be embedded in some Pn with equations of bounded degree.

Fano’s

This simple recipe produces too many Fano’s that they cannot be parametrised by finitely many parameters.

Because their singularities get deeper and deeper. All such varieties can then be embedded in some Pn with equations of bounded degree.

Fano’s

This simple recipe produces too many Fano’s that they cannot be parametrised by finitely many parameters.

Because their singularities get deeper and deeper.

Theorem (B, 2016)

For each d ∈ N the set {X | X is Fano of dimension d with "good" singularities}

is a bounded family. Fano’s

This simple recipe produces too many Fano’s that they cannot be parametrised by finitely many parameters.

Because their singularities get deeper and deeper.

Theorem (B, 2016)

For each d ∈ N the set {X | X is Fano of dimension d with "good" singularities}

is a bounded family.

All such varieties can then be embedded in some Pn with equations of bounded degree. Fano’s

This simple recipe produces too many Fano’s that they cannot be parametrised by finitely many parameters.

Because their singularities get deeper and deeper.

Theorem (B, 2016)

For each d ∈ N the set {X | X is Fano of dimension d with "good" singularities}

is a bounded family.

All such varieties can then be embedded in some Pn with equations of bounded degree. Such Y are Calabi-Yau varieties.

Other examples include abelian varieties.

Theorem (B-Cascini-Hacon-McKernan, Kawamata, 2006)

0 Assume Y 99K Y is a birational map between Calabi-Yau varieties. 0 Then Y 99K Y can be decomposed into a finite sequence of flops:

0 Y = Y1 99K Y2 99K ··· 99K Yr−1 99K Yr = Y .

Calabi-Yau’s

If X is a Fano variety, then often one finds a nice subvariety Y ∼ −KX . Other examples include abelian varieties.

Theorem (B-Cascini-Hacon-McKernan, Kawamata, 2006)

0 Assume Y 99K Y is a birational map between Calabi-Yau varieties. 0 Then Y 99K Y can be decomposed into a finite sequence of flops:

0 Y = Y1 99K Y2 99K ··· 99K Yr−1 99K Yr = Y .

Calabi-Yau’s

If X is a Fano variety, then often one finds a nice subvariety Y ∼ −KX .

Such Y are Calabi-Yau varieties. Theorem (B-Cascini-Hacon-McKernan, Kawamata, 2006)

0 Assume Y 99K Y is a birational map between Calabi-Yau varieties. 0 Then Y 99K Y can be decomposed into a finite sequence of flops:

0 Y = Y1 99K Y2 99K ··· 99K Yr−1 99K Yr = Y .

Calabi-Yau’s

If X is a Fano variety, then often one finds a nice subvariety Y ∼ −KX .

Such Y are Calabi-Yau varieties.

Other examples include abelian varieties. 0 Then Y 99K Y can be decomposed into a finite sequence of flops:

0 Y = Y1 99K Y2 99K ··· 99K Yr−1 99K Yr = Y .

Calabi-Yau’s

If X is a Fano variety, then often one finds a nice subvariety Y ∼ −KX .

Such Y are Calabi-Yau varieties.

Other examples include abelian varieties.

Theorem (B-Cascini-Hacon-McKernan, Kawamata, 2006)

0 Assume Y 99K Y is a birational map between Calabi-Yau varieties. Calabi-Yau’s

If X is a Fano variety, then often one finds a nice subvariety Y ∼ −KX .

Such Y are Calabi-Yau varieties.

Other examples include abelian varieties.

Theorem (B-Cascini-Hacon-McKernan, Kawamata, 2006)

0 Assume Y 99K Y is a birational map between Calabi-Yau varieties. 0 Then Y 99K Y can be decomposed into a finite sequence of flops:

0 Y = Y1 99K Y2 99K ··· 99K Yr−1 99K Yr = Y . Calabi-Yau’s

If X is a Fano variety, then often one finds a nice subvariety Y ∼ −KX .

