Birational geometry and arithmetic - Durham University
Transcript of Birational geometry and arithmetic - Durham University
![Page 1: Birational geometry and arithmetic - Durham University](https://reader030.fdocuments.us/reader030/viewer/2022012418/617328d702cb5f7eb01f19e4/html5/thumbnails/1.jpg)
Birational geometry and arithmetic
July 2012
![Page 2: Birational geometry and arithmetic - Durham University](https://reader030.fdocuments.us/reader030/viewer/2022012418/617328d702cb5f7eb01f19e4/html5/thumbnails/2.jpg)
Basic questions
Let F be a field and X a smooth projective algebraic variety over F .
We are interested in rational points X (F ). Specifically,
Existence
Density
Distribution with respect to heights
Of particular interest are small fields:
F = Fq, Q, Fq(t), C(t), ...
Main idea
Arithmetic properties are governed by global geometric invariants andthe properties of the ground field F .
Introduction
![Page 3: Birational geometry and arithmetic - Durham University](https://reader030.fdocuments.us/reader030/viewer/2022012418/617328d702cb5f7eb01f19e4/html5/thumbnails/3.jpg)
Basic questions
Let F be a field and X a smooth projective algebraic variety over F .We are interested in rational points X (F ). Specifically,
Existence
Density
Distribution with respect to heights
Of particular interest are small fields:
F = Fq, Q, Fq(t), C(t), ...
Main idea
Arithmetic properties are governed by global geometric invariants andthe properties of the ground field F .
Introduction
![Page 4: Birational geometry and arithmetic - Durham University](https://reader030.fdocuments.us/reader030/viewer/2022012418/617328d702cb5f7eb01f19e4/html5/thumbnails/4.jpg)
Basic questions
Let F be a field and X a smooth projective algebraic variety over F .We are interested in rational points X (F ). Specifically,
Existence
Density
Distribution with respect to heights
Of particular interest are small fields:
F = Fq, Q, Fq(t), C(t), ...
Main idea
Arithmetic properties are governed by global geometric invariants andthe properties of the ground field F .
Introduction
![Page 5: Birational geometry and arithmetic - Durham University](https://reader030.fdocuments.us/reader030/viewer/2022012418/617328d702cb5f7eb01f19e4/html5/thumbnails/5.jpg)
Basic questions
Let F be a field and X a smooth projective algebraic variety over F .We are interested in rational points X (F ). Specifically,
Existence
Density
Distribution with respect to heights
Of particular interest are small fields:
F = Fq, Q, Fq(t), C(t), ...
Main idea
Arithmetic properties are governed by global geometric invariants andthe properties of the ground field F .
Introduction
![Page 6: Birational geometry and arithmetic - Durham University](https://reader030.fdocuments.us/reader030/viewer/2022012418/617328d702cb5f7eb01f19e4/html5/thumbnails/6.jpg)
Classification schemes
via dimension: curves, surfaces, ...
via degree: Fano, general type, intermediate typeXd ⊂ Pn, with d < n + 1, d > n + 1 or d = n + 1
how close to Pn: rational, unirational, rationally connectedIn small dimensions some of these notions coincide, e.g., indimension 2 and over algebraically closed fields of characteristiczero
rational = unirational = rationally connected
Small degree surfaces (Del Pezzo surfaces) over algebraicallyclosed fields are rational. Cubic surfaces with a rational point areunirational. A Del Pezzo surface of degree d = 1 always has apoint. Is it unirational?
Introduction
![Page 7: Birational geometry and arithmetic - Durham University](https://reader030.fdocuments.us/reader030/viewer/2022012418/617328d702cb5f7eb01f19e4/html5/thumbnails/7.jpg)
Classification schemes
via dimension: curves, surfaces, ...
via degree: Fano, general type, intermediate type
Xd ⊂ Pn, with d < n + 1, d > n + 1 or d = n + 1
how close to Pn: rational, unirational, rationally connectedIn small dimensions some of these notions coincide, e.g., indimension 2 and over algebraically closed fields of characteristiczero
rational = unirational = rationally connected
Small degree surfaces (Del Pezzo surfaces) over algebraicallyclosed fields are rational. Cubic surfaces with a rational point areunirational. A Del Pezzo surface of degree d = 1 always has apoint. Is it unirational?
Introduction
![Page 8: Birational geometry and arithmetic - Durham University](https://reader030.fdocuments.us/reader030/viewer/2022012418/617328d702cb5f7eb01f19e4/html5/thumbnails/8.jpg)
Classification schemes
via dimension: curves, surfaces, ...
via degree: Fano, general type, intermediate typeXd ⊂ Pn, with d < n + 1, d > n + 1 or d = n + 1
how close to Pn: rational, unirational, rationally connected
In small dimensions some of these notions coincide, e.g., indimension 2 and over algebraically closed fields of characteristiczero
rational = unirational = rationally connected
Small degree surfaces (Del Pezzo surfaces) over algebraicallyclosed fields are rational. Cubic surfaces with a rational point areunirational. A Del Pezzo surface of degree d = 1 always has apoint. Is it unirational?
Introduction
![Page 9: Birational geometry and arithmetic - Durham University](https://reader030.fdocuments.us/reader030/viewer/2022012418/617328d702cb5f7eb01f19e4/html5/thumbnails/9.jpg)
Classification schemes
via dimension: curves, surfaces, ...
via degree: Fano, general type, intermediate typeXd ⊂ Pn, with d < n + 1, d > n + 1 or d = n + 1
how close to Pn: rational, unirational, rationally connectedIn small dimensions some of these notions coincide, e.g., indimension 2 and over algebraically closed fields of characteristiczero
rational = unirational = rationally connected
Small degree surfaces (Del Pezzo surfaces) over algebraicallyclosed fields are rational. Cubic surfaces with a rational point areunirational. A Del Pezzo surface of degree d = 1 always has apoint. Is it unirational?
Introduction
![Page 10: Birational geometry and arithmetic - Durham University](https://reader030.fdocuments.us/reader030/viewer/2022012418/617328d702cb5f7eb01f19e4/html5/thumbnails/10.jpg)
Classification schemes
via dimension: curves, surfaces, ...
via degree: Fano, general type, intermediate typeXd ⊂ Pn, with d < n + 1, d > n + 1 or d = n + 1
how close to Pn: rational, unirational, rationally connectedIn small dimensions some of these notions coincide, e.g., indimension 2 and over algebraically closed fields of characteristiczero
rational = unirational = rationally connected
Small degree surfaces (Del Pezzo surfaces) over algebraicallyclosed fields are rational. Cubic surfaces with a rational point areunirational.
A Del Pezzo surface of degree d = 1 always has apoint. Is it unirational?
Introduction
![Page 11: Birational geometry and arithmetic - Durham University](https://reader030.fdocuments.us/reader030/viewer/2022012418/617328d702cb5f7eb01f19e4/html5/thumbnails/11.jpg)
Classification schemes
via dimension: curves, surfaces, ...
via degree: Fano, general type, intermediate typeXd ⊂ Pn, with d < n + 1, d > n + 1 or d = n + 1
how close to Pn: rational, unirational, rationally connectedIn small dimensions some of these notions coincide, e.g., indimension 2 and over algebraically closed fields of characteristiczero
rational = unirational = rationally connected
Small degree surfaces (Del Pezzo surfaces) over algebraicallyclosed fields are rational. Cubic surfaces with a rational point areunirational. A Del Pezzo surface of degree d = 1 always has apoint. Is it unirational?
Introduction
![Page 12: Birational geometry and arithmetic - Durham University](https://reader030.fdocuments.us/reader030/viewer/2022012418/617328d702cb5f7eb01f19e4/html5/thumbnails/12.jpg)
Basic invariants
Picard group Pic(X ), canonical class KX , cones of ample andeffective divisors
Fibration structures: Conic bundles, elliptic fibrations, ...
Over nonclosed fields F :
Forms and Galois cohomology
Brauer group Br(X )
In some cases, these are effectively computable.
Introduction
![Page 13: Birational geometry and arithmetic - Durham University](https://reader030.fdocuments.us/reader030/viewer/2022012418/617328d702cb5f7eb01f19e4/html5/thumbnails/13.jpg)
Basic invariants
Picard group Pic(X ), canonical class KX , cones of ample andeffective divisors
Fibration structures: Conic bundles, elliptic fibrations, ...
Over nonclosed fields F :
Forms and Galois cohomology
Brauer group Br(X )
In some cases, these are effectively computable.
Introduction
![Page 14: Birational geometry and arithmetic - Durham University](https://reader030.fdocuments.us/reader030/viewer/2022012418/617328d702cb5f7eb01f19e4/html5/thumbnails/14.jpg)
Basic invariants
Picard group Pic(X ), canonical class KX , cones of ample andeffective divisors
Fibration structures: Conic bundles, elliptic fibrations, ...
Over nonclosed fields F :
Forms and Galois cohomology
Brauer group Br(X )
In some cases, these are effectively computable.
Introduction
![Page 15: Birational geometry and arithmetic - Durham University](https://reader030.fdocuments.us/reader030/viewer/2022012418/617328d702cb5f7eb01f19e4/html5/thumbnails/15.jpg)
Basic invariants
Picard group Pic(X ), canonical class KX , cones of ample andeffective divisors
Fibration structures: Conic bundles, elliptic fibrations, ...
Over nonclosed fields F :
Forms and Galois cohomology
Brauer group Br(X )
In some cases, these are effectively computable.
Introduction
![Page 16: Birational geometry and arithmetic - Durham University](https://reader030.fdocuments.us/reader030/viewer/2022012418/617328d702cb5f7eb01f19e4/html5/thumbnails/16.jpg)
Starting point: Curves over number fields
What do we know about curves over number fields?
g = 0: one can decide when X (F ) 6= ∅ (local-global principle), ifX (F ) 6= ∅, then X (F ) is infinite and one has a goodunderstanding of how X (F ) is distributed
g = 1: either X (F ) = ∅, or X (F ) is finite, or infinite; no effectivealgorithms to decide, or to describe X (F ) (at present)
g ≥ 2: #X (F ) <∞, no effective algorithm to determine X (F )(effective Mordell?)
Curves
![Page 17: Birational geometry and arithmetic - Durham University](https://reader030.fdocuments.us/reader030/viewer/2022012418/617328d702cb5f7eb01f19e4/html5/thumbnails/17.jpg)
Starting point: Curves over number fields
What do we know about curves over number fields?
g = 0: one can decide when X (F ) 6= ∅ (local-global principle), ifX (F ) 6= ∅, then X (F ) is infinite and one has a goodunderstanding of how X (F ) is distributed
g = 1: either X (F ) = ∅, or X (F ) is finite, or infinite; no effectivealgorithms to decide, or to describe X (F ) (at present)
g ≥ 2: #X (F ) <∞, no effective algorithm to determine X (F )(effective Mordell?)
