Finite generation of canonical rings II - Math
Transcript of Finite generation of canonical rings II - Math
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IntroductionSurfaces
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Finite generation of canonical rings II
Christopher Hacon
University of Utah
March, 2008
Christopher Hacon Finite generation of canonical rings II
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IntroductionSurfaces
Higher dimensional preview
Outline of the talk
1 Introduction
2 Surfaces
3 Higher dimensional preview
Christopher Hacon Finite generation of canonical rings II
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IntroductionSurfaces
Higher dimensional preview
Outline of the talk
1 Introduction
2 Surfaces
3 Higher dimensional preview
Christopher Hacon Finite generation of canonical rings II
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IntroductionSurfaces
Higher dimensional preview
Outline of the talk
1 Introduction
2 Surfaces
3 Higher dimensional preview
Christopher Hacon Finite generation of canonical rings II
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IntroductionSurfaces
Higher dimensional preview
Outline of the talk
1 Introduction
2 Surfaces
3 Higher dimensional preview
Christopher Hacon Finite generation of canonical rings II
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IntroductionSurfaces
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Notation
I am interested in the birational geometry of complexprojective varieties.
X � PN
C is de�ned by homogeneous polynomialsP1; � � � ;Pt 2 C[x0; � � � ; xN ].
We will assume that X is irreducible and smooth.
In the previous talk I discussed the geometry of curves i.e.dimX = 1.
In this case X is a smooth orientable compact manifold ofdimR X = 2 and so X is a Riemann Surface.
Christopher Hacon Finite generation of canonical rings II
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IntroductionSurfaces
Higher dimensional preview
Notation
I am interested in the birational geometry of complexprojective varieties.
X � PN
C is de�ned by homogeneous polynomialsP1; � � � ;Pt 2 C[x0; � � � ; xN ].
We will assume that X is irreducible and smooth.
In the previous talk I discussed the geometry of curves i.e.dimX = 1.
In this case X is a smooth orientable compact manifold ofdimR X = 2 and so X is a Riemann Surface.
Christopher Hacon Finite generation of canonical rings II
![Page 8: Finite generation of canonical rings II - Math](https://reader031.fdocuments.in/reader031/viewer/2022012021/6168a424d394e9041f716e35/html5/thumbnails/8.jpg)
IntroductionSurfaces
Higher dimensional preview
Notation
I am interested in the birational geometry of complexprojective varieties.
X � PN
C is de�ned by homogeneous polynomialsP1; � � � ;Pt 2 C[x0; � � � ; xN ].
We will assume that X is irreducible and smooth.
In the previous talk I discussed the geometry of curves i.e.dimX = 1.
In this case X is a smooth orientable compact manifold ofdimR X = 2 and so X is a Riemann Surface.
Christopher Hacon Finite generation of canonical rings II
![Page 9: Finite generation of canonical rings II - Math](https://reader031.fdocuments.in/reader031/viewer/2022012021/6168a424d394e9041f716e35/html5/thumbnails/9.jpg)
IntroductionSurfaces
Higher dimensional preview
Notation
I am interested in the birational geometry of complexprojective varieties.
X � PN
C is de�ned by homogeneous polynomialsP1; � � � ;Pt 2 C[x0; � � � ; xN ].
We will assume that X is irreducible and smooth.
In the previous talk I discussed the geometry of curves i.e.dimX = 1.
In this case X is a smooth orientable compact manifold ofdimR X = 2 and so X is a Riemann Surface.
Christopher Hacon Finite generation of canonical rings II
![Page 10: Finite generation of canonical rings II - Math](https://reader031.fdocuments.in/reader031/viewer/2022012021/6168a424d394e9041f716e35/html5/thumbnails/10.jpg)
IntroductionSurfaces
Higher dimensional preview
Notation
I am interested in the birational geometry of complexprojective varieties.
X � PN
C is de�ned by homogeneous polynomialsP1; � � � ;Pt 2 C[x0; � � � ; xN ].
We will assume that X is irreducible and smooth.
In the previous talk I discussed the geometry of curves i.e.dimX = 1.
In this case X is a smooth orientable compact manifold ofdimR X = 2 and so X is a Riemann Surface.
Christopher Hacon Finite generation of canonical rings II
![Page 11: Finite generation of canonical rings II - Math](https://reader031.fdocuments.in/reader031/viewer/2022012021/6168a424d394e9041f716e35/html5/thumbnails/11.jpg)
IntroductionSurfaces
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Riemann Surfaces
Christopher Hacon Finite generation of canonical rings II
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IntroductionSurfaces
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Curves
Curves are topologically determined by their genus.
g = 0: X is a rational curve i.e. X = P1.
g = 1: There is a 1-parameter family of elliptic curves.
g � 2: Curves of general type. These form a3g � 3-dimensional family.
There are in�nitely many ways to embed X � PN
C .
Christopher Hacon Finite generation of canonical rings II
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IntroductionSurfaces
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Curves
Curves are topologically determined by their genus.
g = 0: X is a rational curve i.e. X = P1.
g = 1: There is a 1-parameter family of elliptic curves.
g � 2: Curves of general type. These form a3g � 3-dimensional family.
There are in�nitely many ways to embed X � PN
C .
Christopher Hacon Finite generation of canonical rings II
![Page 14: Finite generation of canonical rings II - Math](https://reader031.fdocuments.in/reader031/viewer/2022012021/6168a424d394e9041f716e35/html5/thumbnails/14.jpg)
IntroductionSurfaces
Higher dimensional preview
Curves
Curves are topologically determined by their genus.
g = 0: X is a rational curve i.e. X = P1.
g = 1: There is a 1-parameter family of elliptic curves.
g � 2: Curves of general type. These form a3g � 3-dimensional family.
There are in�nitely many ways to embed X � PN
C .
Christopher Hacon Finite generation of canonical rings II
![Page 15: Finite generation of canonical rings II - Math](https://reader031.fdocuments.in/reader031/viewer/2022012021/6168a424d394e9041f716e35/html5/thumbnails/15.jpg)
IntroductionSurfaces
Higher dimensional preview
Curves
Curves are topologically determined by their genus.
g = 0: X is a rational curve i.e. X = P1.
g = 1: There is a 1-parameter family of elliptic curves.
g � 2: Curves of general type. These form a3g � 3-dimensional family.