Such Y are Calabi-Yau varieties.

Other examples include abelian varieties.

Theorem (B-Cascini-Hacon-McKernan, Kawamata, 2006)

0 Assume Y 99K Y is a birational map between Calabi-Yau varieties. 0 Then Y 99K Y can be decomposed into a finite sequence of flops:

0 Y = Y1 99K Y2 99K ··· 99K Yr−1 99K Yr = Y . Each smooth projective variety X has a canonical divisor denoted KX .

KX is the divisor (zeros and poles) of a top degree rational differential form.

d The sheaf OX (KX ) associated to KX is the canonical sheaf ωX . In fact, ωX = ∧ ΩX .

n When X = P we have KX = (−n − 1)H where H is a hypersurface of degree 1.

n When X ⊂ P is a hypersurface of degree r, then KX = (r − n − 1)H ∩ X.

Birational classification: canonical divisor KX is the divisor (zeros and poles) of a top degree rational differential form.

d The sheaf OX (KX ) associated to KX is the canonical sheaf ωX . In fact, ωX = ∧ ΩX .

n When X = P we have KX = (−n − 1)H where H is a hypersurface of degree 1.

n When X ⊂ P is a hypersurface of degree r, then KX = (r − n − 1)H ∩ X.

Birational classification: canonical divisor

Each smooth projective variety X has a canonical divisor denoted KX . d The sheaf OX (KX ) associated to KX is the canonical sheaf ωX . In fact, ωX = ∧ ΩX .

n When X = P we have KX = (−n − 1)H where H is a hypersurface of degree 1.

n When X ⊂ P is a hypersurface of degree r, then KX = (r − n − 1)H ∩ X.

Birational classification: canonical divisor

Each smooth projective variety X has a canonical divisor denoted KX .

KX is the divisor (zeros and poles) of a top degree rational differential form. n When X = P we have KX = (−n − 1)H where H is a hypersurface of degree 1.

n When X ⊂ P is a hypersurface of degree r, then KX = (r − n − 1)H ∩ X.

Birational classification: canonical divisor

Each smooth projective variety X has a canonical divisor denoted KX .

KX is the divisor (zeros and poles) of a top degree rational differential form.

d The sheaf OX (KX ) associated to KX is the canonical sheaf ωX . In fact, ωX = ∧ ΩX . n When X ⊂ P is a hypersurface of degree r, then KX = (r − n − 1)H ∩ X.

Birational classification: canonical divisor

Each smooth projective variety X has a canonical divisor denoted KX .

KX is the divisor (zeros and poles) of a top degree rational differential form.

d The sheaf OX (KX ) associated to KX is the canonical sheaf ωX . In fact, ωX = ∧ ΩX .

n When X = P we have KX = (−n − 1)H where H is a hypersurface of degree 1. Birational classification: canonical divisor

Each smooth projective variety X has a canonical divisor denoted KX .

KX is the divisor (zeros and poles) of a top degree rational differential form.

d The sheaf OX (KX ) associated to KX is the canonical sheaf ωX . In fact, ωX = ∧ ΩX .

n When X = P we have KX = (−n − 1)H where H is a hypersurface of degree 1.

n When X ⊂ P is a hypersurface of degree r, then KX = (r − n − 1)H ∩ X. Birational classification: canonical divisor

Each smooth projective variety X has a canonical divisor denoted KX .

KX is the divisor (zeros and poles) of a top degree rational differential form.

d The sheaf OX (KX ) associated to KX is the canonical sheaf ωX . In fact, ωX = ∧ ΩX .

n When X = P we have KX = (−n − 1)H where H is a hypersurface of degree 1.

n When X ⊂ P is a hypersurface of degree r, then KX = (r − n − 1)H ∩ X. One of the central goals of birational geometry is to establish:

Conjecture (Minimal model and abundance) Each variety X is birational to a projective variety Y with good singularities such that either • Y admits a Fano fibration, or • Y admits a Calabi-Yau fibration, or • Y is canonically polarised.