Curves
![Page 18: Birational geometry and arithmetic - Durham University](https://reader030.fdocuments.us/reader030/viewer/2022012418/617328d702cb5f7eb01f19e4/html5/thumbnails/18.jpg)
Starting point: Curves over number fields
What do we know about curves over number fields?
g = 0: one can decide when X (F ) 6= ∅ (local-global principle), ifX (F ) 6= ∅, then X (F ) is infinite and one has a goodunderstanding of how X (F ) is distributed
g = 1: either X (F ) = ∅, or X (F ) is finite, or infinite; no effectivealgorithms to decide, or to describe X (F ) (at present)
g ≥ 2: #X (F ) <∞, no effective algorithm to determine X (F )(effective Mordell?)
Curves
![Page 19: Birational geometry and arithmetic - Durham University](https://reader030.fdocuments.us/reader030/viewer/2022012418/617328d702cb5f7eb01f19e4/html5/thumbnails/19.jpg)
Curve covers (joint with F. Bogomolov)
We say that C ⇒ C ′ if there exist an etale cover C → C and asurjection C → C ′.
Theorem (2001)
Let C be a hyperelliptic curve over Fp of genus ≥ 2 and let C ′ beany curve. Then C ⇒ C ′.
Let C be a hyperelliptic curve over Q of genus ≥ 2 and C6 thecurve y2 = x6 − 1. Then C ⇒ C6.
The cover C → C is explicit, so that effective Mordell for C6 implieseffective Mordell for all hyperbolic hyperelliptic curves.
Conjecture
If C ,C ′ are curves of genus ≥ 2 over Fp or Q then C ⇔ C ′.
Curves
![Page 20: Birational geometry and arithmetic - Durham University](https://reader030.fdocuments.us/reader030/viewer/2022012418/617328d702cb5f7eb01f19e4/html5/thumbnails/20.jpg)
Curve covers (joint with F. Bogomolov)
We say that C ⇒ C ′ if there exist an etale cover C → C and asurjection C → C ′.
Theorem (2001)
Let C be a hyperelliptic curve over Fp of genus ≥ 2 and let C ′ beany curve. Then C ⇒ C ′.
Let C be a hyperelliptic curve over Q of genus ≥ 2 and C6 thecurve y2 = x6 − 1. Then C ⇒ C6.
The cover C → C is explicit, so that effective Mordell for C6 implieseffective Mordell for all hyperbolic hyperelliptic curves.
Conjecture
If C ,C ′ are curves of genus ≥ 2 over Fp or Q then C ⇔ C ′.
Curves
![Page 21: Birational geometry and arithmetic - Durham University](https://reader030.fdocuments.us/reader030/viewer/2022012418/617328d702cb5f7eb01f19e4/html5/thumbnails/21.jpg)
Curve covers (joint with F. Bogomolov)
We say that C ⇒ C ′ if there exist an etale cover C → C and asurjection C → C ′.
Theorem (2001)
Let C be a hyperelliptic curve over Fp of genus ≥ 2 and let C ′ beany curve. Then C ⇒ C ′.
Let C be a hyperelliptic curve over Q of genus ≥ 2 and C6 thecurve y2 = x6 − 1. Then C ⇒ C6.
The cover C → C is explicit, so that effective Mordell for C6 implieseffective Mordell for all hyperbolic hyperelliptic curves.
Conjecture
If C ,C ′ are curves of genus ≥ 2 over Fp or Q then C ⇔ C ′.
Curves
![Page 22: Birational geometry and arithmetic - Durham University](https://reader030.fdocuments.us/reader030/viewer/2022012418/617328d702cb5f7eb01f19e4/html5/thumbnails/22.jpg)
Curve covers (joint with F. Bogomolov)
We say that C ⇒ C ′ if there exist an etale cover C → C and asurjection C → C ′.
Theorem (2001)
Let C be a hyperelliptic curve over Fp of genus ≥ 2 and let C ′ beany curve. Then C ⇒ C ′.
Let C be a hyperelliptic curve over Q of genus ≥ 2 and C6 thecurve y2 = x6 − 1. Then C ⇒ C6.
The cover C → C is explicit, so that effective Mordell for C6 implieseffective Mordell for all hyperbolic hyperelliptic curves.
Conjecture
If C ,C ′ are curves of genus ≥ 2 over Fp or Q then C ⇔ C ′.
Curves
![Page 23: Birational geometry and arithmetic - Durham University](https://reader030.fdocuments.us/reader030/viewer/2022012418/617328d702cb5f7eb01f19e4/html5/thumbnails/23.jpg)
Curve covers (joint with F. Bogomolov)
We say that C ⇒ C ′ if there exist an etale cover C → C and asurjection C → C ′.
Theorem (2001)
Let C be a hyperelliptic curve over Fp of genus ≥ 2 and let C ′ beany curve. Then C ⇒ C ′.
Let C be a hyperelliptic curve over Q of genus ≥ 2 and C6 thecurve y2 = x6 − 1. Then C ⇒ C6.
The cover C → C is explicit, so that effective Mordell for C6 implieseffective Mordell for all hyperbolic hyperelliptic curves.
Conjecture
If C ,C ′ are curves of genus ≥ 2 over Fp or Q then C ⇔ C ′.
Curves
![Page 24: Birational geometry and arithmetic - Durham University](https://reader030.fdocuments.us/reader030/viewer/2022012418/617328d702cb5f7eb01f19e4/html5/thumbnails/24.jpg)
Prototype: hypersurfaces Xf ⊂ Pn over Q
Birch 1961
If n� 2deg(f ) and Xf is smooth then:
if there are solutions in Qp and in R then there are solutions in Qasymptotic formulas
a positive proportion of hypersurfaces over Q have no localobstructions (Poonen-Voloch + Katz, 2003)
the method works over Fq[t] as well
Introduction: rational points on hypersurfaces
![Page 25: Birational geometry and arithmetic - Durham University](https://reader030.fdocuments.us/reader030/viewer/2022012418/617328d702cb5f7eb01f19e4/html5/thumbnails/25.jpg)
Prototype: hypersurfaces Xf ⊂ Pn over Q
Birch 1961
If n� 2deg(f ) and Xf is smooth then:
if there are solutions in Qp and in R then there are solutions in Qasymptotic formulas
a positive proportion of hypersurfaces over Q have no localobstructions (Poonen-Voloch + Katz, 2003)
the method works over Fq[t] as well
Introduction: rational points on hypersurfaces
![Page 26: Birational geometry and arithmetic - Durham University](https://reader030.fdocuments.us/reader030/viewer/2022012418/617328d702cb5f7eb01f19e4/html5/thumbnails/26.jpg)
Prototype: hypersurfaces Xf ⊂ Pn over Q
Birch 1961
If n� 2deg(f ) and Xf is smooth then:
if there are solutions in Qp and in R then there are solutions in Qasymptotic formulas
a positive proportion of hypersurfaces over Q have no localobstructions (Poonen-Voloch + Katz, 2003)
the method works over Fq[t] as well
Introduction: rational points on hypersurfaces
![Page 27: Birational geometry and arithmetic - Durham University](https://reader030.fdocuments.us/reader030/viewer/2022012418/617328d702cb5f7eb01f19e4/html5/thumbnails/27.jpg)
Heuristic
Given: f ∈ Z[x0, . . . , xn] homogeneous of degree deg(f ).
We have |f (x)| = O(Bdeg(f )), for ‖x‖ := maxj(|xj |) ≤ B.
May (?) assume that the probability of f (x) = 0 is B−deg(f ).
There are Bn+1 “events” with ‖x‖ ≤ B.
We expect Bn+1−d solutions with ‖x‖ ≤ B.
Hope: reasonable at least when n + 1− d ≥ 0.
Introduction: rational points on hypersurfaces
![Page 28: Birational geometry and arithmetic - Durham University](https://reader030.fdocuments.us/reader030/viewer/2022012418/617328d702cb5f7eb01f19e4/html5/thumbnails/28.jpg)
Xf ⊂ Pn over C(t)
Theorem
If deg(f ) ≤ n then Xf (C(t)) 6= ∅.
Proof: Insert xj = xj(t) ∈ C[t], of degree e, into
f =∑J
fJxJ = 0, |J| = deg(f ).
This gives a system of e · deg(f ) + const equations in (e + 1)(n + 1)variables. This system is solvable for e � 0, provided deg(f ) ≤ n.
Introduction: rational points on hypersurfaces
![Page 29: Birational geometry and arithmetic - Durham University](https://reader030.fdocuments.us/reader030/viewer/2022012418/617328d702cb5f7eb01f19e4/html5/thumbnails/29.jpg)
Existence of rational points
Theorem (Esnault 2001)
Every smooth rationally connected variety over a finite field has arational point.
Theorem (Graber-Harris-Starr 2001)
Every smooth rationally connected variety over the function field of acurve over an algebraically closed field has a rational point.
Over number fields and higher dimensional function fields, there existlocal and global obstructions to the existence of rational points.
Existence of points
![Page 30: Birational geometry and arithmetic - Durham University](https://reader030.fdocuments.us/reader030/viewer/2022012418/617328d702cb5f7eb01f19e4/html5/thumbnails/30.jpg)
Existence of rational points
Theorem (Esnault 2001)
Every smooth rationally connected variety over a finite field has arational point.
Theorem (Graber-Harris-Starr 2001)
Every smooth rationally connected variety over the function field of acurve over an algebraically closed field has a rational point.
Over number fields and higher dimensional function fields, there existlocal and global obstructions to the existence of rational points.
Existence of points
![Page 31: Birational geometry and arithmetic - Durham University](https://reader030.fdocuments.us/reader030/viewer/2022012418/617328d702cb5f7eb01f19e4/html5/thumbnails/31.jpg)
Existence of rational points
Theorem (Esnault 2001)
Every smooth rationally connected variety over a finite field has arational point.
Theorem (Graber-Harris-Starr 2001)
Every smooth rationally connected variety over the function field of acurve over an algebraically closed field has a rational point.
Over number fields and higher dimensional function fields, there existlocal and global obstructions to the existence of rational points.