There are in�nitely many ways to embed X � PN
C .
Christopher Hacon Finite generation of canonical rings II
![Page 16: Finite generation of canonical rings II - Math](https://reader031.fdocuments.in/reader031/viewer/2022012021/6168a424d394e9041f716e35/html5/thumbnails/16.jpg)
IntroductionSurfaces
Higher dimensional preview
Curves
Curves are topologically determined by their genus.
g = 0: X is a rational curve i.e. X = P1.
g = 1: There is a 1-parameter family of elliptic curves.
g � 2: Curves of general type. These form a3g � 3-dimensional family.
There are in�nitely many ways to embed X � PN
C .
Christopher Hacon Finite generation of canonical rings II
![Page 17: Finite generation of canonical rings II - Math](https://reader031.fdocuments.in/reader031/viewer/2022012021/6168a424d394e9041f716e35/html5/thumbnails/17.jpg)
IntroductionSurfaces
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Curves II
Embeddings X � PN
C are obtained as follows.
If L is a line bundle and s0; : : : ; sN are a basis of H0(X ; L),then x ! [s0(x) : : : : : sN(x)] de�nes a map �L : X 99K P
N
C onthe complement of the common zeroes of the si .
If s0; : : : ; sN separate points and tangent directions on X ,then �L is an embedding.
One natural choice for L is !kX
, where !X is the line bundlewhose sections s 2 H0(X ; !X ) may be locally written assjU = f (x)dx . (One may think of these asrational/meromorhic functions on X with prescribed poles.)
Note that the genus of X is then given by g = dimH0(X ; !X ).
Christopher Hacon Finite generation of canonical rings II
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IntroductionSurfaces
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Curves II
Embeddings X � PN
C are obtained as follows.
If L is a line bundle and s0; : : : ; sN are a basis of H0(X ; L),then x ! [s0(x) : : : : : sN(x)] de�nes a map �L : X 99K P
N
C onthe complement of the common zeroes of the si .
If s0; : : : ; sN separate points and tangent directions on X ,then �L is an embedding.
One natural choice for L is !kX
, where !X is the line bundlewhose sections s 2 H0(X ; !X ) may be locally written assjU = f (x)dx . (One may think of these asrational/meromorhic functions on X with prescribed poles.)
Note that the genus of X is then given by g = dimH0(X ; !X ).
Christopher Hacon Finite generation of canonical rings II
![Page 19: Finite generation of canonical rings II - Math](https://reader031.fdocuments.in/reader031/viewer/2022012021/6168a424d394e9041f716e35/html5/thumbnails/19.jpg)
IntroductionSurfaces
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Curves II
Embeddings X � PN
C are obtained as follows.
If L is a line bundle and s0; : : : ; sN are a basis of H0(X ; L),then x ! [s0(x) : : : : : sN(x)] de�nes a map �L : X 99K P
N
C onthe complement of the common zeroes of the si .
If s0; : : : ; sN separate points and tangent directions on X ,then �L is an embedding.
One natural choice for L is !kX
, where !X is the line bundlewhose sections s 2 H0(X ; !X ) may be locally written assjU = f (x)dx . (One may think of these asrational/meromorhic functions on X with prescribed poles.)
Note that the genus of X is then given by g = dimH0(X ; !X ).
Christopher Hacon Finite generation of canonical rings II
![Page 20: Finite generation of canonical rings II - Math](https://reader031.fdocuments.in/reader031/viewer/2022012021/6168a424d394e9041f716e35/html5/thumbnails/20.jpg)
IntroductionSurfaces
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Curves II
Embeddings X � PN
C are obtained as follows.
If L is a line bundle and s0; : : : ; sN are a basis of H0(X ; L),then x ! [s0(x) : : : : : sN(x)] de�nes a map �L : X 99K P
N
C onthe complement of the common zeroes of the si .
If s0; : : : ; sN separate points and tangent directions on X ,then �L is an embedding.
One natural choice for L is !kX
, where !X is the line bundlewhose sections s 2 H0(X ; !X ) may be locally written assjU = f (x)dx . (One may think of these asrational/meromorhic functions on X with prescribed poles.)
Note that the genus of X is then given by g = dimH0(X ; !X ).
Christopher Hacon Finite generation of canonical rings II
![Page 21: Finite generation of canonical rings II - Math](https://reader031.fdocuments.in/reader031/viewer/2022012021/6168a424d394e9041f716e35/html5/thumbnails/21.jpg)
IntroductionSurfaces
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Curves II
Embeddings X � PN
C are obtained as follows.
If L is a line bundle and s0; : : : ; sN are a basis of H0(X ; L),then x ! [s0(x) : : : : : sN(x)] de�nes a map �L : X 99K P
N
C onthe complement of the common zeroes of the si .
If s0; : : : ; sN separate points and tangent directions on X ,then �L is an embedding.
One natural choice for L is !kX
, where !X is the line bundlewhose sections s 2 H0(X ; !X ) may be locally written assjU = f (x)dx . (One may think of these asrational/meromorhic functions on X with prescribed poles.)
Note that the genus of X is then given by g = dimH0(X ; !X ).
Christopher Hacon Finite generation of canonical rings II
![Page 22: Finite generation of canonical rings II - Math](https://reader031.fdocuments.in/reader031/viewer/2022012021/6168a424d394e9041f716e35/html5/thumbnails/22.jpg)
IntroductionSurfaces
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Curves III
The canonical ring R(X ) =L
k�0 H0(X ; !k
X) is a
fundamental invariant of X .
If g = 0, then !X = OP1
C(�2) so that H0(X ; !k
X) = 0 for all
k > 0 i.e. R(X ) �= C.
If g = 1, then !X = OX so that H0(X ; !kX
) = C for allk > 0 i.e. R(X ) �= C[t].
If g � 2 then !3X
de�nes an embedding � : X ! PN
C .
