Birational geometry and arithmetic. July 2012

Transcription

1 Birational geometry and arithmetic July 2012

2 Basic questions Let F be a field and X a smooth projective algebraic variety over F. Introduction

3 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 Introduction

4 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 = F q, Q, F q (t), C(t),... Introduction

5 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 = F q, Q, F q (t), C(t),... Main idea Arithmetic properties are governed by global geometric invariants and the properties of the ground field F. Introduction

6 Classification schemes via dimension: curves, surfaces,... Introduction

7 Classification schemes via dimension: curves, surfaces,... via degree: Fano, general type, intermediate type Introduction

8 Classification schemes via dimension: curves, surfaces,... via degree: Fano, general type, intermediate type X d P n, with d < n + 1, d > n + 1 or d = n + 1 how close to P n : rational, unirational, rationally connected Introduction

9 Classification schemes via dimension: curves, surfaces,... via degree: Fano, general type, intermediate type X d P n, with d < n + 1, d > n + 1 or d = n + 1 how close to P n : rational, unirational, rationally connected In small dimensions some of these notions coincide, e.g., in dimension 2 and over algebraically closed fields of characteristic zero rational = unirational = rationally connected Introduction

10 Classification schemes via dimension: curves, surfaces,... via degree: Fano, general type, intermediate type X d P n, with d < n + 1, d > n + 1 or d = n + 1 how close to P n : rational, unirational, rationally connected In small dimensions some of these notions coincide, e.g., in dimension 2 and over algebraically closed fields of characteristic zero rational = unirational = rationally connected Small degree surfaces (Del Pezzo surfaces) over algebraically closed fields are rational. Cubic surfaces with a rational point are unirational. Introduction

11 Classification schemes via dimension: curves, surfaces,... via degree: Fano, general type, intermediate type X d P n, with d < n + 1, d > n + 1 or d = n + 1 how close to P n : rational, unirational, rationally connected In small dimensions some of these notions coincide, e.g., in dimension 2 and over algebraically closed fields of characteristic zero rational = unirational = rationally connected Small degree surfaces (Del Pezzo surfaces) over algebraically closed fields are rational. Cubic surfaces with a rational point are unirational. A Del Pezzo surface of degree d = 1 always has a point. Is it unirational? Introduction

12 Basic invariants Picard group Pic(X ), canonical class K X, cones of ample and effective divisors Introduction

13 Basic invariants Picard group Pic(X ), canonical class K X, cones of ample and effective divisors Fibration structures: Conic bundles, elliptic fibrations,... Introduction

14 Basic invariants Picard group Pic(X ), canonical class K X, cones of ample and effective divisors Fibration structures: Conic bundles, elliptic fibrations,... Over nonclosed fields F : Forms and Galois cohomology Brauer group Br(X ) Introduction

15 Basic invariants Picard group Pic(X ), canonical class K X, cones of ample and effective 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

16 Starting point: Curves over number fields What do we know about curves over number fields? g = 0: one can decide when X (F ) (local-global principle), if X (F ), then X (F ) is infinite and one has a good understanding of how X (F ) is distributed Curves

17 Starting point: Curves over number fields What do we know about curves over number fields? g = 0: one can decide when X (F ) (local-global principle), if X (F ), then X (F ) is infinite and one has a good understanding of how X (F ) is distributed g = 1: either X (F ) =, or X (F ) is finite, or infinite; no effective algorithms to decide, or to describe X (F ) (at present) Curves

18 Starting point: Curves over number fields What do we know about curves over number fields? g = 0: one can decide when X (F ) (local-global principle), if X (F ), then X (F ) is infinite and one has a good understanding of how X (F ) is distributed g = 1: either X (F ) =, or X (F ) is finite, or infinite; no effective algorithms to decide, or to describe X (F ) (at present) g 2: #X (F ) <, no effective algorithm to determine X (F ) (effective Mordell?) Curves

19 Curve covers (joint with F. Bogomolov) We say that C C if there exist an étale cover C C and a surjection C C. Curves

20 Curve covers (joint with F. Bogomolov) We say that C C if there exist an étale cover C C and a surjection C C. Theorem (2001) Let C be a hyperelliptic curve over F p of genus 2 and let C be any curve. Then C C. Curves