Existence of points
![Page 32: Birational geometry and arithmetic - Durham University](https://reader030.fdocuments.us/reader030/viewer/2022012418/617328d702cb5f7eb01f19e4/html5/thumbnails/32.jpg)
Hasse principle
HP
X (Fv ) 6= ∅ ∀v ⇒ X (F ) 6= ∅
Basic examples: Quadrics, hypersurfaces of small degree
Counterexamples:
Iskovskikh 1971: The conic bundle X → P1 given by
x2 + y2 = f (t), f (t) = (t2 − 2)(3− t2).
Cassels, Guy 1966: The cubic surface
5x3 + 9y3 + 10z3 + 12t3 = 0.
The proofs use basic algebraic number theory: quadratic and cubicreciprocity, divisibility of class numbers.
Existence of points
![Page 33: Birational geometry and arithmetic - Durham University](https://reader030.fdocuments.us/reader030/viewer/2022012418/617328d702cb5f7eb01f19e4/html5/thumbnails/33.jpg)
Hasse principle
HP
X (Fv ) 6= ∅ ∀v ⇒ X (F ) 6= ∅
Basic examples: Quadrics, hypersurfaces of small degree
Counterexamples:
Iskovskikh 1971: The conic bundle X → P1 given by
x2 + y2 = f (t), f (t) = (t2 − 2)(3− t2).
Cassels, Guy 1966: The cubic surface
5x3 + 9y3 + 10z3 + 12t3 = 0.
The proofs use basic algebraic number theory: quadratic and cubicreciprocity, divisibility of class numbers.
Existence of points
![Page 34: Birational geometry and arithmetic - Durham University](https://reader030.fdocuments.us/reader030/viewer/2022012418/617328d702cb5f7eb01f19e4/html5/thumbnails/34.jpg)
Hasse principle
HP
X (Fv ) 6= ∅ ∀v ⇒ X (F ) 6= ∅
Basic examples: Quadrics, hypersurfaces of small degree
Counterexamples:
Iskovskikh 1971: The conic bundle X → P1 given by
x2 + y2 = f (t), f (t) = (t2 − 2)(3− t2).
Cassels, Guy 1966: The cubic surface
5x3 + 9y3 + 10z3 + 12t3 = 0.
The proofs use basic algebraic number theory: quadratic and cubicreciprocity, divisibility of class numbers.
Existence of points
![Page 35: Birational geometry and arithmetic - Durham University](https://reader030.fdocuments.us/reader030/viewer/2022012418/617328d702cb5f7eb01f19e4/html5/thumbnails/35.jpg)
Hasse principle
HP
X (Fv ) 6= ∅ ∀v ⇒ X (F ) 6= ∅
Basic examples: Quadrics, hypersurfaces of small degree
Counterexamples:
Iskovskikh 1971: The conic bundle X → P1 given by
x2 + y2 = f (t), f (t) = (t2 − 2)(3− t2).
Cassels, Guy 1966: The cubic surface
5x3 + 9y3 + 10z3 + 12t3 = 0.
The proofs use basic algebraic number theory: quadratic and cubicreciprocity, divisibility of class numbers.
Existence of points
![Page 36: Birational geometry and arithmetic - Durham University](https://reader030.fdocuments.us/reader030/viewer/2022012418/617328d702cb5f7eb01f19e4/html5/thumbnails/36.jpg)
Hasse principle
HP
X (Fv ) 6= ∅ ∀v ⇒ X (F ) 6= ∅
Basic examples: Quadrics, hypersurfaces of small degree
Counterexamples:
Iskovskikh 1971: The conic bundle X → P1 given by
x2 + y2 = f (t), f (t) = (t2 − 2)(3− t2).
Cassels, Guy 1966: The cubic surface
5x3 + 9y3 + 10z3 + 12t3 = 0.
The proofs use basic algebraic number theory: quadratic and cubicreciprocity, divisibility of class numbers.
Existence of points
![Page 37: Birational geometry and arithmetic - Durham University](https://reader030.fdocuments.us/reader030/viewer/2022012418/617328d702cb5f7eb01f19e4/html5/thumbnails/37.jpg)
Brauer-Manin obstruction
Br(XF ) //
x
��
⊕v Br(XFv )
(xv )v��
0 // Br(F ) //⊕
v Br(Fv )
∑v invv // Q/Z // 0,
We haveX (F ) ⊂ X (F ) ⊆ X (AF )Br ⊆ X (AF ),
where
X (AF )Br := ∩A∈Br(X ){(xv )v ∈ X (AF ) |∑v
inv(A(xv )) = 0}.
Manin’s formulation gives a more systematic approach to identifyingthe algebraic structure behind the obstruction.
Existence of points
![Page 38: Birational geometry and arithmetic - Durham University](https://reader030.fdocuments.us/reader030/viewer/2022012418/617328d702cb5f7eb01f19e4/html5/thumbnails/38.jpg)
Brauer-Manin obstruction
Br(XF ) //
x
��
⊕v Br(XFv )
(xv )v��
0 // Br(F ) //⊕
v Br(Fv )
∑v invv // Q/Z // 0,
We haveX (F ) ⊂ X (F ) ⊆ X (AF )Br ⊆ X (AF ),
where
X (AF )Br := ∩A∈Br(X ){(xv )v ∈ X (AF ) |∑v
inv(A(xv )) = 0}.
Manin’s formulation gives a more systematic approach to identifyingthe algebraic structure behind the obstruction.
Existence of points
![Page 39: Birational geometry and arithmetic - Durham University](https://reader030.fdocuments.us/reader030/viewer/2022012418/617328d702cb5f7eb01f19e4/html5/thumbnails/39.jpg)
Brauer-Manin obstruction
Br(XF ) //
x
��
⊕v Br(XFv )
(xv )v��
0 // Br(F ) //⊕
v Br(Fv )
∑v invv // Q/Z // 0,
We haveX (F ) ⊂ X (F ) ⊆ X (AF )Br ⊆ X (AF ),
where
X (AF )Br := ∩A∈Br(X ){(xv )v ∈ X (AF ) |∑v
inv(A(xv )) = 0}.
Manin’s formulation gives a more systematic approach to identifyingthe algebraic structure behind the obstruction.
Existence of points
![Page 40: Birational geometry and arithmetic - Durham University](https://reader030.fdocuments.us/reader030/viewer/2022012418/617328d702cb5f7eb01f19e4/html5/thumbnails/40.jpg)
Effectivity of Brauer-Manin obstructions
The obstruction group for geometrically rational surfaces
Br(XF )/Br(F ) = H1(Gal(F/F ),Pic(X )).
Kresch-T. 2006
Let X ⊂ Pn be a geometrically rational surface over a number field F .Then there is an effective algorithm to compute X (AF )Br.
Kresch-T. 2010
Let X ⊂ Pn be a surface over a number field F . Assume that
the geometric Picard group of X is torsion free and is generated byfinitely many divisors, each with a given set of defining equations
Br(X ) can be bounded effectively.
Then there is an effective algorithm to compute X (AF )Br.
Existence of points
![Page 41: Birational geometry and arithmetic - Durham University](https://reader030.fdocuments.us/reader030/viewer/2022012418/617328d702cb5f7eb01f19e4/html5/thumbnails/41.jpg)
Effectivity of Brauer-Manin obstructions
The obstruction group for geometrically rational surfaces
Br(XF )/Br(F ) = H1(Gal(F/F ),Pic(X )).
Kresch-T. 2006
Let X ⊂ Pn be a geometrically rational surface over a number field F .Then there is an effective algorithm to compute X (AF )Br.
Kresch-T. 2010
Let X ⊂ Pn be a surface over a number field F . Assume that
the geometric Picard group of X is torsion free and is generated byfinitely many divisors, each with a given set of defining equations
Br(X ) can be bounded effectively.
Then there is an effective algorithm to compute X (AF )Br.
Existence of points
![Page 42: Birational geometry and arithmetic - Durham University](https://reader030.fdocuments.us/reader030/viewer/2022012418/617328d702cb5f7eb01f19e4/html5/thumbnails/42.jpg)
Effectivity of Brauer-Manin obstructions
The obstruction group for geometrically rational surfaces
Br(XF )/Br(F ) = H1(Gal(F/F ),Pic(X )).
Kresch-T. 2006
Let X ⊂ Pn be a geometrically rational surface over a number field F .Then there is an effective algorithm to compute X (AF )Br.
Kresch-T. 2010
Let X ⊂ Pn be a surface over a number field F . Assume that
the geometric Picard group of X is torsion free and is generated byfinitely many divisors, each with a given set of defining equations
Br(X ) can be bounded effectively.
Then there is an effective algorithm to compute X (AF )Br.
Existence of points
![Page 43: Birational geometry and arithmetic - Durham University](https://reader030.fdocuments.us/reader030/viewer/2022012418/617328d702cb5f7eb01f19e4/html5/thumbnails/43.jpg)
Effectivity of Brauer-Manin obstructions
Hassett-Kresch-T. 2012
Let X be a K3 surface over a number field F of degree 2. Then thereexists an effective algorithm to compute:
Pic(X ), together with the Galois action;
Br(X )/Br(F ) and X (AF )Br.
Previously:
computation of Pic(X ) on some Kummer K3 surfaces: Elsenhans,Jahnel, van Luijk
computation of X (AF )Br for some Kummer surfaces (diagonalquartics): Bright, Skorobogatov, Swinnerton-Dyer, Ieronymou, ...
computations with Br(X )[2] on general degree two K3 surfaces:Hassett, Varilly-Alvarado
Finiteness of Br(X )/Br(F ) for all K3 surfaces over number fields(Skorobogatov-Zarhin 2007)
Existence of points
![Page 44: Birational geometry and arithmetic - Durham University](https://reader030.fdocuments.us/reader030/viewer/2022012418/617328d702cb5f7eb01f19e4/html5/thumbnails/44.jpg)
Effectivity of Brauer-Manin obstructions
Hassett-Kresch-T. 2012
Let X be a K3 surface over a number field F of degree 2. Then thereexists an effective algorithm to compute:
Pic(X ), together with the Galois action;
Br(X )/Br(F ) and X (AF )Br.
Previously:
computation of Pic(X ) on some Kummer K3 surfaces: Elsenhans,Jahnel, van Luijk
computation of X (AF )Br for some Kummer surfaces (diagonalquartics): Bright, Skorobogatov, Swinnerton-Dyer, Ieronymou, ...
computations with Br(X )[2] on general degree two K3 surfaces:Hassett, Varilly-Alvarado
Finiteness of Br(X )/Br(F ) for all K3 surfaces over number fields(Skorobogatov-Zarhin 2007)
Existence of points
![Page 45: Birational geometry and arithmetic - Durham University](https://reader030.fdocuments.us/reader030/viewer/2022012418/617328d702cb5f7eb01f19e4/html5/thumbnails/45.jpg)
Effectivity of Brauer-Manin obstructions
Hassett-Kresch-T. 2012
Let X be a K3 surface over a number field F of degree 2. Then thereexists an effective algorithm to compute:
Pic(X ), together with the Galois action;
Br(X )/Br(F ) and X (AF )Br.
