Supersingular K3 Surfaces are Unirational

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1 Supersingular K3 Surfaces are Unirational Christian Liedtke (Technical University Munich) August 7, 2014 Seoul ICM 2014 Satellite Conference, Daejeon

2 Lüroth s theorem Given a field k a field,

3 Lüroth s theorem Given a field k a field, and a field extension k L k(t) with tr.deg. k L = 1

4 Lüroth s theorem Given a field k a field, and a field extension k L k(t) with tr.deg. k L = 1 we have Theorem (Lüroth) L = k(u)

5 Unirationality X an n-dimensional variety over a field k

6 Unirationality X an n-dimensional variety over a field k Definition X is called unirational if there exists dominant and rational. P n X

7 Unirationality X an n-dimensional variety over a field k Definition X is called unirational if there exists dominant and rational. P n X Equivalently, there exist inclusions of function fields k k(x ) k(t 1,..., t n ).

8 Curves Geometric version of Lüroth s theorem:

9 Curves Geometric version of Lüroth s theorem: Theorem (Lüroth, version 2) Let X be a smooth and proper curve over a field k. TFAE: X is unirational. X is birational to P 1 k.

10 Curves Geometric version of Lüroth s theorem: Theorem (Lüroth, version 2) Let X be a smooth and proper curve over a field k. TFAE: X is unirational. X is birational to P 1 k. In fact, we then even have X = P 1 k.

11 Surfaces Theorem (Castelnuovo) Let X be a smooth and proper surface over an algebraically closed field k

12 Surfaces Theorem (Castelnuovo) Let X be a smooth and proper surface over an algebraically closed field k of characteristic zero. TFAE: X is unirational. X is birational to P 2 k.

13 Surfaces Theorem (Castelnuovo) Let X be a smooth and proper surface over an algebraically closed field k of characteristic zero. TFAE: X is unirational. X is birational to P 2 k. Note: non-trivial birational geometry in dimension 2:

14 Surfaces Theorem (Castelnuovo) Let X be a smooth and proper surface over an algebraically closed field k of characteristic zero. TFAE: X is unirational. X is birational to P 2 k. Note: non-trivial birational geometry in dimension 2: P 2 and P 1 P 1 are birational, but not isomorphic.

15 Surfaces Theorem (Castelnuovo) Let X be a smooth and proper surface over an algebraically closed field k of characteristic zero. TFAE: X is unirational. X is birational to P 2 k. Note: non-trivial birational geometry in dimension 2: P 2 and P 1 P 1 are birational, but not isomorphic.

16 Surfaces Zariski: if char(k) = p > 0, then there exist unirational surfaces that are not rational

17 Surfaces Zariski: if char(k) = p > 0, then there exist unirational surfaces that are not rational Theorem (Zariski) Let X be a smooth and proper surface over an algebraically closed field k

18 Surfaces Zariski: if char(k) = p > 0, then there exist unirational surfaces that are not rational Theorem (Zariski) Let X be a smooth and proper surface over an algebraically closed field k of positive characteristic. TFAE: X is birational to P 2 k. X is separably unirational over k.

19 Threefolds Theorem ((Fano), Iskovskikh Manin, Clemens Griffiths, Artin Mumford) There do exist smooth and projective 3-folds over C that are

20 Threefolds Theorem ((Fano), Iskovskikh Manin, Clemens Griffiths, Artin Mumford) There do exist smooth and projective 3-folds over C that are unirational, but not rational.

21 Unirational surfaces coming back to surfaces:

22 Unirational surfaces coming back to surfaces: Question When is a surface in positive characteristic unirational?

23 Necessary conditions for unirationality Theorem (Shioda) Let X be a unirational surface. Then, ρ(x ) = b 2 (X ), where ρ(x ) = rank NS(X ) b 2 (X ) = dim Ql Hét 2 (X, Q l)

24 Necessary conditions for unirationality Definition A surface X is called Shioda supersingular if ρ(x ) = b 2 (X ).

25 Necessary conditions for unirationality Definition A surface X is called Shioda supersingular if ρ(x ) = b 2 (X ). Thus, unirational Shioda supersingular

26 Necessary conditions for unirationality Proposition Let X be a Shioda supersingular surface. Then, the F -crystal Hcris(X 2 /W ) is of slope 1

27 Necessary conditions for unirationality Definition A surface X is called (Artin )supersingular if is of slope 1. H 2 cris(x /W )

28 Necessary conditions for unirationality Definition A surface X is called (Artin )supersingular if is of slope 1. H 2 cris(x /W ) Thus, unirational Shioda supersingular (Artin )supersingular

29 Converse? The Tate conjecture gives (Artin )supersingular Shioda supersingular

30 K3 surfaces Definition A K3 surface is a smooth projective surface X over a field k such that ω X = OX and b 1 (X ) = 0.