21 Curve covers (joint with F. Bogomolov) We say that C C if there exist an étale cover C C and a surjection C C. Theorem (2001) Let C be a hyperelliptic curve over F p of genus 2 and let C be any curve. Then C C. Let C be a hyperelliptic curve over Q of genus 2 and C 6 the curve y 2 = x 6 1. Then C C 6. Curves

22 Curve covers (joint with F. Bogomolov) We say that C C if there exist an étale cover C C and a surjection C C. Theorem (2001) Let C be a hyperelliptic curve over F p of genus 2 and let C be any curve. Then C C. Let C be a hyperelliptic curve over Q of genus 2 and C 6 the curve y 2 = x 6 1. Then C C 6. The cover C C is explicit, so that effective Mordell for C 6 implies effective Mordell for all hyperbolic hyperelliptic curves. Curves

23 Curve covers (joint with F. Bogomolov) We say that C C if there exist an étale cover C C and a surjection C C. Theorem (2001) Let C be a hyperelliptic curve over F p of genus 2 and let C be any curve. Then C C. Let C be a hyperelliptic curve over Q of genus 2 and C 6 the curve y 2 = x 6 1. Then C C 6. The cover C C is explicit, so that effective Mordell for C 6 implies effective Mordell for all hyperbolic hyperelliptic curves. Conjecture If C, C are curves of genus 2 over F p or Q then C C. Curves

24 Prototype: hypersurfaces X f P n over Q Birch 1961 If n 2 deg(f ) and X f is smooth then: if there are solutions in Q p and in R then there are solutions in Q asymptotic formulas Introduction: rational points on hypersurfaces

25 Prototype: hypersurfaces X f P n over Q Birch 1961 If n 2 deg(f ) and X f is smooth then: if there are solutions in Q p and in R then there are solutions in Q asymptotic formulas a positive proportion of hypersurfaces over Q have no local obstructions (Poonen-Voloch + Katz, 2003) Introduction: rational points on hypersurfaces

26 Prototype: hypersurfaces X f P n over Q Birch 1961 If n 2 deg(f ) and X f is smooth then: if there are solutions in Q p and in R then there are solutions in Q asymptotic formulas a positive proportion of hypersurfaces over Q have no local obstructions (Poonen-Voloch + Katz, 2003) the method works over F q [t] as well Introduction: rational points on hypersurfaces

27 Heuristic Given: f Z[x 0,..., x n ] homogeneous of degree deg(f ). We have f (x) = O(B deg(f ) ), for x := max j ( x j ) B. May (?) assume that the probability of f (x) = 0 is B deg(f ). There are B n+1 events with x B. We expect B n+1 d solutions with x B. Hope: reasonable at least when n + 1 d 0. Introduction: rational points on hypersurfaces

28 X f P n over C(t) Theorem If deg(f ) n then X f (C(t)). Proof: Insert x j = x j (t) C[t], of degree e, into f = J f J x J = 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

29 Existence of rational points Theorem (Esnault 2001) Every smooth rationally connected variety over a finite field has a rational point. Existence of points

30 Existence of rational points Theorem (Esnault 2001) Every smooth rationally connected variety over a finite field has a rational point. Theorem (Graber-Harris-Starr 2001) Every smooth rationally connected variety over the function field of a curve over an algebraically closed field has a rational point. Existence of points

31 Existence of rational points Theorem (Esnault 2001) Every smooth rationally connected variety over a finite field has a rational point. Theorem (Graber-Harris-Starr 2001) Every smooth rationally connected variety over the function field of a curve over an algebraically closed field has a rational point. Over number fields and higher dimensional function fields, there exist local and global obstructions to the existence of rational points. Existence of points

32 Hasse principle HP X (F v ) v X (F ) Existence of points

33 Hasse principle HP X (F v ) v X (F ) Basic examples: Quadrics, hypersurfaces of small degree Existence of points

34 Hasse principle HP X (F v ) v X (F ) Basic examples: Quadrics, hypersurfaces of small degree Counterexamples: Iskovskikh 1971: The conic bundle X P 1 given by x 2 + y 2 = f (t), f (t) = (t 2 2)(3 t 2 ). Existence of points

35 Hasse principle HP X (F v ) v X (F ) Basic examples: Quadrics, hypersurfaces of small degree Counterexamples: Iskovskikh 1971: The conic bundle X P 1 given by x 2 + y 2 = f (t), f (t) = (t 2 2)(3 t 2 ). Cassels, Guy 1966: The cubic surface 5x 3 + 9y z t 3 = 0. Existence of points