Previously:
computation of Pic(X ) on some Kummer K3 surfaces: Elsenhans,Jahnel, van Luijk
computation of X (AF )Br for some Kummer surfaces (diagonalquartics): Bright, Skorobogatov, Swinnerton-Dyer, Ieronymou, ...
computations with Br(X )[2] on general degree two K3 surfaces:Hassett, Varilly-Alvarado
Finiteness of Br(X )/Br(F ) for all K3 surfaces over number fields(Skorobogatov-Zarhin 2007)
Existence of points
![Page 46: Birational geometry and arithmetic - Durham University](https://reader030.fdocuments.us/reader030/viewer/2022012418/617328d702cb5f7eb01f19e4/html5/thumbnails/46.jpg)
Effectivity of Brauer-Manin obstructions
Hassett-Kresch-T. 2012
Let X be a K3 surface over a number field F of degree 2. Then thereexists an effective algorithm to compute:
Pic(X ), together with the Galois action;
Br(X )/Br(F ) and X (AF )Br.
Previously:
computation of Pic(X ) on some Kummer K3 surfaces: Elsenhans,Jahnel, van Luijk
computation of X (AF )Br for some Kummer surfaces (diagonalquartics): Bright, Skorobogatov, Swinnerton-Dyer, Ieronymou, ...
computations with Br(X )[2] on general degree two K3 surfaces:Hassett, Varilly-Alvarado
Finiteness of Br(X )/Br(F ) for all K3 surfaces over number fields(Skorobogatov-Zarhin 2007)
Existence of points
![Page 47: Birational geometry and arithmetic - Durham University](https://reader030.fdocuments.us/reader030/viewer/2022012418/617328d702cb5f7eb01f19e4/html5/thumbnails/47.jpg)
Effectivity of Brauer-Manin obstructions
Hassett-Kresch-T. 2012
Let X be a K3 surface over a number field F of degree 2. Then thereexists an effective algorithm to compute:
Pic(X ), together with the Galois action;
Br(X )/Br(F ) and X (AF )Br.
Previously:
computation of Pic(X ) on some Kummer K3 surfaces: Elsenhans,Jahnel, van Luijk
computation of X (AF )Br for some Kummer surfaces (diagonalquartics): Bright, Skorobogatov, Swinnerton-Dyer, Ieronymou, ...
computations with Br(X )[2] on general degree two K3 surfaces:Hassett, Varilly-Alvarado
Finiteness of Br(X )/Br(F ) for all K3 surfaces over number fields(Skorobogatov-Zarhin 2007)
Existence of points
![Page 48: Birational geometry and arithmetic - Durham University](https://reader030.fdocuments.us/reader030/viewer/2022012418/617328d702cb5f7eb01f19e4/html5/thumbnails/48.jpg)
Effectivity of Brauer-Manin obstructions
Hassett-Kresch-T. 2012
Let X be a K3 surface over a number field F of degree 2. Then thereexists an effective algorithm to compute:
Pic(X ), together with the Galois action;
Br(X )/Br(F ) and X (AF )Br.
Previously:
computation of Pic(X ) on some Kummer K3 surfaces: Elsenhans,Jahnel, van Luijk
computation of X (AF )Br for some Kummer surfaces (diagonalquartics): Bright, Skorobogatov, Swinnerton-Dyer, Ieronymou, ...
computations with Br(X )[2] on general degree two K3 surfaces:Hassett, Varilly-Alvarado
Finiteness of Br(X )/Br(F ) for all K3 surfaces over number fields(Skorobogatov-Zarhin 2007)
Existence of points
![Page 49: Birational geometry and arithmetic - Durham University](https://reader030.fdocuments.us/reader030/viewer/2022012418/617328d702cb5f7eb01f19e4/html5/thumbnails/49.jpg)
Effectivity of Brauer-Manin obstructions
Main ingredients:
effective Kuga-Satake correspondence, relies on an effectiveconstruction of the Bailey-Borel compactification of the modulispace of polarized K3 surfaces;
the work of Masser-Wustholz on the effective Tate conjecture forabelian varieties;
effective GIT, Matsusaka, Hilbert Nullstellensatz, etc..
Existence of points
![Page 50: Birational geometry and arithmetic - Durham University](https://reader030.fdocuments.us/reader030/viewer/2022012418/617328d702cb5f7eb01f19e4/html5/thumbnails/50.jpg)
Effectivity of Brauer-Manin obstructions
Main ingredients:
effective Kuga-Satake correspondence, relies on an effectiveconstruction of the Bailey-Borel compactification of the modulispace of polarized K3 surfaces;
the work of Masser-Wustholz on the effective Tate conjecture forabelian varieties;
effective GIT, Matsusaka, Hilbert Nullstellensatz, etc..
Existence of points
![Page 51: Birational geometry and arithmetic - Durham University](https://reader030.fdocuments.us/reader030/viewer/2022012418/617328d702cb5f7eb01f19e4/html5/thumbnails/51.jpg)
Effectivity of Brauer-Manin obstructions
Main ingredients:
effective Kuga-Satake correspondence, relies on an effectiveconstruction of the Bailey-Borel compactification of the modulispace of polarized K3 surfaces;
the work of Masser-Wustholz on the effective Tate conjecture forabelian varieties;
effective GIT, Matsusaka, Hilbert Nullstellensatz, etc..
Existence of points
![Page 52: Birational geometry and arithmetic - Durham University](https://reader030.fdocuments.us/reader030/viewer/2022012418/617328d702cb5f7eb01f19e4/html5/thumbnails/52.jpg)
Computing the obstruction group
Let X be a smooth intersection of two quadrics in P4 over Q (a DelPezzo surface of degree 4).
The Galois action on the 16 lines factorsthrough the Weyl group W (D5) (a group of order 1920).
Bright, Bruin, Flynn, Logan 2007
If the degree of the splitting field over Q is > 96 then
H1(Gal(F/F ),Pic(X )) = 0.
In all other cases, the obstruction group is either
1,Z/2Z, or (Z/2Z)2.
Implement an algorithm to compute the BM obstruction andprovide more examples of Iskovskikh type.
Existence of points
![Page 53: Birational geometry and arithmetic - Durham University](https://reader030.fdocuments.us/reader030/viewer/2022012418/617328d702cb5f7eb01f19e4/html5/thumbnails/53.jpg)
Computing the obstruction group
Let X be a smooth intersection of two quadrics in P4 over Q (a DelPezzo surface of degree 4). The Galois action on the 16 lines factorsthrough the Weyl group W (D5) (a group of order 1920).
Bright, Bruin, Flynn, Logan 2007
If the degree of the splitting field over Q is > 96 then
H1(Gal(F/F ),Pic(X )) = 0.
In all other cases, the obstruction group is either
1,Z/2Z, or (Z/2Z)2.
Implement an algorithm to compute the BM obstruction andprovide more examples of Iskovskikh type.
Existence of points
![Page 54: Birational geometry and arithmetic - Durham University](https://reader030.fdocuments.us/reader030/viewer/2022012418/617328d702cb5f7eb01f19e4/html5/thumbnails/54.jpg)
Computing the obstruction group
Let X be a smooth intersection of two quadrics in P4 over Q (a DelPezzo surface of degree 4). The Galois action on the 16 lines factorsthrough the Weyl group W (D5) (a group of order 1920).
Bright, Bruin, Flynn, Logan 2007
If the degree of the splitting field over Q is > 96 then
H1(Gal(F/F ),Pic(X )) = 0.
In all other cases, the obstruction group is either
1,Z/2Z, or (Z/2Z)2.
Implement an algorithm to compute the BM obstruction andprovide more examples of Iskovskikh type.
Existence of points
![Page 55: Birational geometry and arithmetic - Durham University](https://reader030.fdocuments.us/reader030/viewer/2022012418/617328d702cb5f7eb01f19e4/html5/thumbnails/55.jpg)
Computing the obstruction group
Let X be a smooth intersection of two quadrics in P4 over Q (a DelPezzo surface of degree 4). The Galois action on the 16 lines factorsthrough the Weyl group W (D5) (a group of order 1920).
Bright, Bruin, Flynn, Logan 2007
If the degree of the splitting field over Q is > 96 then
H1(Gal(F/F ),Pic(X )) = 0.
In all other cases, the obstruction group is either
1,Z/2Z, or (Z/2Z)2.
Implement an algorithm to compute the BM obstruction andprovide more examples of Iskovskikh type.
Existence of points
![Page 56: Birational geometry and arithmetic - Durham University](https://reader030.fdocuments.us/reader030/viewer/2022012418/617328d702cb5f7eb01f19e4/html5/thumbnails/56.jpg)
Uniqueness of the Brauer-Manin obstruction
Conjecture (Colliot-Thelene–Sansuc 1980)
Let X be a smooth projective rationally-connected surface over anumber field F , e.g., an intersection of two quadrics in P4 or a cubic inP3. Then
X (F ) = X (AF )Br.
In particular, existence of rational points on rationally-connectedsurfaces would be decidable.
Colliot-Thelene–Sansuc–Swinnerton-Dyer 1987:Degree 4 Del Pezzo surfaces admitting a conic bundle X → P1.Theconjecture is open for general degree 4 Del Pezzo surfaces.
Existence of points
![Page 57: Birational geometry and arithmetic - Durham University](https://reader030.fdocuments.us/reader030/viewer/2022012418/617328d702cb5f7eb01f19e4/html5/thumbnails/57.jpg)
Uniqueness of the Brauer-Manin obstruction
Conjecture (Colliot-Thelene–Sansuc 1980)
Let X be a smooth projective rationally-connected surface over anumber field F , e.g., an intersection of two quadrics in P4 or a cubic inP3. Then
X (F ) = X (AF )Br.
In particular, existence of rational points on rationally-connectedsurfaces would be decidable.
Colliot-Thelene–Sansuc–Swinnerton-Dyer 1987:Degree 4 Del Pezzo surfaces admitting a conic bundle X → P1.Theconjecture is open for general degree 4 Del Pezzo surfaces.