It follows that !3X
= ��OPN
C(1). It is then easy to see that
R(X ) is �nitely generated and X = ProjR(X ).
Christopher Hacon Finite generation of canonical rings II
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IntroductionSurfaces
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Curves III
The canonical ring R(X ) =L
k�0 H0(X ; !k
X) is a
fundamental invariant of X .
If g = 0, then !X = OP1
C(�2) so that H0(X ; !k
X) = 0 for all
k > 0 i.e. R(X ) �= C.
If g = 1, then !X = OX so that H0(X ; !kX
) = C for allk > 0 i.e. R(X ) �= C[t].
If g � 2 then !3X
de�nes an embedding � : X ! PN
C .
It follows that !3X
= ��OPN
C(1). It is then easy to see that
R(X ) is �nitely generated and X = ProjR(X ).
Christopher Hacon Finite generation of canonical rings II
![Page 24: Finite generation of canonical rings II - Math](https://reader031.fdocuments.in/reader031/viewer/2022012021/6168a424d394e9041f716e35/html5/thumbnails/24.jpg)
IntroductionSurfaces
Higher dimensional preview
Curves III
The canonical ring R(X ) =L
k�0 H0(X ; !k
X) is a
fundamental invariant of X .
If g = 0, then !X = OP1
C(�2) so that H0(X ; !k
X) = 0 for all
k > 0 i.e. R(X ) �= C.
If g = 1, then !X = OX so that H0(X ; !kX
) = C for allk > 0 i.e. R(X ) �= C[t].
If g � 2 then !3X
de�nes an embedding � : X ! PN
C .
It follows that !3X
= ��OPN
C(1). It is then easy to see that
R(X ) is �nitely generated and X = ProjR(X ).
Christopher Hacon Finite generation of canonical rings II
![Page 25: Finite generation of canonical rings II - Math](https://reader031.fdocuments.in/reader031/viewer/2022012021/6168a424d394e9041f716e35/html5/thumbnails/25.jpg)
IntroductionSurfaces
Higher dimensional preview
Curves III
The canonical ring R(X ) =L
k�0 H0(X ; !k
X) is a
fundamental invariant of X .
If g = 0, then !X = OP1
C(�2) so that H0(X ; !k
X) = 0 for all
k > 0 i.e. R(X ) �= C.
If g = 1, then !X = OX so that H0(X ; !kX
) = C for allk > 0 i.e. R(X ) �= C[t].
If g � 2 then !3X
de�nes an embedding � : X ! PN
C .
It follows that !3X
= ��OPN
C(1). It is then easy to see that
R(X ) is �nitely generated and X = ProjR(X ).
Christopher Hacon Finite generation of canonical rings II
![Page 26: Finite generation of canonical rings II - Math](https://reader031.fdocuments.in/reader031/viewer/2022012021/6168a424d394e9041f716e35/html5/thumbnails/26.jpg)
IntroductionSurfaces
Higher dimensional preview
Curves III
The canonical ring R(X ) =L
k�0 H0(X ; !k
X) is a
fundamental invariant of X .
If g = 0, then !X = OP1
C(�2) so that H0(X ; !k
X) = 0 for all
k > 0 i.e. R(X ) �= C.
If g = 1, then !X = OX so that H0(X ; !kX
) = C for allk > 0 i.e. R(X ) �= C[t].
If g � 2 then !3X
de�nes an embedding � : X ! PN
C .
It follows that !3X
= ��OPN
C(1). It is then easy to see that
R(X ) is �nitely generated and X = ProjR(X ).
Christopher Hacon Finite generation of canonical rings II
![Page 27: Finite generation of canonical rings II - Math](https://reader031.fdocuments.in/reader031/viewer/2022012021/6168a424d394e9041f716e35/html5/thumbnails/27.jpg)
IntroductionSurfaces
Higher dimensional preview
Outline of the talk
1 Introduction
2 Surfaces
3 Higher dimensional preview
Christopher Hacon Finite generation of canonical rings II
![Page 28: Finite generation of canonical rings II - Math](https://reader031.fdocuments.in/reader031/viewer/2022012021/6168a424d394e9041f716e35/html5/thumbnails/28.jpg)
IntroductionSurfaces
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Surfaces
One would naturally like to generalize these results to higherdimensions.
The next case of interest are surfaces i.e. dimX = 2. (Notethat dimR X = 4!)
The �rst di�culty is that given any point on a surface x 2 X ,one can produce a new surface
� : ~X = Blx(X )! X
known as the blow up of X at x .
The morphism � may be viewed as a surgery that replaces thepoint x by a rational curve E �= P
1 = P(TxX ).
Christopher Hacon Finite generation of canonical rings II
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IntroductionSurfaces
Higher dimensional preview
Surfaces
One would naturally like to generalize these results to higherdimensions.
The next case of interest are surfaces i.e. dimX = 2. (Notethat dimR X = 4!)
The �rst di�culty is that given any point on a surface x 2 X ,one can produce a new surface
� : ~X = Blx(X )! X
known as the blow up of X at x .
The morphism � may be viewed as a surgery that replaces thepoint x by a rational curve E �= P
1 = P(TxX ).
Christopher Hacon Finite generation of canonical rings II
![Page 30: Finite generation of canonical rings II - Math](https://reader031.fdocuments.in/reader031/viewer/2022012021/6168a424d394e9041f716e35/html5/thumbnails/30.jpg)
IntroductionSurfaces
Higher dimensional preview
Surfaces
One would naturally like to generalize these results to higherdimensions.
The next case of interest are surfaces i.e. dimX = 2. (Notethat dimR X = 4!)
The �rst di�culty is that given any point on a surface x 2 X ,one can produce a new surface
� : ~X = Blx(X )! X
known as the blow up of X at x .
The morphism � may be viewed as a surgery that replaces thepoint x by a rational curve E �= P
1 = P(TxX ).
Christopher Hacon Finite generation of canonical rings II
![Page 31: Finite generation of canonical rings II - Math](https://reader031.fdocuments.in/reader031/viewer/2022012021/6168a424d394e9041f716e35/html5/thumbnails/31.jpg)
IntroductionSurfaces
Higher dimensional preview
Surfaces
One would naturally like to generalize these results to higherdimensions.