36 Hasse principle HP X (F v ) v X (F ) Basic examples: Quadrics, hypersurfaces of small degree Counterexamples: Iskovskikh 1971: The conic bundle X P 1 given by x 2 + y 2 = f (t), f (t) = (t 2 2)(3 t 2 ). Cassels, Guy 1966: The cubic surface 5x 3 + 9y z t 3 = 0. The proofs use basic algebraic number theory: quadratic and cubic reciprocity, divisibility of class numbers. Existence of points

37 Brauer-Manin obstruction Br(X F ) v Br(X F v ) x (x v ) v 0 Br(F ) v Br(F v ) v invv Q/Z 0, Existence of points

38 Brauer-Manin obstruction Br(X F ) v Br(X F v ) We have where x (x v ) v 0 Br(F ) v Br(F v ) v invv Q/Z 0, X (F ) X (F ) X (A F ) Br X (A F ), X (A F ) Br := A Br(X ) {(x v ) v X (A F ) v inv(a(x v )) = 0}. Existence of points

39 Brauer-Manin obstruction Br(X F ) v Br(X F v ) We have where x (x v ) v 0 Br(F ) v Br(F v ) v invv Q/Z 0, X (F ) X (F ) X (A F ) Br X (A F ), X (A F ) Br := A Br(X ) {(x v ) v X (A F ) v inv(a(x v )) = 0}. Manin s formulation gives a more systematic approach to identifying the algebraic structure behind the obstruction. Existence of points

40 Effectivity of Brauer-Manin obstructions The obstruction group for geometrically rational surfaces Br(X F )/Br(F ) = H 1 (Gal( F /F ), Pic( X )). Existence of points

41 Effectivity of Brauer-Manin obstructions The obstruction group for geometrically rational surfaces Br(X F )/Br(F ) = H 1 (Gal( F /F ), Pic( X )). Kresch-T Let X P n be a geometrically rational surface over a number field F. Then there is an effective algorithm to compute X (A F ) Br. Existence of points

42 Effectivity of Brauer-Manin obstructions The obstruction group for geometrically rational surfaces Br(X F )/Br(F ) = H 1 (Gal( F /F ), Pic( X )). Kresch-T Let X P n be a geometrically rational surface over a number field F. Then there is an effective algorithm to compute X (A F ) Br. Kresch-T Let X P n be a surface over a number field F. Assume that the geometric Picard group of X is torsion free and is generated by finitely 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 (A F ) Br. Existence of points

43 Effectivity of Brauer-Manin obstructions Hassett-Kresch-T Let X be a K3 surface over a number field F of degree 2. Then there exists an effective algorithm to compute: Pic( X ), together with the Galois action; Existence of points

44 Effectivity of Brauer-Manin obstructions Hassett-Kresch-T Let X be a K3 surface over a number field F of degree 2. Then there exists an effective algorithm to compute: Pic( X ), together with the Galois action; Br(X )/Br(F ) and X (A F ) Br. Existence of points

45 Effectivity of Brauer-Manin obstructions Hassett-Kresch-T Let X be a K3 surface over a number field F of degree 2. Then there exists an effective algorithm to compute: Pic( X ), together with the Galois action; Br(X )/Br(F ) and X (A F ) Br. Previously: computation of Pic( X ) on some Kummer K3 surfaces: Elsenhans, Jahnel, van Luijk Existence of points

46 Effectivity of Brauer-Manin obstructions Hassett-Kresch-T Let X be a K3 surface over a number field F of degree 2. Then there exists an effective algorithm to compute: Pic( X ), together with the Galois action; Br(X )/Br(F ) and X (A F ) Br. Previously: computation of Pic( X ) on some Kummer K3 surfaces: Elsenhans, Jahnel, van Luijk computation of X (A F ) Br for some Kummer surfaces (diagonal quartics): Bright, Skorobogatov, Swinnerton-Dyer, Ieronymou,... Existence of points

47 Effectivity of Brauer-Manin obstructions Hassett-Kresch-T Let X be a K3 surface over a number field F of degree 2. Then there exists an effective algorithm to compute: Pic( X ), together with the Galois action; Br(X )/Br(F ) and X (A F ) Br. Previously: computation of Pic( X ) on some Kummer K3 surfaces: Elsenhans, Jahnel, van Luijk computation of X (A F ) Br for some Kummer surfaces (diagonal quartics): Bright, Skorobogatov, Swinnerton-Dyer, Ieronymou,... computations with Br(X )[2] on general degree two K3 surfaces: Hassett, Varilly-Alvarado Existence of points