Existence of points
![Page 58: Birational geometry and arithmetic - Durham University](https://reader030.fdocuments.us/reader030/viewer/2022012418/617328d702cb5f7eb01f19e4/html5/thumbnails/58.jpg)
Uniqueness of the Brauer-Manin obstruction
Conjecture (Colliot-Thelene–Sansuc 1980)
Let X be a smooth projective rationally-connected surface over anumber field F , e.g., an intersection of two quadrics in P4 or a cubic inP3. Then
X (F ) = X (AF )Br.
In particular, existence of rational points on rationally-connectedsurfaces would be decidable.
Colliot-Thelene–Sansuc–Swinnerton-Dyer 1987:Degree 4 Del Pezzo surfaces admitting a conic bundle X → P1.
Theconjecture is open for general degree 4 Del Pezzo surfaces.
Existence of points
![Page 59: Birational geometry and arithmetic - Durham University](https://reader030.fdocuments.us/reader030/viewer/2022012418/617328d702cb5f7eb01f19e4/html5/thumbnails/59.jpg)
Uniqueness of the Brauer-Manin obstruction
Conjecture (Colliot-Thelene–Sansuc 1980)
Let X be a smooth projective rationally-connected surface over anumber field F , e.g., an intersection of two quadrics in P4 or a cubic inP3. Then
X (F ) = X (AF )Br.
In particular, existence of rational points on rationally-connectedsurfaces would be decidable.
Colliot-Thelene–Sansuc–Swinnerton-Dyer 1987:Degree 4 Del Pezzo surfaces admitting a conic bundle X → P1.Theconjecture is open for general degree 4 Del Pezzo surfaces.
Existence of points
![Page 60: Birational geometry and arithmetic - Durham University](https://reader030.fdocuments.us/reader030/viewer/2022012418/617328d702cb5f7eb01f19e4/html5/thumbnails/60.jpg)
Do we believe this conjecture?
Recall that a general Del Pezzo surface X has points locally, and that
Br(X )/Br(F ) = 1.
The Galois group action on the exceptional curves has to be small toallow an obstruction; this is counterintuitive.
Elsenhans–Jahnel 2007: Thousands of examples of cubic surfacesover Q with different Galois actions, the conjecture holds in all cases.
Existence of points
![Page 61: Birational geometry and arithmetic - Durham University](https://reader030.fdocuments.us/reader030/viewer/2022012418/617328d702cb5f7eb01f19e4/html5/thumbnails/61.jpg)
Do we believe this conjecture?
Recall that a general Del Pezzo surface X has points locally, and that
Br(X )/Br(F ) = 1.
The Galois group action on the exceptional curves has to be small toallow an obstruction; this is counterintuitive.
Elsenhans–Jahnel 2007: Thousands of examples of cubic surfacesover Q with different Galois actions, the conjecture holds in all cases.
Existence of points
![Page 62: Birational geometry and arithmetic - Durham University](https://reader030.fdocuments.us/reader030/viewer/2022012418/617328d702cb5f7eb01f19e4/html5/thumbnails/62.jpg)
Do we believe this conjecture?
Recall that a general Del Pezzo surface X has points locally, and that
Br(X )/Br(F ) = 1.
The Galois group action on the exceptional curves has to be small toallow an obstruction; this is counterintuitive.
Elsenhans–Jahnel 2007: Thousands of examples of cubic surfacesover Q with different Galois actions, the conjecture holds in all cases.
Existence of points
![Page 63: Birational geometry and arithmetic - Durham University](https://reader030.fdocuments.us/reader030/viewer/2022012418/617328d702cb5f7eb01f19e4/html5/thumbnails/63.jpg)
Del Pezzo surfaces over Fq(t)
Theorem (Hassett-T. 2011)
Let k be a finite field with at least 22 · 174 elements and X a generalDel Pezzo surface of degree 4 over F = k(t) such that its integralmodel
X → P1
is a complete intersection in P1 × P4 of two general forms of bi-degree(1, 2). Then X (F ) 6= ∅.
The idea of proof will follow....
Existence of points
![Page 64: Birational geometry and arithmetic - Durham University](https://reader030.fdocuments.us/reader030/viewer/2022012418/617328d702cb5f7eb01f19e4/html5/thumbnails/64.jpg)
Del Pezzo surfaces over Fq(t)
Theorem (Hassett-T. 2011)
Let k be a finite field with at least 22 · 174 elements and X a generalDel Pezzo surface of degree 4 over F = k(t) such that its integralmodel
X → P1
is a complete intersection in P1 × P4 of two general forms of bi-degree(1, 2). Then X (F ) 6= ∅.
The idea of proof will follow....
Existence of points
![Page 65: Birational geometry and arithmetic - Durham University](https://reader030.fdocuments.us/reader030/viewer/2022012418/617328d702cb5f7eb01f19e4/html5/thumbnails/65.jpg)
K3 surfaces
Potential density: Zariski density after a finite extension of the field.
Bogomolov-T. 1999
If X is a K3 surface which is either elliptic or has infiniteautomorphisms then potential density holds for X .
What about general K3 surfaces, i.e., those with Picard rank one?
Let X ⊂ P1 × P3 be a general hypersurface of bidegree (1, 4).Then the K3 surface fibration X → P1 has a Zariski dense set ofsections, i.e., such K3 surfaces over F = C(t) have Zariski denserational points; more generally, this holds for general pencils of K3surfaces of degree ≤ 18 (Hassett-T. 2008)
Same holds, if X ⊂ P1 × P3 is given by a general form of bidegree(2, 4) (Zhiyuan Li 2011)
Zariski density
![Page 66: Birational geometry and arithmetic - Durham University](https://reader030.fdocuments.us/reader030/viewer/2022012418/617328d702cb5f7eb01f19e4/html5/thumbnails/66.jpg)
K3 surfaces
Potential density: Zariski density after a finite extension of the field.
Bogomolov-T. 1999
If X is a K3 surface which is either elliptic or has infiniteautomorphisms then potential density holds for X .
What about general K3 surfaces, i.e., those with Picard rank one?
Let X ⊂ P1 × P3 be a general hypersurface of bidegree (1, 4).Then the K3 surface fibration X → P1 has a Zariski dense set ofsections, i.e., such K3 surfaces over F = C(t) have Zariski denserational points; more generally, this holds for general pencils of K3surfaces of degree ≤ 18 (Hassett-T. 2008)
Same holds, if X ⊂ P1 × P3 is given by a general form of bidegree(2, 4) (Zhiyuan Li 2011)
Zariski density
![Page 67: Birational geometry and arithmetic - Durham University](https://reader030.fdocuments.us/reader030/viewer/2022012418/617328d702cb5f7eb01f19e4/html5/thumbnails/67.jpg)
K3 surfaces
Potential density: Zariski density after a finite extension of the field.
Bogomolov-T. 1999
If X is a K3 surface which is either elliptic or has infiniteautomorphisms then potential density holds for X .
What about general K3 surfaces, i.e., those with Picard rank one?
Let X ⊂ P1 × P3 be a general hypersurface of bidegree (1, 4).Then the K3 surface fibration X → P1 has a Zariski dense set ofsections, i.e., such K3 surfaces over F = C(t) have Zariski denserational points; more generally, this holds for general pencils of K3surfaces of degree ≤ 18 (Hassett-T. 2008)
Same holds, if X ⊂ P1 × P3 is given by a general form of bidegree(2, 4) (Zhiyuan Li 2011)
Zariski density
![Page 68: Birational geometry and arithmetic - Durham University](https://reader030.fdocuments.us/reader030/viewer/2022012418/617328d702cb5f7eb01f19e4/html5/thumbnails/68.jpg)
K3 surfaces
Potential density: Zariski density after a finite extension of the field.
Bogomolov-T. 1999
If X is a K3 surface which is either elliptic or has infiniteautomorphisms then potential density holds for X .
What about general K3 surfaces, i.e., those with Picard rank one?
Let X ⊂ P1 × P3 be a general hypersurface of bidegree (1, 4).Then the K3 surface fibration X → P1 has a Zariski dense set ofsections, i.e., such K3 surfaces over F = C(t) have Zariski denserational points; more generally, this holds for general pencils of K3surfaces of degree ≤ 18 (Hassett-T. 2008)
Same holds, if X ⊂ P1 × P3 is given by a general form of bidegree(2, 4) (Zhiyuan Li 2011)
Zariski density
![Page 69: Birational geometry and arithmetic - Durham University](https://reader030.fdocuments.us/reader030/viewer/2022012418/617328d702cb5f7eb01f19e4/html5/thumbnails/69.jpg)
K3 surfaces
Potential density: Zariski density after a finite extension of the field.
Bogomolov-T. 1999
If X is a K3 surface which is either elliptic or has infiniteautomorphisms then potential density holds for X .
What about general K3 surfaces, i.e., those with Picard rank one?