The next case of interest are surfaces i.e. dimX = 2. (Notethat dimR X = 4!)
The �rst di�culty is that given any point on a surface x 2 X ,one can produce a new surface
� : ~X = Blx(X )! X
known as the blow up of X at x .
The morphism � may be viewed as a surgery that replaces thepoint x by a rational curve E �= P
1 = P(TxX ).
Christopher Hacon Finite generation of canonical rings II
![Page 32: Finite generation of canonical rings II - Math](https://reader031.fdocuments.in/reader031/viewer/2022012021/6168a424d394e9041f716e35/html5/thumbnails/32.jpg)
IntroductionSurfaces
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Blowing up a point
Christopher Hacon Finite generation of canonical rings II
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Birational Surfaces
The exceptional curve E is called a �1 curve since
!~X � E = deg(!~X jE ) = �1:
One says that X ;X 0 are birational if they have isomorphicopen subsets U �= U 0.
For example X is birational to Blx(X ).
It turns out that two surfaces are birational if and only if thereis a �nite sequence of blow ups such that
BlxtBlxt�1� � �Blx1(X ) = Blx 0sBlx 0s�1
� � �Blx 01(X 0):
Christopher Hacon Finite generation of canonical rings II
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IntroductionSurfaces
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Birational Surfaces
The exceptional curve E is called a �1 curve since
!~X � E = deg(!~X jE ) = �1:
One says that X ;X 0 are birational if they have isomorphicopen subsets U �= U 0.
For example X is birational to Blx(X ).
It turns out that two surfaces are birational if and only if thereis a �nite sequence of blow ups such that
BlxtBlxt�1� � �Blx1(X ) = Blx 0sBlx 0s�1
� � �Blx 01(X 0):
Christopher Hacon Finite generation of canonical rings II
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IntroductionSurfaces
Higher dimensional preview
Birational Surfaces
The exceptional curve E is called a �1 curve since
!~X � E = deg(!~X jE ) = �1:
One says that X ;X 0 are birational if they have isomorphicopen subsets U �= U 0.
For example X is birational to Blx(X ).
It turns out that two surfaces are birational if and only if thereis a �nite sequence of blow ups such that
BlxtBlxt�1� � �Blx1(X ) = Blx 0sBlx 0s�1
� � �Blx 01(X 0):
Christopher Hacon Finite generation of canonical rings II
![Page 36: Finite generation of canonical rings II - Math](https://reader031.fdocuments.in/reader031/viewer/2022012021/6168a424d394e9041f716e35/html5/thumbnails/36.jpg)
IntroductionSurfaces
Higher dimensional preview
Birational Surfaces
The exceptional curve E is called a �1 curve since
!~X � E = deg(!~X jE ) = �1:
One says that X ;X 0 are birational if they have isomorphicopen subsets U �= U 0.
For example X is birational to Blx(X ).
It turns out that two surfaces are birational if and only if thereis a �nite sequence of blow ups such that
BlxtBlxt�1� � �Blx1(X ) = Blx 0sBlx 0s�1
� � �Blx 01(X 0):
Christopher Hacon Finite generation of canonical rings II
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Example of a birational map
Christopher Hacon Finite generation of canonical rings II
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IntroductionSurfaces
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Birational Surfaces II
Since the space C(X ) of rational functions on X aredetermined by any open subset U � X , two surfaces X ;X 0 arebirational if and only if C(X ) �= C(X 0).
For example a surface X is rational (i.e. birational to P2C if
and only if C(X ) �= C(x ; y).
It is then natural to study surfaces modulo birationalequivalence.
Given X one would like to identify a unique representative X 0
birational to X . Ideally this representative X 0 has niceproperties that are useful in understanding the geometry of X 0
(and hence of X ).
Christopher Hacon Finite generation of canonical rings II
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IntroductionSurfaces
Higher dimensional preview
Birational Surfaces II
Since the space C(X ) of rational functions on X aredetermined by any open subset U � X , two surfaces X ;X 0 arebirational if and only if C(X ) �= C(X 0).
For example a surface X is rational (i.e. birational to P2C if
and only if C(X ) �= C(x ; y).
It is then natural to study surfaces modulo birationalequivalence.
Given X one would like to identify a unique representative X 0
birational to X . Ideally this representative X 0 has niceproperties that are useful in understanding the geometry of X 0
(and hence of X ).
Christopher Hacon Finite generation of canonical rings II
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IntroductionSurfaces
Higher dimensional preview
Birational Surfaces II
Since the space C(X ) of rational functions on X aredetermined by any open subset U � X , two surfaces X ;X 0 arebirational if and only if C(X ) �= C(X 0).
For example a surface X is rational (i.e. birational to P2C if
and only if C(X ) �= C(x ; y).
It is then natural to study surfaces modulo birationalequivalence.
Given X one would like to identify a unique representative X 0
birational to X . Ideally this representative X 0 has niceproperties that are useful in understanding the geometry of X 0
(and hence of X ).
Christopher Hacon Finite generation of canonical rings II
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IntroductionSurfaces
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Birational Surfaces II
Since the space C(X ) of rational functions on X aredetermined by any open subset U � X , two surfaces X ;X 0 arebirational if and only if C(X ) �= C(X 0).
For example a surface X is rational (i.e. birational to P2C if
and only if C(X ) �= C(x ; y).
It is then natural to study surfaces modulo birationalequivalence.
Given X one would like to identify a unique representative X 0
birational to X . Ideally this representative X 0 has niceproperties that are useful in understanding the geometry of X 0
(and hence of X ).
Christopher Hacon Finite generation of canonical rings II
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IntroductionSurfaces
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Castelnuovo's Theorem
The answer is given by a result of Castelnuovo. It is easiest todescribe this in terms of the canonical bundle.
Let !X = �2T_Xbe the canonical line bundle.
If � : ~X = Blx(X )! X is the blow up of a point, andE = ��1(x) is the exceptional curve, then E is a �1 curve i.e.E �= P
1C and E � !X = deg!X jE = �1.