48 Effectivity of Brauer-Manin obstructions Hassett-Kresch-T Let X be a K3 surface over a number field F of degree 2. Then there exists an effective algorithm to compute: Pic( X ), together with the Galois action; Br(X )/Br(F ) and X (A F ) Br. Previously: Existence of points computation of Pic( X ) on some Kummer K3 surfaces: Elsenhans, Jahnel, van Luijk computation of X (A F ) Br for some Kummer surfaces (diagonal quartics): 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)

49 Effectivity of Brauer-Manin obstructions Main ingredients: effective Kuga-Satake correspondence, relies on an effective construction of the Bailey-Borel compactification of the moduli space of polarized K3 surfaces; Existence of points

50 Effectivity of Brauer-Manin obstructions Main ingredients: effective Kuga-Satake correspondence, relies on an effective construction of the Bailey-Borel compactification of the moduli space of polarized K3 surfaces; the work of Masser-Wüstholz on the effective Tate conjecture for abelian varieties; Existence of points

51 Effectivity of Brauer-Manin obstructions Main ingredients: effective Kuga-Satake correspondence, relies on an effective construction of the Bailey-Borel compactification of the moduli space of polarized K3 surfaces; the work of Masser-Wüstholz on the effective Tate conjecture for abelian varieties; effective GIT, Matsusaka, Hilbert Nullstellensatz, etc.. Existence of points

52 Computing the obstruction group Let X be a smooth intersection of two quadrics in P 4 over Q (a Del Pezzo surface of degree 4). Existence of points

53 Computing the obstruction group Let X be a smooth intersection of two quadrics in P 4 over Q (a Del Pezzo surface of degree 4). The Galois action on the 16 lines factors through the Weyl group W (D 5 ) (a group of order 1920). Bright, Bruin, Flynn, Logan 2007 If the degree of the splitting field over Q is > 96 then H 1 (Gal( F /F ), Pic( X )) = 0. Existence of points

54 Computing the obstruction group Let X be a smooth intersection of two quadrics in P 4 over Q (a Del Pezzo surface of degree 4). The Galois action on the 16 lines factors through the Weyl group W (D 5 ) (a group of order 1920). Bright, Bruin, Flynn, Logan 2007 If the degree of the splitting field over Q is > 96 then H 1 (Gal( F /F ), Pic( X )) = 0. In all other cases, the obstruction group is either 1, Z/2Z, or (Z/2Z) 2. Existence of points

55 Computing the obstruction group Let X be a smooth intersection of two quadrics in P 4 over Q (a Del Pezzo surface of degree 4). The Galois action on the 16 lines factors through the Weyl group W (D 5 ) (a group of order 1920). Bright, Bruin, Flynn, Logan 2007 If the degree of the splitting field over Q is > 96 then H 1 (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 and provide more examples of Iskovskikh type. Existence of points

56 Uniqueness of the Brauer-Manin obstruction Conjecture (Colliot-Thélène Sansuc 1980) Let X be a smooth projective rationally-connected surface over a number field F, e.g., an intersection of two quadrics in P 4 or a cubic in P 3. Then X (F ) = X (A F ) Br. Existence of points

57 Uniqueness of the Brauer-Manin obstruction Conjecture (Colliot-Thélène Sansuc 1980) Let X be a smooth projective rationally-connected surface over a number field F, e.g., an intersection of two quadrics in P 4 or a cubic in P 3. Then X (F ) = X (A F ) Br. In particular, existence of rational points on rationally-connected surfaces would be decidable. Existence of points

58 Uniqueness of the Brauer-Manin obstruction Conjecture (Colliot-Thélène Sansuc 1980) Let X be a smooth projective rationally-connected surface over a number field F, e.g., an intersection of two quadrics in P 4 or a cubic in P 3. Then X (F ) = X (A F ) Br. In particular, existence of rational points on rationally-connected surfaces would be decidable. Colliot-Thélène Sansuc Swinnerton-Dyer 1987: Degree 4 Del Pezzo surfaces admitting a conic bundle X P 1. Existence of points