Let X ⊂ P1 × P3 be a general hypersurface of bidegree (1, 4).Then the K3 surface fibration X → P1 has a Zariski dense set ofsections, i.e., such K3 surfaces over F = C(t) have Zariski denserational points; more generally, this holds for general pencils of K3surfaces of degree ≤ 18 (Hassett-T. 2008)
Same holds, if X ⊂ P1 × P3 is given by a general form of bidegree(2, 4) (Zhiyuan Li 2011)
Zariski density
![Page 70: Birational geometry and arithmetic - Durham University](https://reader030.fdocuments.us/reader030/viewer/2022012418/617328d702cb5f7eb01f19e4/html5/thumbnails/70.jpg)
Higher dimensions
Potential density holds for:
all smooth Fano threefolds, with the possible exception of
W22:1−→ P3, ramified in a degree 6 surface S6 (Harris-T. 1998,
Bogomolov-T. 1998)
W2, provided S6 is singular; similar results in dimension 4(Chelsov-Park 2004, Cheltsov 2004)
varieties of lines on general cubic fourfolds (Amerik-Voisin 2008,Amerik-Bogomolov-Rovinski)
varieties of lines of some special cubic fourfolds, i.e., those notcontaining a plane and admitting a hyperplane section with 6ordinary double points in general linear position (Hassett-T. 2008)
Zariski density
![Page 71: Birational geometry and arithmetic - Durham University](https://reader030.fdocuments.us/reader030/viewer/2022012418/617328d702cb5f7eb01f19e4/html5/thumbnails/71.jpg)
Higher dimensions
Potential density holds for:
all smooth Fano threefolds, with the possible exception of
W22:1−→ P3, ramified in a degree 6 surface S6 (Harris-T. 1998,
Bogomolov-T. 1998)
W2, provided S6 is singular; similar results in dimension 4(Chelsov-Park 2004, Cheltsov 2004)
varieties of lines on general cubic fourfolds (Amerik-Voisin 2008,Amerik-Bogomolov-Rovinski)
varieties of lines of some special cubic fourfolds, i.e., those notcontaining a plane and admitting a hyperplane section with 6ordinary double points in general linear position (Hassett-T. 2008)
Zariski density
![Page 72: Birational geometry and arithmetic - Durham University](https://reader030.fdocuments.us/reader030/viewer/2022012418/617328d702cb5f7eb01f19e4/html5/thumbnails/72.jpg)
Higher dimensions
Potential density holds for:
all smooth Fano threefolds, with the possible exception of
W22:1−→ P3, ramified in a degree 6 surface S6 (Harris-T. 1998,
Bogomolov-T. 1998)
W2, provided S6 is singular; similar results in dimension 4(Chelsov-Park 2004, Cheltsov 2004)
varieties of lines on general cubic fourfolds (Amerik-Voisin 2008,Amerik-Bogomolov-Rovinski)
varieties of lines of some special cubic fourfolds, i.e., those notcontaining a plane and admitting a hyperplane section with 6ordinary double points in general linear position (Hassett-T. 2008)
Zariski density
![Page 73: Birational geometry and arithmetic - Durham University](https://reader030.fdocuments.us/reader030/viewer/2022012418/617328d702cb5f7eb01f19e4/html5/thumbnails/73.jpg)
Higher dimensions
Potential density holds for:
all smooth Fano threefolds, with the possible exception of
W22:1−→ P3, ramified in a degree 6 surface S6 (Harris-T. 1998,
Bogomolov-T. 1998)
W2, provided S6 is singular; similar results in dimension 4(Chelsov-Park 2004, Cheltsov 2004)
varieties of lines on general cubic fourfolds (Amerik-Voisin 2008,Amerik-Bogomolov-Rovinski)
varieties of lines of some special cubic fourfolds, i.e., those notcontaining a plane and admitting a hyperplane section with 6ordinary double points in general linear position (Hassett-T. 2008)
Zariski density
![Page 74: Birational geometry and arithmetic - Durham University](https://reader030.fdocuments.us/reader030/viewer/2022012418/617328d702cb5f7eb01f19e4/html5/thumbnails/74.jpg)
Counting rational points
Counting problems depend on:
a projective embedding X ↪→ Pn;
a choice of X ◦ ⊂ X ;
a choice of a height function H : Pn(F )→ R>0.
Main problem
N(X ◦(F ),B) = #{x ∈ X ◦(F ) | H(x) ≤ B} ?∼ c · Ba log(B)b−1
Counting points
![Page 75: Birational geometry and arithmetic - Durham University](https://reader030.fdocuments.us/reader030/viewer/2022012418/617328d702cb5f7eb01f19e4/html5/thumbnails/75.jpg)
Counting rational points
Counting problems depend on:
a projective embedding X ↪→ Pn;
a choice of X ◦ ⊂ X ;
a choice of a height function H : Pn(F )→ R>0.
Main problem
N(X ◦(F ),B) = #{x ∈ X ◦(F ) | H(x) ≤ B} ?∼ c · Ba log(B)b−1
Counting points
![Page 76: Birational geometry and arithmetic - Durham University](https://reader030.fdocuments.us/reader030/viewer/2022012418/617328d702cb5f7eb01f19e4/html5/thumbnails/76.jpg)
The geometric framework
Conjecture (Manin 1989)
Let X ⊂ Pn be a smooth projective Fano variety over a number fieldF , in its anticanonical embedding.
Then there exists a Zariski opensubset X ◦ ⊂ X such that
N(X ◦(F ),B) ∼ c · B log(B)b−1, B→∞,
where b = rkPic(X ).
We do not know, in general, whether or not X (F ) is Zariski dense,even after a finite extension of F . Potential density of rational pointshas been proved for some families of Fano varieties, but is still open,e.g., for hypersurfaces Xd ⊂ Pd , with d ≥ 5.
Counting points
![Page 77: Birational geometry and arithmetic - Durham University](https://reader030.fdocuments.us/reader030/viewer/2022012418/617328d702cb5f7eb01f19e4/html5/thumbnails/77.jpg)
The geometric framework
Conjecture (Manin 1989)
Let X ⊂ Pn be a smooth projective Fano variety over a number fieldF , in its anticanonical embedding. Then there exists a Zariski opensubset X ◦ ⊂ X such that
N(X ◦(F ),B) ∼ c · B log(B)b−1, B→∞,
where b = rkPic(X ).
We do not know, in general, whether or not X (F ) is Zariski dense,even after a finite extension of F . Potential density of rational pointshas been proved for some families of Fano varieties, but is still open,e.g., for hypersurfaces Xd ⊂ Pd , with d ≥ 5.
Counting points
![Page 78: Birational geometry and arithmetic - Durham University](https://reader030.fdocuments.us/reader030/viewer/2022012418/617328d702cb5f7eb01f19e4/html5/thumbnails/78.jpg)
The geometric framework
Conjecture (Manin 1989)
Let X ⊂ Pn be a smooth projective Fano variety over a number fieldF , in its anticanonical embedding. Then there exists a Zariski opensubset X ◦ ⊂ X such that
N(X ◦(F ),B) ∼ c · B log(B)b−1, B→∞,
where b = rkPic(X ).
We do not know, in general, whether or not X (F ) is Zariski dense,even after a finite extension of F . Potential density of rational pointshas been proved for some families of Fano varieties, but is still open,e.g., for hypersurfaces Xd ⊂ Pd , with d ≥ 5.
Counting points
![Page 79: Birational geometry and arithmetic - Durham University](https://reader030.fdocuments.us/reader030/viewer/2022012418/617328d702cb5f7eb01f19e4/html5/thumbnails/79.jpg)
The function field case / Batyrev 1987
Let F = Fq(B) be a global function field and X/F a smooth Fanovariety. Let
π : X → B
be a model. A point x ∈ X (F ) gives rise to a section x of π. Let L bea very ample line bundle on X . The height zeta function takes theform
Z(s) =∑x
q−(L,x)s
=∑d
Md(Fq)q−ds ,
where d = (L, x) and Md is the space of sections of degree d .
Counting points
![Page 80: Birational geometry and arithmetic - Durham University](https://reader030.fdocuments.us/reader030/viewer/2022012418/617328d702cb5f7eb01f19e4/html5/thumbnails/80.jpg)
The function field case / Batyrev 1987
The dimension of Md can be estimated, provided x is unobstructed:
dimMd ∼ (−KX , x), x ∈Md .
Heuristic assumption:
Md(Fq) = qdim(Md )
leads to a modified zeta function
Zmod(s) =∑
q−(L,x)s+(−KX ,x),
its analytic properties are governed by the ratio of the linear forms
(−KX , ·) and (L, ·)
Counting points
![Page 81: Birational geometry and arithmetic - Durham University](https://reader030.fdocuments.us/reader030/viewer/2022012418/617328d702cb5f7eb01f19e4/html5/thumbnails/81.jpg)
The function field case / Batyrev 1987
The dimension of Md can be estimated, provided x is unobstructed:
dimMd ∼ (−KX , x), x ∈Md .
Heuristic assumption:
Md(Fq) = qdim(Md )
leads to a modified zeta function
Zmod(s) =∑
q−(L,x)s+(−KX ,x),
its analytic properties are governed by the ratio of the linear forms
(−KX , ·) and (L, ·)
Counting points
![Page 82: Birational geometry and arithmetic - Durham University](https://reader030.fdocuments.us/reader030/viewer/2022012418/617328d702cb5f7eb01f19e4/html5/thumbnails/82.jpg)
The Batyrev–Manin conjecture
N(X ◦,L,B) = c · Ba(L) · log(B)b(L)−1(1 + o(1)), B→∞
a(L) = inf{a | a[L] + [KX ] ∈ Λeff(X )},b(L) = codimension of the face of Λeff(X ) containinga(L)[L] + [KX ],
c(−KX ) = α(X ) · β(X ) · τ(KX ) – “volume” of the effective cone,nontrivial part of the Brauer group Br(X )/Br(F ), Peyre’sTamagawa type number,
c(L) =∑
y c(L|Xy ), where X → Y is a “Mori fiber space” –L-primitive fibrations of Batyrev–T.
Counting points
![Page 83: Birational geometry and arithmetic - Durham University](https://reader030.fdocuments.us/reader030/viewer/2022012418/617328d702cb5f7eb01f19e4/html5/thumbnails/83.jpg)
The Batyrev–Manin conjecture
N(X ◦,L,B) = c · Ba(L) · log(B)b(L)−1(1 + o(1)), B→∞
a(L) = inf{a | a[L] + [KX ] ∈ Λeff(X )},
b(L) = codimension of the face of Λeff(X ) containinga(L)[L] + [KX ],
c(−KX ) = α(X ) · β(X ) · τ(KX ) – “volume” of the effective cone,nontrivial part of the Brauer group Br(X )/Br(F ), Peyre’sTamagawa type number,
c(L) =∑
y c(L|Xy ), where X → Y is a “Mori fiber space” –L-primitive fibrations of Batyrev–T.
Counting points
![Page 84: Birational geometry and arithmetic - Durham University](https://reader030.fdocuments.us/reader030/viewer/2022012418/617328d702cb5f7eb01f19e4/html5/thumbnails/84.jpg)
The Batyrev–Manin conjecture
N(X ◦,L,B) = c · Ba(L) · log(B)b(L)−1(1 + o(1)), B→∞
a(L) = inf{a | a[L] + [KX ] ∈ Λeff(X )},b(L) = codimension of the face of Λeff(X ) containinga(L)[L] + [KX ],
c(−KX ) = α(X ) · β(X ) · τ(KX ) – “volume” of the effective cone,nontrivial part of the Brauer group Br(X )/Br(F ), Peyre’sTamagawa type number,
c(L) =∑
y c(L|Xy ), where X → Y is a “Mori fiber space” –L-primitive fibrations of Batyrev–T.
Counting points
![Page 85: Birational geometry and arithmetic - Durham University](https://reader030.fdocuments.us/reader030/viewer/2022012418/617328d702cb5f7eb01f19e4/html5/thumbnails/85.jpg)
The Batyrev–Manin conjecture
N(X ◦,L,B) = c · Ba(L) · log(B)b(L)−1(1 + o(1)), B→∞
a(L) = inf{a | a[L] + [KX ] ∈ Λeff(X )},b(L) = codimension of the face of Λeff(X ) containinga(L)[L] + [KX ],
c(−KX ) = α(X ) · β(X ) · τ(KX ) – “volume” of the effective cone,nontrivial part of the Brauer group Br(X )/Br(F ), Peyre’sTamagawa type number,
c(L) =∑
y c(L|Xy ), where X → Y is a “Mori fiber space” –L-primitive fibrations of Batyrev–T.