Castelnuovo's Theorem says that one may reverse thisprocedure i.e. given any surface X and any �1 curve E � X ,there exists a morphism � : X ! X1 such that X = Blx(X1)for some point x 2 X1.
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IntroductionSurfaces
Higher dimensional preview
Castelnuovo's Theorem
The answer is given by a result of Castelnuovo. It is easiest todescribe this in terms of the canonical bundle.
Let !X = �2T_Xbe the canonical line bundle.
If � : ~X = Blx(X )! X is the blow up of a point, andE = ��1(x) is the exceptional curve, then E is a �1 curve i.e.E �= P
1C and E � !X = deg!X jE = �1.
Castelnuovo's Theorem says that one may reverse thisprocedure i.e. given any surface X and any �1 curve E � X ,there exists a morphism � : X ! X1 such that X = Blx(X1)for some point x 2 X1.
Christopher Hacon Finite generation of canonical rings II
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IntroductionSurfaces
Higher dimensional preview
Castelnuovo's Theorem
The answer is given by a result of Castelnuovo. It is easiest todescribe this in terms of the canonical bundle.
Let !X = �2T_Xbe the canonical line bundle.
If � : ~X = Blx(X )! X is the blow up of a point, andE = ��1(x) is the exceptional curve, then E is a �1 curve i.e.E �= P
1C and E � !X = deg!X jE = �1.
Castelnuovo's Theorem says that one may reverse thisprocedure i.e. given any surface X and any �1 curve E � X ,there exists a morphism � : X ! X1 such that X = Blx(X1)for some point x 2 X1.
Christopher Hacon Finite generation of canonical rings II
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IntroductionSurfaces
Higher dimensional preview
Castelnuovo's Theorem
The answer is given by a result of Castelnuovo. It is easiest todescribe this in terms of the canonical bundle.
Let !X = �2T_Xbe the canonical line bundle.
If � : ~X = Blx(X )! X is the blow up of a point, andE = ��1(x) is the exceptional curve, then E is a �1 curve i.e.E �= P
1C and E � !X = deg!X jE = �1.
Castelnuovo's Theorem says that one may reverse thisprocedure i.e. given any surface X and any �1 curve E � X ,there exists a morphism � : X ! X1 such that X = Blx(X1)for some point x 2 X1.
Christopher Hacon Finite generation of canonical rings II
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IntroductionSurfaces
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Minimal Surfaces
A minimal surface is a surface that contains no �1 curves.
Note that �(Blx(X )) = �(X ) + 1 so that one can only blowdown �nitely many curves.
So, given a surface X , after blowing down �nitely many �1curves, one obtains a morphism X ! Xmin where Xmin is aminimal surface.
The natural question is then: Is Xmin unique? Does Xmin haveany interesting special properties?
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IntroductionSurfaces
Higher dimensional preview
Minimal Surfaces
A minimal surface is a surface that contains no �1 curves.
Note that �(Blx(X )) = �(X ) + 1 so that one can only blowdown �nitely many curves.
So, given a surface X , after blowing down �nitely many �1curves, one obtains a morphism X ! Xmin where Xmin is aminimal surface.
The natural question is then: Is Xmin unique? Does Xmin haveany interesting special properties?
Christopher Hacon Finite generation of canonical rings II
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IntroductionSurfaces
Higher dimensional preview
Minimal Surfaces
A minimal surface is a surface that contains no �1 curves.
Note that �(Blx(X )) = �(X ) + 1 so that one can only blowdown �nitely many curves.
So, given a surface X , after blowing down �nitely many �1curves, one obtains a morphism X ! Xmin where Xmin is aminimal surface.
The natural question is then: Is Xmin unique? Does Xmin haveany interesting special properties?
Christopher Hacon Finite generation of canonical rings II
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IntroductionSurfaces
Higher dimensional preview
Minimal Surfaces
A minimal surface is a surface that contains no �1 curves.
Note that �(Blx(X )) = �(X ) + 1 so that one can only blowdown �nitely many curves.
So, given a surface X , after blowing down �nitely many �1curves, one obtains a morphism X ! Xmin where Xmin is aminimal surface.
The natural question is then: Is Xmin unique? Does Xmin haveany interesting special properties?
Christopher Hacon Finite generation of canonical rings II
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IntroductionSurfaces
Higher dimensional preview
Canonical Ring
The answer is given in terms of the canonical ring
R(X ) =M
k�0
H0(X ; !kX
):
The Kodaira dimension of X is
�(X ) := tr:deg:CR(X )� 1:
We have �(X ) := maxk>0fdim�!k
X
(X )g.
If �(X ) = �1, then the minimal model X ! Xmin is notunique (but it is well understood how two di�erent minimalmodels are related).
If �(X ) � 0, then the minimal model X ! Xmin is unique.
Christopher Hacon Finite generation of canonical rings II
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IntroductionSurfaces
Higher dimensional preview
Canonical Ring
The answer is given in terms of the canonical ring
R(X ) =M
k�0
H0(X ; !kX
):
The Kodaira dimension of X is
�(X ) := tr:deg:CR(X )� 1:
We have �(X ) := maxk>0fdim�!k
X
(X )g.
If �(X ) = �1, then the minimal model X ! Xmin is notunique (but it is well understood how two di�erent minimalmodels are related).
If �(X ) � 0, then the minimal model X ! Xmin is unique.
Christopher Hacon Finite generation of canonical rings II
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IntroductionSurfaces
Higher dimensional preview
Canonical Ring
The answer is given in terms of the canonical ring
R(X ) =M
k�0
H0(X ; !kX
):
The Kodaira dimension of X is
�(X ) := tr:deg:CR(X )� 1:
We have �(X ) := maxk>0fdim�!k
X
(X )g.
If �(X ) = �1, then the minimal model X ! Xmin is notunique (but it is well understood how two di�erent minimalmodels are related).
If �(X ) � 0, then the minimal model X ! Xmin is unique.