59 Uniqueness of the Brauer-Manin obstruction Conjecture (Colliot-Thélène Sansuc 1980) Let X be a smooth projective rationally-connected surface over a number field F, e.g., an intersection of two quadrics in P 4 or a cubic in P 3. Then X (F ) = X (A F ) Br. In particular, existence of rational points on rationally-connected surfaces would be decidable. Colliot-Thélène Sansuc Swinnerton-Dyer 1987: Degree 4 Del Pezzo surfaces admitting a conic bundle X P 1.The conjecture is open for general degree 4 Del Pezzo surfaces. Existence of points

60 Do we believe this conjecture? Recall that a general Del Pezzo surface X has points locally, and that Br(X )/Br(F ) = 1. Existence of points

61 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 to allow an obstruction; this is counterintuitive. Existence of points

62 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 to allow an obstruction; this is counterintuitive. Elsenhans Jahnel 2007: Thousands of examples of cubic surfaces over Q with different Galois actions, the conjecture holds in all cases. Existence of points

63 Del Pezzo surfaces over F q (t) Theorem (Hassett-T. 2011) Let k be a finite field with at least elements and X a general Del Pezzo surface of degree 4 over F = k(t) such that its integral model X P 1 is a complete intersection in P 1 P 4 of two general forms of bi-degree (1, 2). Then X (F ). Existence of points

64 Del Pezzo surfaces over F q (t) Theorem (Hassett-T. 2011) Let k be a finite field with at least elements and X a general Del Pezzo surface of degree 4 over F = k(t) such that its integral model X P 1 is a complete intersection in P 1 P 4 of two general forms of bi-degree (1, 2). Then X (F ). The idea of proof will follow... Existence of points

65 K3 surfaces Potential density: Zariski density after a finite extension of the field. Zariski density

66 K3 surfaces Potential density: Zariski density after a finite extension of the field. Bogomolov-T If X is a K3 surface which is either elliptic or has infinite automorphisms then potential density holds for X. Zariski density

67 K3 surfaces Potential density: Zariski density after a finite extension of the field. Bogomolov-T If X is a K3 surface which is either elliptic or has infinite automorphisms then potential density holds for X. What about general K3 surfaces, i.e., those with Picard rank one? Zariski density

68 K3 surfaces Potential density: Zariski density after a finite extension of the field. Bogomolov-T If X is a K3 surface which is either elliptic or has infinite automorphisms then potential density holds for X. What about general K3 surfaces, i.e., those with Picard rank one? Let X P 1 P 3 be a general hypersurface of bidegree (1, 4). Then the K3 surface fibration X P 1 has a Zariski dense set of sections, i.e., such K3 surfaces over F = C(t) have Zariski dense rational points; more generally, this holds for general pencils of K3 surfaces of degree 18 (Hassett-T. 2008) Zariski density

69 K3 surfaces Potential density: Zariski density after a finite extension of the field. Bogomolov-T If X is a K3 surface which is either elliptic or has infinite automorphisms then potential density holds for X. What about general K3 surfaces, i.e., those with Picard rank one? Let X P 1 P 3 be a general hypersurface of bidegree (1, 4). Then the K3 surface fibration X P 1 has a Zariski dense set of sections, i.e., such K3 surfaces over F = C(t) have Zariski dense rational points; more generally, this holds for general pencils of K3 surfaces of degree 18 (Hassett-T. 2008) Same holds, if X P 1 P 3 is given by a general form of bidegree (2, 4) (Zhiyuan Li 2011) Zariski density

70 Higher dimensions Potential density holds for: all smooth Fano threefolds, with the possible exception of 2:1 W 2 P 3, ramified in a degree 6 surface S 6 (Harris-T. 1998, Bogomolov-T. 1998) Zariski density

71 Higher dimensions Potential density holds for: all smooth Fano threefolds, with the possible exception of 2:1 W 2 P 3, ramified in a degree 6 surface S 6 (Harris-T. 1998, Bogomolov-T. 1998) W 2, provided S 6 is singular; similar results in dimension 4 (Chelsov-Park 2004, Cheltsov 2004) Zariski density

72 Higher dimensions Potential density holds for: all smooth Fano threefolds, with the possible exception of 2:1 W 2 P 3, ramified in a degree 6 surface S 6 (Harris-T. 1998, Bogomolov-T. 1998) W 2, provided S 6 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) Zariski density