Counting points
![Page 86: Birational geometry and arithmetic - Durham University](https://reader030.fdocuments.us/reader030/viewer/2022012418/617328d702cb5f7eb01f19e4/html5/thumbnails/86.jpg)
The Batyrev–Manin conjecture
N(X ◦,L,B) = c · Ba(L) · log(B)b(L)−1(1 + o(1)), B→∞
a(L) = inf{a | a[L] + [KX ] ∈ Λeff(X )},b(L) = codimension of the face of Λeff(X ) containinga(L)[L] + [KX ],
c(−KX ) = α(X ) · β(X ) · τ(KX ) – “volume” of the effective cone,nontrivial part of the Brauer group Br(X )/Br(F ), Peyre’sTamagawa type number,
c(L) =∑
y c(L|Xy ), where X → Y is a “Mori fiber space” –L-primitive fibrations of Batyrev–T.
Counting points
![Page 87: Birational geometry and arithmetic - Durham University](https://reader030.fdocuments.us/reader030/viewer/2022012418/617328d702cb5f7eb01f19e4/html5/thumbnails/87.jpg)
Justification
G. Segal 1979
Contd(S2,Pn(C)) ∼ Hold(S2,Pn(C)), d →∞
(isomorphism ofπi , for i ≤ d).
H∗(Hold(S2,Pn(C))) stabilizes, for d →∞.
This was generalized to Grassmannians and toric varieties as targetspaces by Kirwan 1986, Guest 1994, and others.
Basic idea
Md(Fq) ∼ qdim(Md ), for d →∞, provided the homology stabilizes.
Counting points
![Page 88: Birational geometry and arithmetic - Durham University](https://reader030.fdocuments.us/reader030/viewer/2022012418/617328d702cb5f7eb01f19e4/html5/thumbnails/88.jpg)
Justification
G. Segal 1979
Contd(S2,Pn(C)) ∼ Hold(S2,Pn(C)), d →∞ (isomorphism ofπi , for i ≤ d).
H∗(Hold(S2,Pn(C))) stabilizes, for d →∞.
This was generalized to Grassmannians and toric varieties as targetspaces by Kirwan 1986, Guest 1994, and others.
Basic idea
Md(Fq) ∼ qdim(Md ), for d →∞, provided the homology stabilizes.
Counting points
![Page 89: Birational geometry and arithmetic - Durham University](https://reader030.fdocuments.us/reader030/viewer/2022012418/617328d702cb5f7eb01f19e4/html5/thumbnails/89.jpg)
Justification
G. Segal 1979
Contd(S2,Pn(C)) ∼ Hold(S2,Pn(C)), d →∞ (isomorphism ofπi , for i ≤ d).
H∗(Hold(S2,Pn(C))) stabilizes, for d →∞.
This was generalized to Grassmannians and toric varieties as targetspaces by Kirwan 1986, Guest 1994, and others.
Basic idea
Md(Fq) ∼ qdim(Md ), for d →∞, provided the homology stabilizes.
Counting points
![Page 90: Birational geometry and arithmetic - Durham University](https://reader030.fdocuments.us/reader030/viewer/2022012418/617328d702cb5f7eb01f19e4/html5/thumbnails/90.jpg)
Justification
G. Segal 1979
Contd(S2,Pn(C)) ∼ Hold(S2,Pn(C)), d →∞ (isomorphism ofπi , for i ≤ d).
H∗(Hold(S2,Pn(C))) stabilizes, for d →∞.
This was generalized to Grassmannians and toric varieties as targetspaces by Kirwan 1986, Guest 1994, and others.
Basic idea
Md(Fq) ∼ qdim(Md ), for d →∞, provided the homology stabilizes.
Counting points
![Page 91: Birational geometry and arithmetic - Durham University](https://reader030.fdocuments.us/reader030/viewer/2022012418/617328d702cb5f7eb01f19e4/html5/thumbnails/91.jpg)
Justification
G. Segal 1979
Contd(S2,Pn(C)) ∼ Hold(S2,Pn(C)), d →∞ (isomorphism ofπi , for i ≤ d).
H∗(Hold(S2,Pn(C))) stabilizes, for d →∞.
This was generalized to Grassmannians and toric varieties as targetspaces by Kirwan 1986, Guest 1994, and others.
Basic idea
Md(Fq) ∼ qdim(Md ), for d →∞, provided the homology stabilizes.
Counting points
![Page 92: Birational geometry and arithmetic - Durham University](https://reader030.fdocuments.us/reader030/viewer/2022012418/617328d702cb5f7eb01f19e4/html5/thumbnails/92.jpg)
Ellenberg–Venkatesh–Westerland 2009
Effective stabilization of homology of Hurwitz spaces
There exist A,B,D such that
dimHd(HurcG ,n) = dimHd(HurcG ,n+D),
for n ≥ Ad + B.
This has applications to Cohen–Lenstra heuristics over function fieldsof curves.
Applications in the context of height zeta functions?
Counting points
![Page 93: Birational geometry and arithmetic - Durham University](https://reader030.fdocuments.us/reader030/viewer/2022012418/617328d702cb5f7eb01f19e4/html5/thumbnails/93.jpg)
Ellenberg–Venkatesh–Westerland 2009
Effective stabilization of homology of Hurwitz spaces
There exist A,B,D such that
dimHd(HurcG ,n) = dimHd(HurcG ,n+D),
for n ≥ Ad + B.
This has applications to Cohen–Lenstra heuristics over function fieldsof curves.
Applications in the context of height zeta functions?
Counting points
![Page 94: Birational geometry and arithmetic - Durham University](https://reader030.fdocuments.us/reader030/viewer/2022012418/617328d702cb5f7eb01f19e4/html5/thumbnails/94.jpg)
Ellenberg–Venkatesh–Westerland 2009
Effective stabilization of homology of Hurwitz spaces
There exist A,B,D such that
dimHd(HurcG ,n) = dimHd(HurcG ,n+D),
for n ≥ Ad + B.
This has applications to Cohen–Lenstra heuristics over function fieldsof curves.
Applications in the context of height zeta functions?
Counting points
![Page 95: Birational geometry and arithmetic - Durham University](https://reader030.fdocuments.us/reader030/viewer/2022012418/617328d702cb5f7eb01f19e4/html5/thumbnails/95.jpg)
Results over Q
Extensive numerical computations confirming Manin’s conjecture, andits refinements, for Del Pezzo surfaces, hypersurfaces of small degreein dimension 3 and 4.
Many recent theoretical results on asymptotics of points of boundedheight on cubic surfaces and other Del Pezzo surfaces, via (uni)versaltorsors (Browning, de la Breteche, Derenthal, Heath-Brown, Peyre,Salberger, Wooley, ...)
Caution: counterexamples to Manin’s conjecture for cubic surfacebundles over P1 (Batyrev-T. 1996). These are compactifications ofaffine spaces.
Counting points
![Page 96: Birational geometry and arithmetic - Durham University](https://reader030.fdocuments.us/reader030/viewer/2022012418/617328d702cb5f7eb01f19e4/html5/thumbnails/96.jpg)
Results over Q
Extensive numerical computations confirming Manin’s conjecture, andits refinements, for Del Pezzo surfaces, hypersurfaces of small degreein dimension 3 and 4.
Many recent theoretical results on asymptotics of points of boundedheight on cubic surfaces and other Del Pezzo surfaces, via (uni)versaltorsors (Browning, de la Breteche, Derenthal, Heath-Brown, Peyre,Salberger, Wooley, ...)
Caution: counterexamples to Manin’s conjecture for cubic surfacebundles over P1 (Batyrev-T. 1996). These are compactifications ofaffine spaces.
Counting points
![Page 97: Birational geometry and arithmetic - Durham University](https://reader030.fdocuments.us/reader030/viewer/2022012418/617328d702cb5f7eb01f19e4/html5/thumbnails/97.jpg)
Results over Q
Extensive numerical computations confirming Manin’s conjecture, andits refinements, for Del Pezzo surfaces, hypersurfaces of small degreein dimension 3 and 4.
Many recent theoretical results on asymptotics of points of boundedheight on cubic surfaces and other Del Pezzo surfaces, via (uni)versaltorsors (Browning, de la Breteche, Derenthal, Heath-Brown, Peyre,Salberger, Wooley, ...)
Caution: counterexamples to Manin’s conjecture for cubic surfacebundles over P1 (Batyrev-T. 1996). These are compactifications ofaffine spaces.
Counting points
![Page 98: Birational geometry and arithmetic - Durham University](https://reader030.fdocuments.us/reader030/viewer/2022012418/617328d702cb5f7eb01f19e4/html5/thumbnails/98.jpg)
Points of smallest height
Legendre: If ax2 + by2 = cz2 is solvable mod p, for all p, then it issolvable in Z.
Moreover, the height of the smallest solution is boundedby√abc.
An effective bound on the error term in the circle method (or in theother asymptotic results) also gives an effective bound on Hmin, theheight of smallest solutions.
In particular, Manin’s and Peyre’s conjecture suggest that
Hmin ≤1
τ
Counting points
![Page 99: Birational geometry and arithmetic - Durham University](https://reader030.fdocuments.us/reader030/viewer/2022012418/617328d702cb5f7eb01f19e4/html5/thumbnails/99.jpg)
Points of smallest height
Legendre: If ax2 + by2 = cz2 is solvable mod p, for all p, then it issolvable in Z. Moreover, the height of the smallest solution is boundedby√abc.
An effective bound on the error term in the circle method (or in theother asymptotic results) also gives an effective bound on Hmin, theheight of smallest solutions.
In particular, Manin’s and Peyre’s conjecture suggest that
Hmin ≤1
τ
Counting points
![Page 100: Birational geometry and arithmetic - Durham University](https://reader030.fdocuments.us/reader030/viewer/2022012418/617328d702cb5f7eb01f19e4/html5/thumbnails/100.jpg)
Points of smallest height
Legendre: If ax2 + by2 = cz2 is solvable mod p, for all p, then it issolvable in Z. Moreover, the height of the smallest solution is boundedby√abc.
An effective bound on the error term in the circle method (or in theother asymptotic results) also gives an effective bound on Hmin, theheight of smallest solutions.
In particular, Manin’s and Peyre’s conjecture suggest that
Hmin ≤1
τ
Counting points
![Page 101: Birational geometry and arithmetic - Durham University](https://reader030.fdocuments.us/reader030/viewer/2022012418/617328d702cb5f7eb01f19e4/html5/thumbnails/101.jpg)
Points of smallest height
Legendre: If ax2 + by2 = cz2 is solvable mod p, for all p, then it issolvable in Z. Moreover, the height of the smallest solution is boundedby√abc.