The conjecture is known in dimension ≤ 3, and lots of cases in higher dimension.

Question: How to find Y ?

Birational classification: main conjecture Conjecture (Minimal model and abundance) Each variety X is birational to a projective variety Y with good singularities such that either • Y admits a Fano fibration, or • Y admits a Calabi-Yau fibration, or • Y is canonically polarised.

The conjecture is known in dimension ≤ 3, and lots of cases in higher dimension.

Question: How to find Y ?

Birational classification: main conjecture

One of the central goals of birational geometry is to establish: The conjecture is known in dimension ≤ 3, and lots of cases in higher dimension.

Question: How to find Y ?

Birational classification: main conjecture

One of the central goals of birational geometry is to establish:

Conjecture (Minimal model and abundance) Each variety X is birational to a projective variety Y with good singularities such that either • Y admits a Fano fibration, or • Y admits a Calabi-Yau fibration, or • Y is canonically polarised. Question: How to find Y ?

Birational classification: main conjecture

One of the central goals of birational geometry is to establish:

Conjecture (Minimal model and abundance) Each variety X is birational to a projective variety Y with good singularities such that either • Y admits a Fano fibration, or • Y admits a Calabi-Yau fibration, or • Y is canonically polarised.

The conjecture is known in dimension ≤ 3, and lots of cases in higher dimension. Birational classification: main conjecture

One of the central goals of birational geometry is to establish:

Conjecture (Minimal model and abundance) Each variety X is birational to a projective variety Y with good singularities such that either • Y admits a Fano fibration, or • Y admits a Calabi-Yau fibration, or • Y is canonically polarised.

The conjecture is known in dimension ≤ 3, and lots of cases in higher dimension.

Question: How to find Y ? Important ingredient:

Theorem (B-Cascini-Hacon-McKernan, Kawamata, 2006) The canonical ring M 0 R = H (mKX ) m≥0 is a finitely generated C-algebra.

When Y is of general type, Y = Proj R.

Otherwise we need the full force of MMP to find Y .

Birational classification: MMP

Run the MMP on X giving a sequence of birational transformations

div contraction flip X = X1 99K X2 99K X3 99K ··· 99K Xt = Y Theorem (B-Cascini-Hacon-McKernan, Kawamata, 2006) The canonical ring M 0 R = H (mKX ) m≥0 is a finitely generated C-algebra.

When Y is of general type, Y = Proj R.

Otherwise we need the full force of MMP to find Y .

Birational classification: MMP

Run the MMP on X giving a sequence of birational transformations

div contraction flip X = X1 99K X2 99K X3 99K ··· 99K Xt = Y

Important ingredient: When Y is of general type, Y = Proj R.

Otherwise we need the full force of MMP to find Y .

Birational classification: MMP

Run the MMP on X giving a sequence of birational transformations

div contraction flip X = X1 99K X2 99K X3 99K ··· 99K Xt = Y

Important ingredient:

Theorem (B-Cascini-Hacon-McKernan, Kawamata, 2006) The canonical ring M 0 R = H (mKX ) m≥0 is a finitely generated C-algebra. Otherwise we need the full force of MMP to find Y .

Birational classification: MMP

Run the MMP on X giving a sequence of birational transformations

div contraction flip X = X1 99K X2 99K X3 99K ··· 99K Xt = Y

Important ingredient:

Theorem (B-Cascini-Hacon-McKernan, Kawamata, 2006) The canonical ring M 0 R = H (mKX ) m≥0 is a finitely generated C-algebra.

When Y is of general type, Y = Proj R. Birational classification: MMP

Run the MMP on X giving a sequence of birational transformations

div contraction flip X = X1 99K X2 99K X3 99K ··· 99K Xt = Y

Important ingredient:

Theorem (B-Cascini-Hacon-McKernan, Kawamata, 2006) The canonical ring M 0 R = H (mKX ) m≥0 is a finitely generated C-algebra.