73 Higher dimensions Potential density holds for: all smooth Fano threefolds, with the possible exception of 2:1 W 2 P 3, ramified in a degree 6 surface S 6 (Harris-T. 1998, Bogomolov-T. 1998) W 2, provided S 6 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 not containing a plane and admitting a hyperplane section with 6 ordinary double points in general linear position (Hassett-T. 2008) Zariski density

74 Counting rational points Counting problems depend on: a projective embedding X P n ; a choice of X X ; a choice of a height function H : P n (F ) R >0. Counting points

75 Counting rational points Counting problems depend on: a projective embedding X P n ; a choice of X X ; a choice of a height function H : P n (F ) R >0. Main problem N(X (F ), B) = #{x X (F ) H(x) B}? c B a log(b) b 1 Counting points

76 The geometric framework Conjecture (Manin 1989) Let X P n be a smooth projective Fano variety over a number field F, in its anticanonical embedding. Counting points

77 The geometric framework Conjecture (Manin 1989) Let X P n be a smooth projective Fano variety over a number field F, in its anticanonical embedding. Then there exists a Zariski open subset X X such that where b = rk Pic(X ). N(X (F ), B) c B log(b) b 1, B, Counting points

78 The geometric framework Conjecture (Manin 1989) Let X P n be a smooth projective Fano variety over a number field F, in its anticanonical embedding. Then there exists a Zariski open subset X X such that where b = rk Pic(X ). N(X (F ), B) c B log(b) b 1, B, 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 points has been proved for some families of Fano varieties, but is still open, e.g., for hypersurfaces X d P d, with d 5. Counting points

79 The function field case / Batyrev 1987 Let F = F q (B) be a global function field and X /F a smooth Fano variety. Let π : X B be a model. A point x X (F ) gives rise to a section x of π. Let L be a very ample line bundle on X. The height zeta function takes the form Z(s) = x = d q (L, x)s M d (F q )q ds, where d = (L, x) and M d is the space of sections of degree d. Counting points

80 The function field case / Batyrev 1987 The dimension of M d can be estimated, provided x is unobstructed: dim M d ( K X, x), x M d. Counting points

81 The function field case / Batyrev 1987 The dimension of M d can be estimated, provided x is unobstructed: dim M d ( K X, x), x M d. Heuristic assumption: M d (F q ) = q dim(m d ) leads to a modified zeta function Z mod (s) = q (L, x)s+( K X, x), its analytic properties are governed by the ratio of the linear forms ( K X, ) and (L, ) Counting points

82 The Batyrev Manin conjecture N(X, L, B) = c B a(l) log(b) b(l) 1 (1 + o(1)), B Counting points

83 The Batyrev Manin conjecture N(X, L, B) = c B a(l) log(b) b(l) 1 (1 + o(1)), B a(l) = inf{a a[l] + [K X ] Λ eff (X )}, Counting points

84 The Batyrev Manin conjecture N(X, L, B) = c B a(l) log(b) b(l) 1 (1 + o(1)), B a(l) = inf{a a[l] + [K X ] Λ eff (X )}, b(l) = codimension of the face of Λ eff (X ) containing a(l)[l] + [K X ], Counting points

85 The Batyrev Manin conjecture N(X, L, B) = c B a(l) log(b) b(l) 1 (1 + o(1)), B a(l) = inf{a a[l] + [K X ] Λ eff (X )}, b(l) = codimension of the face of Λ eff (X ) containing a(l)[l] + [K X ], c( K X ) = α(x ) β(x ) τ(k X ) volume of the effective cone, nontrivial part of the Brauer group Br(X )/Br(F ), Peyre s Tamagawa type number, Counting points

86 The Batyrev Manin conjecture N(X, L, B) = c B a(l) log(b) b(l) 1 (1 + o(1)), B a(l) = inf{a a[l] + [K X ] Λ eff (X )}, b(l) = codimension of the face of Λ eff (X ) containing a(l)[l] + [K X ], c( K X ) = α(x ) β(x ) τ(k X ) volume of the effective cone, nontrivial part of the Brauer group Br(X )/Br(F ), Peyre s Tamagawa type number, c(l) = y c(l X y ), where X Y is a Mori fiber space L-primitive fibrations of Batyrev T. Counting points

87 Justification G. Segal 1979 Cont d (S 2, P n (C)) Hol d (S 2, P n (C)), d Counting points

88 Justification G. Segal 1979 Cont d (S 2, P n (C)) Hol d (S 2, P n (C)), d (isomorphism of π i, for i d). Counting points