Christopher Hacon Finite generation of canonical rings II
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IntroductionSurfaces
Higher dimensional preview
Canonical Ring
The answer is given in terms of the canonical ring
R(X ) =M
k�0
H0(X ; !kX
):
The Kodaira dimension of X is
�(X ) := tr:deg:CR(X )� 1:
We have �(X ) := maxk>0fdim�!k
X
(X )g.
If �(X ) = �1, then the minimal model X ! Xmin is notunique (but it is well understood how two di�erent minimalmodels are related).
If �(X ) � 0, then the minimal model X ! Xmin is unique.
Christopher Hacon Finite generation of canonical rings II
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IntroductionSurfaces
Higher dimensional preview
Canonical Ring
The answer is given in terms of the canonical ring
R(X ) =M
k�0
H0(X ; !kX
):
The Kodaira dimension of X is
�(X ) := tr:deg:CR(X )� 1:
We have �(X ) := maxk>0fdim�!k
X
(X )g.
If �(X ) = �1, then the minimal model X ! Xmin is notunique (but it is well understood how two di�erent minimalmodels are related).
If �(X ) � 0, then the minimal model X ! Xmin is unique.
Christopher Hacon Finite generation of canonical rings II
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Birational geometry of Surfaces
However, if �(X ) = �1, then Xmin is either P2C or a ruled
surface so that X is covered by rational curves (i.e. by P1's).The sections of H0(!m
X) must vanish along these curves for
m > 0. Hence R(X ) �= C.
If �(X ) � 0, then the minimal model X ! Xmin is unique and!Xmin
is nef. This means that for any curve C � Xmin we have
!Xmin� C � 0:
In fact one can show that !Xminis semiample i.e. there is an
integer m > 0 such that the sections of H0(!mXmin
) have nocommon zeroes.
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IntroductionSurfaces
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Birational geometry of Surfaces
However, if �(X ) = �1, then Xmin is either P2C or a ruled
surface so that X is covered by rational curves (i.e. by P1's).The sections of H0(!m
X) must vanish along these curves for
m > 0. Hence R(X ) �= C.
If �(X ) � 0, then the minimal model X ! Xmin is unique and!Xmin
is nef. This means that for any curve C � Xmin we have
!Xmin� C � 0:
In fact one can show that !Xminis semiample i.e. there is an
integer m > 0 such that the sections of H0(!mXmin
) have nocommon zeroes.
Christopher Hacon Finite generation of canonical rings II
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Birational geometry of Surfaces
However, if �(X ) = �1, then Xmin is either P2C or a ruled
surface so that X is covered by rational curves (i.e. by P1's).The sections of H0(!m
X) must vanish along these curves for
m > 0. Hence R(X ) �= C.
If �(X ) � 0, then the minimal model X ! Xmin is unique and!Xmin
is nef. This means that for any curve C � Xmin we have
!Xmin� C � 0:
In fact one can show that !Xminis semiample i.e. there is an
integer m > 0 such that the sections of H0(!mXmin
) have nocommon zeroes.
Christopher Hacon Finite generation of canonical rings II
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Birational geometry of surfaces II
Therefore, '!m
Xmin
: Xmin ! PN is a morphism,
!mXmin
�= '�OPN (1).
It is then easy to show that R(X ) is �nitely generated.
Conclusion: If X is not covered by rational curves, then afterperforming a �nite sequence of geometrically meaningfuloperations (contracting �1 curves) we obtain a birationalmodel on which ! is positive (nef and semiample).
Finite generation of R(X ) then follows.
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Birational geometry of surfaces II
Therefore, '!m
Xmin
: Xmin ! PN is a morphism,
!mXmin
�= '�OPN (1).
It is then easy to show that R(X ) is �nitely generated.
Conclusion: If X is not covered by rational curves, then afterperforming a �nite sequence of geometrically meaningfuloperations (contracting �1 curves) we obtain a birationalmodel on which ! is positive (nef and semiample).
Finite generation of R(X ) then follows.
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Birational geometry of surfaces II
Therefore, '!m
Xmin
: Xmin ! PN is a morphism,
!mXmin
�= '�OPN (1).
It is then easy to show that R(X ) is �nitely generated.
Conclusion: If X is not covered by rational curves, then afterperforming a �nite sequence of geometrically meaningfuloperations (contracting �1 curves) we obtain a birationalmodel on which ! is positive (nef and semiample).
Finite generation of R(X ) then follows.
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Birational geometry of surfaces II
Therefore, '!m
Xmin
: Xmin ! PN is a morphism,
!mXmin
�= '�OPN (1).
It is then easy to show that R(X ) is �nitely generated.
Conclusion: If X is not covered by rational curves, then afterperforming a �nite sequence of geometrically meaningfuloperations (contracting �1 curves) we obtain a birationalmodel on which ! is positive (nef and semiample).
Finite generation of R(X ) then follows.
Christopher Hacon Finite generation of canonical rings II
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Birational geometry of surfaces III
More precisely we have the following:
If �(X ) = 0, then Xmin is either an abelian, hyperellyptic, K3or Enriques surface. !12
Xmin= OX so that
R(X )(12) = �m�0H0(!12m
X) �= C[t].
If �(X ) = 1, then !12Xmin
de�nes a morphism f : Xmin ! C
whose general �ber is an elliptic curve. We have !12Xmin
= f �Hwhere H is an ample line bundle on C .
If �(X ) = 2, then !5Xmin
de�nes a morphism
' : Xmin ! Xcan � PN where Xcan = Proj(R(X )) is the
canonical model of X . The map X ! Xcan is birational andXcan has �nitely many mildly singular points (rational doublepoints).
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IntroductionSurfaces
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Birational geometry of surfaces III
More precisely we have the following:
If �(X ) = 0, then Xmin is either an abelian, hyperellyptic, K3or Enriques surface. !12
Xmin= OX so that
R(X )(12) = �m�0H0(!12m
X) �= C[t].
If �(X ) = 1, then !12Xmin
de�nes a morphism f : Xmin ! C
whose general �ber is an elliptic curve. We have !12Xmin
= f �Hwhere H is an ample line bundle on C .