When Y is of general type, Y = Proj R.

Otherwise we need the full force of MMP to find Y . If KX is semi-positive, we stop.

Otherwise there is a curve E ⊂ X such that E ' P1 and E 2 = −1.

Castelnuovo: we can contract E via a morphism X := X1 → X2.

Repeating the process we get a sequence of birational transformations

X = X1 → X2 → X3 → · · · → Xt = Y ending with the anticipated Y .

Even in this case it is hard to show Y satisfies the properties listed in the conjecture.

Birational classification: MMP - dim 2

Lets see how MMP works when X is smooth of dimension 2. Otherwise there is a curve E ⊂ X such that E ' P1 and E 2 = −1.

Castelnuovo: we can contract E via a morphism X := X1 → X2.

Repeating the process we get a sequence of birational transformations

X = X1 → X2 → X3 → · · · → Xt = Y ending with the anticipated Y .

Even in this case it is hard to show Y satisfies the properties listed in the conjecture.

Birational classification: MMP - dim 2

Lets see how MMP works when X is smooth of dimension 2.

If KX is semi-positive, we stop. Castelnuovo: we can contract E via a morphism X := X1 → X2.

Repeating the process we get a sequence of birational transformations

X = X1 → X2 → X3 → · · · → Xt = Y ending with the anticipated Y .

Even in this case it is hard to show Y satisfies the properties listed in the conjecture.

Birational classification: MMP - dim 2

Lets see how MMP works when X is smooth of dimension 2.

If KX is semi-positive, we stop.

Otherwise there is a curve E ⊂ X such that E ' P1 and E 2 = −1. Repeating the process we get a sequence of birational transformations

X = X1 → X2 → X3 → · · · → Xt = Y ending with the anticipated Y .

Even in this case it is hard to show Y satisfies the properties listed in the conjecture.

Birational classification: MMP - dim 2

Lets see how MMP works when X is smooth of dimension 2.

If KX is semi-positive, we stop.

Otherwise there is a curve E ⊂ X such that E ' P1 and E 2 = −1.

Castelnuovo: we can contract E via a morphism X := X1 → X2. Even in this case it is hard to show Y satisfies the properties listed in the conjecture.

Birational classification: MMP - dim 2

Lets see how MMP works when X is smooth of dimension 2.

If KX is semi-positive, we stop.

Otherwise there is a curve E ⊂ X such that E ' P1 and E 2 = −1.

Castelnuovo: we can contract E via a morphism X := X1 → X2.

Repeating the process we get a sequence of birational transformations

X = X1 → X2 → X3 → · · · → Xt = Y ending with the anticipated Y . Birational classification: MMP - dim 2

Lets see how MMP works when X is smooth of dimension 2.

If KX is semi-positive, we stop.

Otherwise there is a curve E ⊂ X such that E ' P1 and E 2 = −1.

Castelnuovo: we can contract E via a morphism X := X1 → X2.

Repeating the process we get a sequence of birational transformations

X = X1 → X2 → X3 → · · · → Xt = Y ending with the anticipated Y .

Even in this case it is hard to show Y satisfies the properties listed in the conjecture. n n n Let Bir(P ) be the set of all birational maps P 99K P . Composition of maps gives it a group structure.

2 2 Example: φ: P 99K P defined by

(a0 : a1 : a2) 7→ (a1a2 : a0a2 : a0a1) is a birational map.

n Bir(P ) contains PGLn+1(C) = set of (n + 1) × (n + 1) invertible matrices.

n n Elements of PGLn+1(C) give isomorphisms P → P .

1 Bir(P ) = PGL2(C).

2 Bir(P ) is generated by PGL3(C) and φ defined above.

Group theory Composition of maps gives it a group structure.

2 2 Example: φ: P 99K P defined by

(a0 : a1 : a2) 7→ (a1a2 : a0a2 : a0a1) is a birational map.

n Bir(P ) contains PGLn+1(C) = set of (n + 1) × (n + 1) invertible matrices.

n n Elements of PGLn+1(C) give isomorphisms P → P .