89 Justification G. Segal 1979 Cont d (S 2, P n (C)) Hol d (S 2, P n (C)), d (isomorphism of π i, for i d). H (Hol d (S 2, P n (C))) stabilizes, for d. Counting points

90 Justification G. Segal 1979 Cont d (S 2, P n (C)) Hol d (S 2, P n (C)), d (isomorphism of π i, for i d). H (Hol d (S 2, P n (C))) stabilizes, for d. This was generalized to Grassmannians and toric varieties as target spaces by Kirwan 1986, Guest 1994, and others. Counting points

91 Justification G. Segal 1979 Cont d (S 2, P n (C)) Hol d (S 2, P n (C)), d (isomorphism of π i, for i d). H (Hol d (S 2, P n (C))) stabilizes, for d. This was generalized to Grassmannians and toric varieties as target spaces by Kirwan 1986, Guest 1994, and others. Basic idea M d (F q ) q dim(m d ), for d, provided the homology stabilizes. Counting points

92 Ellenberg Venkatesh Westerland 2009 Effective stabilization of homology of Hurwitz spaces There exist A, B, D such that dim H d (Hur c G,n ) = dim H d(hur c G,n+D ), for n Ad + B. Counting points

93 Ellenberg Venkatesh Westerland 2009 Effective stabilization of homology of Hurwitz spaces There exist A, B, D such that for n Ad + B. dim H d (Hur c G,n ) = dim H d(hur c G,n+D ), This has applications to Cohen Lenstra heuristics over function fields of curves. Counting points

94 Ellenberg Venkatesh Westerland 2009 Effective stabilization of homology of Hurwitz spaces There exist A, B, D such that for n Ad + B. dim H d (Hur c G,n ) = dim H d(hur c G,n+D ), This has applications to Cohen Lenstra heuristics over function fields of curves. Applications in the context of height zeta functions? Counting points

95 Results over Q Extensive numerical computations confirming Manin s conjecture, and its refinements, for Del Pezzo surfaces, hypersurfaces of small degree in dimension 3 and 4. Counting points

96 Results over Q Extensive numerical computations confirming Manin s conjecture, and its refinements, for Del Pezzo surfaces, hypersurfaces of small degree in dimension 3 and 4. Many recent theoretical results on asymptotics of points of bounded height on cubic surfaces and other Del Pezzo surfaces, via (uni)versal torsors (Browning, de la Breteche, Derenthal, Heath-Brown, Peyre, Salberger, Wooley,...) Counting points

97 Results over Q Extensive numerical computations confirming Manin s conjecture, and its refinements, for Del Pezzo surfaces, hypersurfaces of small degree in dimension 3 and 4. Many recent theoretical results on asymptotics of points of bounded height on cubic surfaces and other Del Pezzo surfaces, via (uni)versal torsors (Browning, de la Breteche, Derenthal, Heath-Brown, Peyre, Salberger, Wooley,...) Caution: counterexamples to Manin s conjecture for cubic surface bundles over P 1 (Batyrev-T. 1996). These are compactifications of affine spaces. Counting points

98 Points of smallest height Legendre: If ax 2 + by 2 = cz 2 is solvable mod p, for all p, then it is solvable in Z. Counting points

99 Points of smallest height Legendre: If ax 2 + by 2 = cz 2 is solvable mod p, for all p, then it is solvable in Z. Moreover, the height of the smallest solution is bounded by abc. Counting points

100 Points of smallest height Legendre: If ax 2 + by 2 = cz 2 is solvable mod p, for all p, then it is solvable in Z. Moreover, the height of the smallest solution is bounded by abc. An effective bound on the error term in the circle method (or in the other asymptotic results) also gives an effective bound on H min, the height of smallest solutions. Counting points

101 Points of smallest height Legendre: If ax 2 + by 2 = cz 2 is solvable mod p, for all p, then it is solvable in Z. Moreover, the height of the smallest solution is bounded by abc. An effective bound on the error term in the circle method (or in the other asymptotic results) also gives an effective bound on H min, the height of smallest solutions. In particular, Manin s and Peyre s conjecture suggest that H min 1 τ Counting points

102 Points of smallest height There are extensive numerical data for smallest points on Del Pezzo surfaces, Fano threefolds. E.g., Elsenhans-Jahnel 2010 Counting points