An effective bound on the error term in the circle method (or in theother asymptotic results) also gives an effective bound on Hmin, theheight of smallest solutions.
In particular, Manin’s and Peyre’s conjecture suggest that
Hmin ≤1
τ
Counting points
![Page 102: Birational geometry and arithmetic - Durham University](https://reader030.fdocuments.us/reader030/viewer/2022012418/617328d702cb5f7eb01f19e4/html5/thumbnails/102.jpg)
Points of smallest height
There are extensive numerical data for smallest points on Del Pezzosurfaces, Fano threefolds. E.g.,
Elsenhans-Jahnel 2010
Counting points
![Page 103: Birational geometry and arithmetic - Durham University](https://reader030.fdocuments.us/reader030/viewer/2022012418/617328d702cb5f7eb01f19e4/html5/thumbnails/103.jpg)
How are all of these related? (Hassett-T.)
Let X be a Del Pezzo surface over F = Fq(t) and
π : X → P1.
its integral model. Fix a height, and consider the spaces Md ofsections of π of height d (degree of the section).
Ideal scenario
Md are geometrically irreducible, for d � 0
Md dominate the intermediate Jacobian IJ(X ) of Xthere is a critical d0, related to the height of X , such that Md0 iseither birational to IJ(X ) or to a P1-bundle over IJ(X )
for d ≥ d0, Md fibers over IJ(X ), with general fiber a rationallyconnected variety
Counting points
![Page 104: Birational geometry and arithmetic - Durham University](https://reader030.fdocuments.us/reader030/viewer/2022012418/617328d702cb5f7eb01f19e4/html5/thumbnails/104.jpg)
How are all of these related? (Hassett-T.)
Let X be a Del Pezzo surface over F = Fq(t) and
π : X → P1.
its integral model. Fix a height, and consider the spaces Md ofsections of π of height d (degree of the section).
Ideal scenario
Md are geometrically irreducible, for d � 0
Md dominate the intermediate Jacobian IJ(X ) of Xthere is a critical d0, related to the height of X , such that Md0 iseither birational to IJ(X ) or to a P1-bundle over IJ(X )
for d ≥ d0, Md fibers over IJ(X ), with general fiber a rationallyconnected variety
Counting points
![Page 105: Birational geometry and arithmetic - Durham University](https://reader030.fdocuments.us/reader030/viewer/2022012418/617328d702cb5f7eb01f19e4/html5/thumbnails/105.jpg)
How are all of these related? (Hassett-T.)
Let X be a Del Pezzo surface over F = Fq(t) and
π : X → P1.
its integral model. Fix a height, and consider the spaces Md ofsections of π of height d (degree of the section).
Ideal scenario
Md are geometrically irreducible, for d � 0
Md dominate the intermediate Jacobian IJ(X ) of X
there is a critical d0, related to the height of X , such that Md0 iseither birational to IJ(X ) or to a P1-bundle over IJ(X )
for d ≥ d0, Md fibers over IJ(X ), with general fiber a rationallyconnected variety
Counting points
![Page 106: Birational geometry and arithmetic - Durham University](https://reader030.fdocuments.us/reader030/viewer/2022012418/617328d702cb5f7eb01f19e4/html5/thumbnails/106.jpg)
How are all of these related? (Hassett-T.)
Let X be a Del Pezzo surface over F = Fq(t) and
π : X → P1.
its integral model. Fix a height, and consider the spaces Md ofsections of π of height d (degree of the section).
Ideal scenario
Md are geometrically irreducible, for d � 0
Md dominate the intermediate Jacobian IJ(X ) of Xthere is a critical d0, related to the height of X , such that Md0 iseither birational to IJ(X ) or to a P1-bundle over IJ(X )
for d ≥ d0, Md fibers over IJ(X ), with general fiber a rationallyconnected variety
Counting points
![Page 107: Birational geometry and arithmetic - Durham University](https://reader030.fdocuments.us/reader030/viewer/2022012418/617328d702cb5f7eb01f19e4/html5/thumbnails/107.jpg)
How are all of these related? (Hassett-T.)
Let X be a Del Pezzo surface over F = Fq(t) and
π : X → P1.
its integral model. Fix a height, and consider the spaces Md ofsections of π of height d (degree of the section).
Ideal scenario
Md are geometrically irreducible, for d � 0
Md dominate the intermediate Jacobian IJ(X ) of Xthere is a critical d0, related to the height of X , such that Md0 iseither birational to IJ(X ) or to a P1-bundle over IJ(X )
for d ≥ d0, Md fibers over IJ(X ), with general fiber a rationallyconnected variety
Counting points
![Page 108: Birational geometry and arithmetic - Durham University](https://reader030.fdocuments.us/reader030/viewer/2022012418/617328d702cb5f7eb01f19e4/html5/thumbnails/108.jpg)
Del Pezzo surfaces over Fq(t)
We consider fibrationsπ : X → P1,
with general fiber a degree-four Del Pezzo surface and with square-freediscriminant. In this situation, we have an embedding
X ⊂ P(π∗ω−1π ).
We haveπ∗ω
−1π = ⊕5
i=1OP1(−ai ),
witha1 ≤ a2 ≤ a3 ≤ a4 ≤ a5,
occurring cases are discussed by Shramov (2006), in his investigationsof rationality properties of such fibrations.
Counting points
![Page 109: Birational geometry and arithmetic - Durham University](https://reader030.fdocuments.us/reader030/viewer/2022012418/617328d702cb5f7eb01f19e4/html5/thumbnails/109.jpg)
Del Pezzo surfaces over Fq(t)
We consider fibrationsπ : X → P1,
with general fiber a degree-four Del Pezzo surface and with square-freediscriminant. In this situation, we have an embedding
X ⊂ P(π∗ω−1π ).
We haveπ∗ω
−1π = ⊕5
i=1OP1(−ai ),
witha1 ≤ a2 ≤ a3 ≤ a4 ≤ a5,
occurring cases are discussed by Shramov (2006), in his investigationsof rationality properties of such fibrations.
Counting points
![Page 110: Birational geometry and arithmetic - Durham University](https://reader030.fdocuments.us/reader030/viewer/2022012418/617328d702cb5f7eb01f19e4/html5/thumbnails/110.jpg)
Del Pezzo surfaces over Fq(t)
We assume that π∗ω−1π is generic, i.e., a5 − a1 ≤ 1; we can realize
X ⊂ P1 × Pd , d = 4, 5, . . . , 8,
as a complete intersection.
Theorem (Hassett-T. 2011)
Let k be a finite field with at least 22 · 174 elements and X a generalDel Pezzo surface of degree 4 over F = k(t) such that its integralmodel
X → P1
is a complete intersection in P1 × P4 of two general forms of bi-degree(1, 2). Then X (F ) 6= ∅.
Counting points
![Page 111: Birational geometry and arithmetic - Durham University](https://reader030.fdocuments.us/reader030/viewer/2022012418/617328d702cb5f7eb01f19e4/html5/thumbnails/111.jpg)
Del Pezzo surfaces over Fq(t)
We assume that π∗ω−1π is generic, i.e., a5 − a1 ≤ 1; we can realize
X ⊂ P1 × Pd , d = 4, 5, . . . , 8,
as a complete intersection.
Theorem (Hassett-T. 2011)
Let k be a finite field with at least 22 · 174 elements and X a generalDel Pezzo surface of degree 4 over F = k(t) such that its integralmodel
X → P1
is a complete intersection in P1 × P4 of two general forms of bi-degree(1, 2). Then X (F ) 6= ∅.
Counting points
![Page 112: Birational geometry and arithmetic - Durham University](https://reader030.fdocuments.us/reader030/viewer/2022012418/617328d702cb5f7eb01f19e4/html5/thumbnails/112.jpg)
Idea of proof
Write X ⊂ P1 × P4 as a complete intersection
P1s + Q1t = P2s + Q2t = 0,
where Pi ,Qi are quadrics in P4.
The projection
π : X → P1
has 16 constant sections corresponding to solutions y1, . . . , y16 of
P1 = Q1 = P2 = Q2 = 0.
Counting points
![Page 113: Birational geometry and arithmetic - Durham University](https://reader030.fdocuments.us/reader030/viewer/2022012418/617328d702cb5f7eb01f19e4/html5/thumbnails/113.jpg)
Idea of proof
Write X ⊂ P1 × P4 as a complete intersection
P1s + Q1t = P2s + Q2t = 0,
where Pi ,Qi are quadrics in P4. The projection
π : X → P1
has 16 constant sections corresponding to solutions y1, . . . , y16 of
P1 = Q1 = P2 = Q2 = 0.
Counting points
![Page 114: Birational geometry and arithmetic - Durham University](https://reader030.fdocuments.us/reader030/viewer/2022012418/617328d702cb5f7eb01f19e4/html5/thumbnails/114.jpg)
Idea of proof
Projection onto the second factor gives a (nonrational) singular quarticthreefold Y:
P1Q2 − Q1P2 = 0,
with nodes at y1, . . . , y16.
The projection X → Y is a small resolutionof the singularities of Y. We analyse lines in the smooth locus of Y.
Main observation
There exists an irreducible curve (of genus 289) of lines l ⊂ Y, givingsections of π.
Counting points
![Page 115: Birational geometry and arithmetic - Durham University](https://reader030.fdocuments.us/reader030/viewer/2022012418/617328d702cb5f7eb01f19e4/html5/thumbnails/115.jpg)
Idea of proof
Projection onto the second factor gives a (nonrational) singular quarticthreefold Y:
P1Q2 − Q1P2 = 0,
with nodes at y1, . . . , y16. The projection X → Y is a small resolutionof the singularities of Y.
We analyse lines in the smooth locus of Y.
Main observation
There exists an irreducible curve (of genus 289) of lines l ⊂ Y, givingsections of π.
Counting points
![Page 116: Birational geometry and arithmetic - Durham University](https://reader030.fdocuments.us/reader030/viewer/2022012418/617328d702cb5f7eb01f19e4/html5/thumbnails/116.jpg)
Idea of proof
Projection onto the second factor gives a (nonrational) singular quarticthreefold Y:
P1Q2 − Q1P2 = 0,
with nodes at y1, . . . , y16. The projection X → Y is a small resolutionof the singularities of Y. We analyse lines in the smooth locus of Y.
Main observation
There exists an irreducible curve (of genus 289) of lines l ⊂ Y, givingsections of π.
Counting points