1 Bir(P ) = PGL2(C).

2 Bir(P ) is generated by PGL3(C) and φ defined above.

Group theory

n n n Let Bir(P ) be the set of all birational maps P 99K P . 2 2 Example: φ: P 99K P defined by

(a0 : a1 : a2) 7→ (a1a2 : a0a2 : a0a1) is a birational map.

n Bir(P ) contains PGLn+1(C) = set of (n + 1) × (n + 1) invertible matrices.

n n Elements of PGLn+1(C) give isomorphisms P → P .

1 Bir(P ) = PGL2(C).

2 Bir(P ) is generated by PGL3(C) and φ defined above.

Group theory

n n n Let Bir(P ) be the set of all birational maps P 99K P . Composition of maps gives it a group structure. n Bir(P ) contains PGLn+1(C) = set of (n + 1) × (n + 1) invertible matrices.

n n Elements of PGLn+1(C) give isomorphisms P → P .

1 Bir(P ) = PGL2(C).

2 Bir(P ) is generated by PGL3(C) and φ defined above.

Group theory

n n n Let Bir(P ) be the set of all birational maps P 99K P . Composition of maps gives it a group structure.

2 2 Example: φ: P 99K P defined by

(a0 : a1 : a2) 7→ (a1a2 : a0a2 : a0a1) is a birational map. n n Elements of PGLn+1(C) give isomorphisms P → P .

1 Bir(P ) = PGL2(C).

2 Bir(P ) is generated by PGL3(C) and φ defined above.

Group theory

n n n Let Bir(P ) be the set of all birational maps P 99K P . Composition of maps gives it a group structure.

2 2 Example: φ: P 99K P defined by

(a0 : a1 : a2) 7→ (a1a2 : a0a2 : a0a1) is a birational map.

n Bir(P ) contains PGLn+1(C) = set of (n + 1) × (n + 1) invertible matrices. 1 Bir(P ) = PGL2(C).

2 Bir(P ) is generated by PGL3(C) and φ defined above.

Group theory

n n n Let Bir(P ) be the set of all birational maps P 99K P . Composition of maps gives it a group structure.

2 2 Example: φ: P 99K P defined by

(a0 : a1 : a2) 7→ (a1a2 : a0a2 : a0a1) is a birational map.

n Bir(P ) contains PGLn+1(C) = set of (n + 1) × (n + 1) invertible matrices.

n n Elements of PGLn+1(C) give isomorphisms P → P . 2 Bir(P ) is generated by PGL3(C) and φ defined above.

Group theory

n n n Let Bir(P ) be the set of all birational maps P 99K P . Composition of maps gives it a group structure.

2 2 Example: φ: P 99K P defined by

(a0 : a1 : a2) 7→ (a1a2 : a0a2 : a0a1) is a birational map.

n Bir(P ) contains PGLn+1(C) = set of (n + 1) × (n + 1) invertible matrices.

n n Elements of PGLn+1(C) give isomorphisms P → P .

1 Bir(P ) = PGL2(C). Group theory

n n n Let Bir(P ) be the set of all birational maps P 99K P . Composition of maps gives it a group structure.

2 2 Example: φ: P 99K P defined by

(a0 : a1 : a2) 7→ (a1a2 : a0a2 : a0a1) is a birational map.

n Bir(P ) contains PGLn+1(C) = set of (n + 1) × (n + 1) invertible matrices.

n n Elements of PGLn+1(C) give isomorphisms P → P .

1 Bir(P ) = PGL2(C).

2 Bir(P ) is generated by PGL3(C) and φ defined above. Group theory

n n n Let Bir(P ) be the set of all birational maps P 99K P . Composition of maps gives it a group structure.