If �(X ) = 2, then !5Xmin
de�nes a morphism
' : Xmin ! Xcan � PN where Xcan = Proj(R(X )) is the
canonical model of X . The map X ! Xcan is birational andXcan has �nitely many mildly singular points (rational doublepoints).
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IntroductionSurfaces
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Birational geometry of surfaces III
More precisely we have the following:
If �(X ) = 0, then Xmin is either an abelian, hyperellyptic, K3or Enriques surface. !12
Xmin= OX so that
R(X )(12) = �m�0H0(!12m
X) �= C[t].
If �(X ) = 1, then !12Xmin
de�nes a morphism f : Xmin ! C
whose general �ber is an elliptic curve. We have !12Xmin
= f �Hwhere H is an ample line bundle on C .
If �(X ) = 2, then !5Xmin
de�nes a morphism
' : Xmin ! Xcan � PN where Xcan = Proj(R(X )) is the
canonical model of X . The map X ! Xcan is birational andXcan has �nitely many mildly singular points (rational doublepoints).
Christopher Hacon Finite generation of canonical rings II
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Birational geometry of surfaces IV
If �(X ) = 2, it is natural to ask if one can understand thegeometry of Xmin more precisely. We have the following.
Theorem (Castelnuovo)
If X is a surface with H0(X ; !2X
) = 0 and H0(X ;1X) = 0, then X
is birational to P2C.
Corollary (L�uroth problem)
Let C � L � C(x ; y) be any �eld then L �= C or L �= C(x) orL �= C(x ; y). (Note that we may have a strict inclusion even ifL �= C(x ; y).)
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IntroductionSurfaces
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Birational geometry of surfaces IV
If �(X ) = 2, it is natural to ask if one can understand thegeometry of Xmin more precisely. We have the following.
Theorem (Castelnuovo)
If X is a surface with H0(X ; !2X
) = 0 and H0(X ;1X) = 0, then X
is birational to P2C.
Corollary (L�uroth problem)
Let C � L � C(x ; y) be any �eld then L �= C or L �= C(x) orL �= C(x ; y). (Note that we may have a strict inclusion even ifL �= C(x ; y).)
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Birational geometry of surfaces IV
If �(X ) = 2, it is natural to ask if one can understand thegeometry of Xmin more precisely. We have the following.
Theorem (Castelnuovo)
If X is a surface with H0(X ; !2X
) = 0 and H0(X ;1X) = 0, then X
is birational to P2C.
Corollary (L�uroth problem)
Let C � L � C(x ; y) be any �eld then L �= C or L �= C(x) orL �= C(x ; y). (Note that we may have a strict inclusion even ifL �= C(x ; y).)
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Birational geometry of surfaces V
Idea of proof. The inclusion L � C(x ; y) corresponds to amorphism X 0 ! X where C(X 0) = L and C(X ) = C(x ; y).
If dimX 0 = 0, then L �= C.If dimX 0 = 1, then H0(!x 0) = H0(1
X 0) = 0 as H0(1
X) = 0.
Therefore X 0 = P1C and L = C(X 0) = C(x).
If dimX 0 = 2, then H0(X 0; !2X 0
) = 0 and H0(X ;1X 0) = 0 so that
X 0 is birational to P2C and hence L �= C(x ; y).
Remark. The above result fails in higher dimension.
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Birational geometry of surfaces V
Idea of proof. The inclusion L � C(x ; y) corresponds to amorphism X 0 ! X where C(X 0) = L and C(X ) = C(x ; y).If dimX 0 = 0, then L �= C.
If dimX 0 = 1, then H0(!x 0) = H0(1X 0) = 0 as H0(1
X) = 0.
Therefore X 0 = P1C and L = C(X 0) = C(x).
If dimX 0 = 2, then H0(X 0; !2X 0
) = 0 and H0(X ;1X 0) = 0 so that
X 0 is birational to P2C and hence L �= C(x ; y).
Remark. The above result fails in higher dimension.
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Birational geometry of surfaces V
Idea of proof. The inclusion L � C(x ; y) corresponds to amorphism X 0 ! X where C(X 0) = L and C(X ) = C(x ; y).If dimX 0 = 0, then L �= C.If dimX 0 = 1, then H0(!x 0) = H0(1
X 0) = 0 as H0(1
X) = 0.
Therefore X 0 = P1C and L = C(X 0) = C(x).
If dimX 0 = 2, then H0(X 0; !2X 0
) = 0 and H0(X ;1X 0) = 0 so that
X 0 is birational to P2C and hence L �= C(x ; y).
Remark. The above result fails in higher dimension.
Christopher Hacon Finite generation of canonical rings II
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IntroductionSurfaces
Higher dimensional preview
Birational geometry of surfaces V
Idea of proof. The inclusion L � C(x ; y) corresponds to amorphism X 0 ! X where C(X 0) = L and C(X ) = C(x ; y).If dimX 0 = 0, then L �= C.If dimX 0 = 1, then H0(!x 0) = H0(1
X 0) = 0 as H0(1
X) = 0.
Therefore X 0 = P1C and L = C(X 0) = C(x).
If dimX 0 = 2, then H0(X 0; !2X 0
) = 0 and H0(X ;1X 0) = 0 so that
X 0 is birational to P2C and hence L �= C(x ; y).
Remark. The above result fails in higher dimension.
Christopher Hacon Finite generation of canonical rings II
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IntroductionSurfaces
Higher dimensional preview
Birational geometry of surfaces VI
In general it is impossible to classify surfaces with �(X ) = 2.
It is not even known if there exists a surface X homeomorphicto P2
C with �(X ) = 2.
Never-the-less we have a good qualitative understanding ofthe birational geometry of X .
In the next lecture, I will illustrate how to extend theKodaira-Enriques surface classi�cation to arbitrary dimension.
Christopher Hacon Finite generation of canonical rings II
![Page 73: Finite generation of canonical rings II - Math](https://reader031.fdocuments.in/reader031/viewer/2022012021/6168a424d394e9041f716e35/html5/thumbnails/73.jpg)
IntroductionSurfaces
Higher dimensional preview
Birational geometry of surfaces VI
In general it is impossible to classify surfaces with �(X ) = 2.