103 How are all of these related? (Hassett-T.) Let X be a Del Pezzo surface over F = F q (t) and π : X P 1. its integral model. Fix a height, and consider the spaces M d of sections of π of height d (degree of the section). Counting points

104 How are all of these related? (Hassett-T.) Let X be a Del Pezzo surface over F = F q (t) and π : X P 1. its integral model. Fix a height, and consider the spaces M d of sections of π of height d (degree of the section). Ideal scenario M d are geometrically irreducible, for d 0 Counting points

105 How are all of these related? (Hassett-T.) Let X be a Del Pezzo surface over F = F q (t) and π : X P 1. its integral model. Fix a height, and consider the spaces M d of sections of π of height d (degree of the section). Ideal scenario M d are geometrically irreducible, for d 0 M d dominate the intermediate Jacobian IJ(X ) of X Counting points

106 How are all of these related? (Hassett-T.) Let X be a Del Pezzo surface over F = F q (t) and π : X P 1. its integral model. Fix a height, and consider the spaces M d of sections of π of height d (degree of the section). Ideal scenario M d are geometrically irreducible, for d 0 M d dominate the intermediate Jacobian IJ(X ) of X there is a critical d 0, related to the height of X, such that M d0 either birational to IJ(X ) or to a P 1 -bundle over IJ(X ) is Counting points

107 How are all of these related? (Hassett-T.) Let X be a Del Pezzo surface over F = F q (t) and π : X P 1. its integral model. Fix a height, and consider the spaces M d of sections of π of height d (degree of the section). Ideal scenario M d are geometrically irreducible, for d 0 M d dominate the intermediate Jacobian IJ(X ) of X there is a critical d 0, related to the height of X, such that M d0 is either birational to IJ(X ) or to a P 1 -bundle over IJ(X ) for d d 0, M d fibers over IJ(X ), with general fiber a rationally connected variety Counting points

108 Del Pezzo surfaces over F q (t) We consider fibrations π : X P 1, with general fiber a degree-four Del Pezzo surface and with square-free discriminant. In this situation, we have an embedding X P(π ω 1 π ). Counting points

109 Del Pezzo surfaces over F q (t) We consider fibrations π : X P 1, with general fiber a degree-four Del Pezzo surface and with square-free discriminant. In this situation, we have an embedding We have with X P(π ω 1 π ). π ω 1 π = 5 i=1o P 1( a i ), a 1 a 2 a 3 a 4 a 5, occurring cases are discussed by Shramov (2006), in his investigations of rationality properties of such fibrations. Counting points

110 Del Pezzo surfaces over F q (t) We assume that π ω 1 π is generic, i.e., a 5 a 1 1; we can realize as a complete intersection. X P 1 P d, d = 4, 5,..., 8, Counting points

111 Del Pezzo surfaces over F q (t) We assume that π ω 1 π is generic, i.e., a 5 a 1 1; we can realize as a complete intersection. X P 1 P d, d = 4, 5,..., 8, Theorem (Hassett-T. 2011) Let k be a finite field with at least elements and X a general Del Pezzo surface of degree 4 over F = k(t) such that its integral model X P 1 is a complete intersection in P 1 P 4 of two general forms of bi-degree (1, 2). Then X (F ). Counting points

112 Idea of proof Write X P 1 P 4 as a complete intersection where P i, Q i are quadrics in P 4. P 1 s + Q 1 t = P 2 s + Q 2 t = 0, Counting points

113 Idea of proof Write X P 1 P 4 as a complete intersection P 1 s + Q 1 t = P 2 s + Q 2 t = 0, where P i, Q i are quadrics in P 4. The projection π : X P 1 has 16 constant sections corresponding to solutions y 1,..., y 16 of P 1 = Q 1 = P 2 = Q 2 = 0. Counting points

114 Idea of proof Projection onto the second factor gives a (nonrational) singular quartic threefold Y: P 1 Q 2 Q 1 P 2 = 0, with nodes at y 1,..., y 16. Counting points

115 Idea of proof Projection onto the second factor gives a (nonrational) singular quartic threefold Y: P 1 Q 2 Q 1 P 2 = 0, with nodes at y 1,..., y 16. The projection X Y is a small resolution of the singularities of Y. Counting points

116 Idea of proof Projection onto the second factor gives a (nonrational) singular quartic threefold Y: P 1 Q 2 Q 1 P 2 = 0, with nodes at y 1,..., y 16. The projection X Y is a small resolution of 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, giving sections of π. Counting points