2 2 Example: φ: P 99K P defined by

(a0 : a1 : a2) 7→ (a1a2 : a0a2 : a0a1) is a birational map.

n Bir(P ) contains PGLn+1(C) = set of (n + 1) × (n + 1) invertible matrices.

n n Elements of PGLn+1(C) give isomorphisms P → P .

1 Bir(P ) = PGL2(C).

2 Bir(P ) is generated by PGL3(C) and φ defined above. i.e. is there h ∈ N such that for any finite subgroup G of Bir(Pn), there is a normal abelian subgroup H of G of index at most h?

Serre: yes for n = 2.

Prokhorov and Shramov studied Serre’s question using minimal model program and Fano varieties.

Theorem (Prokhorov-Shramov 2012 + B 2016) Bir(Pn) is Jordan.

Group theory

Question (Serre): Is Bir(Pn) Jordan? for any finite subgroup G of Bir(Pn), there is a normal abelian subgroup H of G of index at most h?

Serre: yes for n = 2.

Prokhorov and Shramov studied Serre’s question using minimal model program and Fano varieties.

Theorem (Prokhorov-Shramov 2012 + B 2016) Bir(Pn) is Jordan.

Group theory

Question (Serre): Is Bir(Pn) Jordan? i.e. is there h ∈ N such that there is a normal abelian subgroup H of G of index at most h?

Serre: yes for n = 2.

Prokhorov and Shramov studied Serre’s question using minimal model program and Fano varieties.

Theorem (Prokhorov-Shramov 2012 + B 2016) Bir(Pn) is Jordan.

Group theory

Question (Serre): Is Bir(Pn) Jordan? i.e. is there h ∈ N such that for any finite subgroup G of Bir(Pn), Serre: yes for n = 2.

Prokhorov and Shramov studied Serre’s question using minimal model program and Fano varieties.

Theorem (Prokhorov-Shramov 2012 + B 2016) Bir(Pn) is Jordan.

Group theory

Question (Serre): Is Bir(Pn) Jordan? i.e. is there h ∈ N such that for any finite subgroup G of Bir(Pn), there is a normal abelian subgroup H of G of index at most h? Prokhorov and Shramov studied Serre’s question using minimal model program and Fano varieties.

Theorem (Prokhorov-Shramov 2012 + B 2016) Bir(Pn) is Jordan.

Group theory

Question (Serre): Is Bir(Pn) Jordan? i.e. is there h ∈ N such that for any finite subgroup G of Bir(Pn), there is a normal abelian subgroup H of G of index at most h?

Serre: yes for n = 2. Theorem (Prokhorov-Shramov 2012 + B 2016) Bir(Pn) is Jordan.

Group theory

Question (Serre): Is Bir(Pn) Jordan? i.e. is there h ∈ N such that for any finite subgroup G of Bir(Pn), there is a normal abelian subgroup H of G of index at most h?

Serre: yes for n = 2.

Prokhorov and Shramov studied Serre’s question using minimal model program and Fano varieties. Group theory

Question (Serre): Is Bir(Pn) Jordan? i.e. is there h ∈ N such that for any finite subgroup G of Bir(Pn), there is a normal abelian subgroup H of G of index at most h?

Serre: yes for n = 2.

Prokhorov and Shramov studied Serre’s question using minimal model program and Fano varieties.

Theorem (Prokhorov-Shramov 2012 + B 2016) Bir(Pn) is Jordan. Group theory

Question (Serre): Is Bir(Pn) Jordan? i.e. is there h ∈ N such that for any finite subgroup G of Bir(Pn), there is a normal abelian subgroup H of G of index at most h?

Serre: yes for n = 2.

Prokhorov and Shramov studied Serre’s question using minimal model program and Fano varieties.

Theorem (Prokhorov-Shramov 2012 + B 2016) Bir(Pn) is Jordan. Pick a finite subgroup G of Bir(Pn).

n There is a birational map W 99K P with W smooth projective such that G is a subgroup of Aut(W ). This means G acts on W as automorphisms.

Next run a G-equivariant MMP, that is, an MMP that respects the action of G.