It is not even known if there exists a surface X homeomorphicto P2
C with �(X ) = 2.
Never-the-less we have a good qualitative understanding ofthe birational geometry of X .
In the next lecture, I will illustrate how to extend theKodaira-Enriques surface classi�cation to arbitrary dimension.
Christopher Hacon Finite generation of canonical rings II
![Page 74: Finite generation of canonical rings II - Math](https://reader031.fdocuments.in/reader031/viewer/2022012021/6168a424d394e9041f716e35/html5/thumbnails/74.jpg)
IntroductionSurfaces
Higher dimensional preview
Birational geometry of surfaces VI
In general it is impossible to classify surfaces with �(X ) = 2.
It is not even known if there exists a surface X homeomorphicto P2
C with �(X ) = 2.
Never-the-less we have a good qualitative understanding ofthe birational geometry of X .
In the next lecture, I will illustrate how to extend theKodaira-Enriques surface classi�cation to arbitrary dimension.
Christopher Hacon Finite generation of canonical rings II
![Page 75: Finite generation of canonical rings II - Math](https://reader031.fdocuments.in/reader031/viewer/2022012021/6168a424d394e9041f716e35/html5/thumbnails/75.jpg)
IntroductionSurfaces
Higher dimensional preview
Birational geometry of surfaces VI
In general it is impossible to classify surfaces with �(X ) = 2.
It is not even known if there exists a surface X homeomorphicto P2
C with �(X ) = 2.
Never-the-less we have a good qualitative understanding ofthe birational geometry of X .
In the next lecture, I will illustrate how to extend theKodaira-Enriques surface classi�cation to arbitrary dimension.
Christopher Hacon Finite generation of canonical rings II
![Page 76: Finite generation of canonical rings II - Math](https://reader031.fdocuments.in/reader031/viewer/2022012021/6168a424d394e9041f716e35/html5/thumbnails/76.jpg)
IntroductionSurfaces
Higher dimensional preview
Outline of the talk
1 Introduction
2 Surfaces
3 Higher dimensional preview
Christopher Hacon Finite generation of canonical rings II
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IntroductionSurfaces
Higher dimensional preview
Higher dimensional varieties
Theorem (Birkar-Cascini-Hacon-McKernan - Siu)
Let X be a smooth complex projective variety of general type (i.e.�(X ) = dim(X )) then X has a canonical model
Xcan = ProjR(X ):
One can also show that X has a minimal model Xmin.
Christopher Hacon Finite generation of canonical rings II
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IntroductionSurfaces
Higher dimensional preview
Higher dimensional varieties
Theorem (Birkar-Cascini-Hacon-McKernan - Siu)
Let X be a smooth complex projective variety of general type (i.e.�(X ) = dim(X )) then X has a canonical model
Xcan = ProjR(X ):
One can also show that X has a minimal model Xmin.
Christopher Hacon Finite generation of canonical rings II
![Page 79: Finite generation of canonical rings II - Math](https://reader031.fdocuments.in/reader031/viewer/2022012021/6168a424d394e9041f716e35/html5/thumbnails/79.jpg)
IntroductionSurfaces
Higher dimensional preview
Higher dimensional varieties
There are several new feaures Xcan and Xmin may have mildsingularities.
Xmin may not be unique.
The map X 99K Xmin is given by a carefully chosen �nitesequence of divisorial contractions (the analog of contractinga �1 curve) and ips (a new operation).
In dimension 3, ips were constructed by S. Mori,in dimension 4 by V. Shokurov, andin dimension � 5 by Birkar-Cascini-Hacon-McKernan usingtechniques of Shokurov and Siu.
Christopher Hacon Finite generation of canonical rings II
![Page 80: Finite generation of canonical rings II - Math](https://reader031.fdocuments.in/reader031/viewer/2022012021/6168a424d394e9041f716e35/html5/thumbnails/80.jpg)
IntroductionSurfaces
Higher dimensional preview
Higher dimensional varieties
There are several new feaures Xcan and Xmin may have mildsingularities.
Xmin may not be unique.
The map X 99K Xmin is given by a carefully chosen �nitesequence of divisorial contractions (the analog of contractinga �1 curve) and ips (a new operation).
In dimension 3, ips were constructed by S. Mori,in dimension 4 by V. Shokurov, andin dimension � 5 by Birkar-Cascini-Hacon-McKernan usingtechniques of Shokurov and Siu.
Christopher Hacon Finite generation of canonical rings II
![Page 81: Finite generation of canonical rings II - Math](https://reader031.fdocuments.in/reader031/viewer/2022012021/6168a424d394e9041f716e35/html5/thumbnails/81.jpg)
IntroductionSurfaces
Higher dimensional preview
Higher dimensional varieties
There are several new feaures Xcan and Xmin may have mildsingularities.
Xmin may not be unique.
The map X 99K Xmin is given by a carefully chosen �nitesequence of divisorial contractions (the analog of contractinga �1 curve) and ips (a new operation).
In dimension 3, ips were constructed by S. Mori,in dimension 4 by V. Shokurov, andin dimension � 5 by Birkar-Cascini-Hacon-McKernan usingtechniques of Shokurov and Siu.
Christopher Hacon Finite generation of canonical rings II
![Page 82: Finite generation of canonical rings II - Math](https://reader031.fdocuments.in/reader031/viewer/2022012021/6168a424d394e9041f716e35/html5/thumbnails/82.jpg)
IntroductionSurfaces
Higher dimensional preview
Higher dimensional varieties
There are several new feaures Xcan and Xmin may have mildsingularities.
Xmin may not be unique.
The map X 99K Xmin is given by a carefully chosen �nitesequence of divisorial contractions (the analog of contractinga �1 curve) and ips (a new operation).
In dimension 3, ips were constructed by S. Mori,in dimension 4 by V. Shokurov, andin dimension � 5 by Birkar-Cascini-Hacon-McKernan usingtechniques of Shokurov and Siu.
Christopher Hacon Finite generation of canonical rings II