This gives a birational map W 99K Y and a G-equivariant Mori-Fano fibration f : Y → Z .

G acts on Y , giving 1 → H → G → E → 1 where H acts on fibres F of f and E acts on Z.

If dim Z > 0, we can use induction on dimension to control these actions.

Group theory

Sketch of proof: n There is a birational map W 99K P with W smooth projective such that G is a subgroup of Aut(W ). This means G acts on W as automorphisms.

Next run a G-equivariant MMP, that is, an MMP that respects the action of G.

This gives a birational map W 99K Y and a G-equivariant Mori-Fano fibration f : Y → Z .

G acts on Y , giving 1 → H → G → E → 1 where H acts on fibres F of f and E acts on Z.

If dim Z > 0, we can use induction on dimension to control these actions.

Group theory

Sketch of proof: Pick a finite subgroup G of Bir(Pn). This means G acts on W as automorphisms.

Next run a G-equivariant MMP, that is, an MMP that respects the action of G.

This gives a birational map W 99K Y and a G-equivariant Mori-Fano fibration f : Y → Z .

G acts on Y , giving 1 → H → G → E → 1 where H acts on fibres F of f and E acts on Z.

If dim Z > 0, we can use induction on dimension to control these actions.

Group theory

Sketch of proof: Pick a finite subgroup G of Bir(Pn).

n There is a birational map W 99K P with W smooth projective such that G is a subgroup of Aut(W ). Next run a G-equivariant MMP, that is, an MMP that respects the action of G.

This gives a birational map W 99K Y and a G-equivariant Mori-Fano fibration f : Y → Z .

G acts on Y , giving 1 → H → G → E → 1 where H acts on fibres F of f and E acts on Z.

If dim Z > 0, we can use induction on dimension to control these actions.

Group theory

Sketch of proof: Pick a finite subgroup G of Bir(Pn).

n There is a birational map W 99K P with W smooth projective such that G is a subgroup of Aut(W ). This means G acts on W as automorphisms. that is, an MMP that respects the action of G.

This gives a birational map W 99K Y and a G-equivariant Mori-Fano fibration f : Y → Z .

G acts on Y , giving 1 → H → G → E → 1 where H acts on fibres F of f and E acts on Z.

If dim Z > 0, we can use induction on dimension to control these actions.

Group theory

Sketch of proof: Pick a finite subgroup G of Bir(Pn).

n There is a birational map W 99K P with W smooth projective such that G is a subgroup of Aut(W ). This means G acts on W as automorphisms.

Next run a G-equivariant MMP, This gives a birational map W 99K Y and a G-equivariant Mori-Fano fibration f : Y → Z .

G acts on Y , giving 1 → H → G → E → 1 where H acts on fibres F of f and E acts on Z.

If dim Z > 0, we can use induction on dimension to control these actions.

Group theory

Sketch of proof: Pick a finite subgroup G of Bir(Pn).

n There is a birational map W 99K P with W smooth projective such that G is a subgroup of Aut(W ). This means G acts on W as automorphisms.

Next run a G-equivariant MMP, that is, an MMP that respects the action of G. G acts on Y , giving 1 → H → G → E → 1 where H acts on fibres F of f and E acts on Z.

If dim Z > 0, we can use induction on dimension to control these actions.

Group theory

Sketch of proof: Pick a finite subgroup G of Bir(Pn).

n There is a birational map W 99K P with W smooth projective such that G is a subgroup of Aut(W ). This means G acts on W as automorphisms.

Next run a G-equivariant MMP, that is, an MMP that respects the action of G.

This gives a birational map W 99K Y and a G-equivariant Mori-Fano fibration f : Y → Z . giving 1 → H → G → E → 1 where H acts on fibres F of f and E acts on Z.

If dim Z > 0, we can use induction on dimension to control these actions.

Group theory

Sketch of proof: Pick a finite subgroup G of Bir(Pn).

n There is a birational map W 99K P with W smooth projective such that G is a subgroup of Aut(W ). This means G acts on W as automorphisms.

Next run a G-equivariant MMP, that is, an MMP that respects the action of G.

This gives a birational map W 99K Y and a G-equivariant Mori-Fano fibration f : Y → Z .

G acts on Y , If dim Z > 0, we can use induction on dimension to control these actions.

Group theory

Sketch of proof: Pick a finite subgroup G of Bir(Pn).

n There is a birational map W 99K P with W smooth projective such that G is a subgroup of Aut(W ). This means G acts on W as automorphisms.

Next run a G-equivariant MMP, that is, an MMP that respects the action of G.

This gives a birational map W 99K Y and a G-equivariant Mori-Fano fibration f : Y → Z .

G acts on Y , giving 1 → H → G → E → 1 where H acts on fibres F of f and E acts on Z. Group theory

Sketch of proof: Pick a finite subgroup G of Bir(Pn).

n There is a birational map W 99K P with W smooth projective such that G is a subgroup of Aut(W ). This means G acts on W as automorphisms.

Next run a G-equivariant MMP, that is, an MMP that respects the action of G.

This gives a birational map W 99K Y and a G-equivariant Mori-Fano fibration f : Y → Z .

G acts on Y , giving 1 → H → G → E → 1 where H acts on fibres F of f and E acts on Z.

If dim Z > 0, we can use induction on dimension to control these actions. Group theory

Sketch of proof: Pick a finite subgroup G of Bir(Pn).

n There is a birational map W 99K P with W smooth projective such that G is a subgroup of Aut(W ). This means G acts on W as automorphisms.

Next run a G-equivariant MMP, that is, an MMP that respects the action of G.

This gives a birational map W 99K Y and a G-equivariant Mori-Fano fibration f : Y → Z .

G acts on Y , giving 1 → H → G → E → 1 where H acts on fibres F of f and E acts on Z.

If dim Z > 0, we can use induction on dimension to control these actions. By boundedness of Fano’s, the Y form a bounded family.

This allows to embed G into Aut(Pd ) = PGL(d, C) for some d.

Therefore, we are reduced to showing PGL(C, d) is Jordan. This is a classical result of Jordan himself.

Group theory

So assume dim Z = 0, hence F = Y is Fano with "good" singularities. This allows to embed G into Aut(Pd ) = PGL(d, C) for some d.

Therefore, we are reduced to showing PGL(C, d) is Jordan. This is a classical result of Jordan himself.

Group theory

So assume dim Z = 0, hence F = Y is Fano with "good" singularities.

By boundedness of Fano’s, the Y form a bounded family. Therefore, we are reduced to showing PGL(C, d) is Jordan. This is a classical result of Jordan himself.

Group theory

So assume dim Z = 0, hence F = Y is Fano with "good" singularities.

By boundedness of Fano’s, the Y form a bounded family.

This allows to embed G into Aut(Pd ) = PGL(d, C) for some d. This is a classical result of Jordan himself.

Group theory

So assume dim Z = 0, hence F = Y is Fano with "good" singularities.

By boundedness of Fano’s, the Y form a bounded family.

This allows to embed G into Aut(Pd ) = PGL(d, C) for some d.

Therefore, we are reduced to showing PGL(C, d) is Jordan. Group theory

So assume dim Z = 0, hence F = Y is Fano with "good" singularities.

By boundedness of Fano’s, the Y form a bounded family.

This allows to embed G into Aut(Pd ) = PGL(d, C) for some d.

Therefore, we are reduced to showing PGL(C, d) is Jordan. This is a classical result of Jordan himself. Group theory

So assume dim Z = 0, hence F = Y is Fano with "good" singularities.

By boundedness of Fano’s, the Y form a bounded family.

This allows to embed G into Aut(Pd ) = PGL(d, C) for some d.

Therefore, we are reduced to showing PGL(C, d) is Jordan. This is a classical result of Jordan himself.