31 K3 surfaces Definition A K3 surface is a smooth projective surface X over a field k such that ω X = OX and b 1 (X ) = 0. Example A smooth projective quartic hypersurface is a K3 surface. X 4 P 3 k

32 Tate conjecture Theorem (Nygaard, Nygaard Ogus, Maulik, Charles, Madapusi-Pera) The Tate conjecture holds for K3 surfaces in odd characteristic.

33 Tate conjecture Theorem (Nygaard, Nygaard Ogus, Maulik, Charles, Madapusi-Pera) The Tate conjecture holds for K3 surfaces in odd characteristic. Corollary For a K3 surface X in odd characteristic, X is Shioda supersingular X is Artin supersingular.

34 Tate conjecture Theorem (Nygaard, Nygaard Ogus, Maulik, Charles, Madapusi-Pera) The Tate conjecture holds for K3 surfaces in odd characteristic. Corollary For a K3 surface X in odd characteristic, X is Shioda supersingular X is Artin supersingular. Thus, for K3 surfaces, only one notion: supersingular.

35 Artin Rudakov Shafarevich Shioda conjecture Conjecture (Artin, Rudakov Shafarevich, Shioda) For a K3 surface X X is unirational X is supersingular.

36 Evidence Theorem The Artin Rudakov Shafarevich Shioda conjecture holds for Shioda-supersingular K3 surfaces in characteristic 2. (Rudakov Shafarevich) Supersingular K3 surfaces with σ 0 6 in characteristic 3. (Rudakov Shafarevich)

37 Evidence Theorem The Artin Rudakov Shafarevich Shioda conjecture holds for Shioda-supersingular K3 surfaces in characteristic 2. (Rudakov Shafarevich) Supersingular K3 surfaces with σ 0 6 in characteristic 3. (Rudakov Shafarevich) Supersingular K3 surfaces with σ 0 3 in characteristic 5. (Pho Shimada)

38 Evidence Theorem The Artin Rudakov Shafarevich Shioda conjecture holds for Shioda-supersingular K3 surfaces in characteristic 2. (Rudakov Shafarevich) Supersingular K3 surfaces with σ 0 6 in characteristic 3. (Rudakov Shafarevich) Supersingular K3 surfaces with σ 0 3 in characteristic 5. (Pho Shimada) Supersingular K3 surfaces with σ 0 2 in odd characteristic. (Shioda)

39 Evidence Corollary For every odd prime p, there exists a supersingular K3 surface in characteristic p, which is unirational.

40 Formal groups arising from algebraic varieties X proper over k Artin and Mazur studied functor Φ i X /k : (Art/k) (Abelian groups) S ker ( H í et (X k S, G m ) H í et (X, G m) )

41 Formal groups arising from algebraic varieties X proper over k Artin and Mazur studied functor Φ i X /k : (Art/k) (Abelian groups) S ker ( H í et (X k S, G m ) H í et (X, G m) ) Example (formal Picard group) If i = 1, then Φ 1 X /k = Pic X /k.

42 Formal Brauer group Fact (Artin Mazur) Have tangent-obstruction theory for Φ i X /k : tangent space: H i (X, O X ) obstruction space: H i+1 (X, O X )

43 Formal Brauer group Fact (Artin Mazur) Have tangent-obstruction theory for Φ i X /k : tangent space: H i (X, O X ) obstruction space: H i+1 (X, O X ) Corollary Definition For a K3 surface, Φ 2 X /k is pro-representable by a 1-dimensional formal group law Br X /k over k,

44 Formal Brauer group Fact (Artin Mazur) Have tangent-obstruction theory for Φ i X /k : tangent space: H i (X, O X ) obstruction space: H i+1 (X, O X ) Corollary Definition For a K3 surface, Φ 2 X /k is pro-representable by a 1-dimensional formal group law Br X /k over k, called the formal Brauer group.

45 1-dimensional formal group laws G = Spf k[[t]] a 1-dimensional formal group law over k, k algebraically closed, char(k) = p > 0

46 1-dimensional formal group laws G = Spf k[[t]] a 1-dimensional formal group law over k, k algebraically closed, char(k) = p > 0 multiplication µ : G G G is given by f (x, y) k[[x, y]]

47 1-dimensional formal group laws G = Spf k[[t]] a 1-dimensional formal group law over k, k algebraically closed, char(k) = p > 0 multiplication µ : G G G is given by f (x, y) k[[x, y]] G is determined by its height h(g)

48 1-dimensional formal group laws G = Spf k[[t]] a 1-dimensional formal group law over k, k algebraically closed, char(k) = p > 0 multiplication µ : G G G is given by f (x, y) k[[x, y]] G is determined by its height h(g) multiplication by p : G G is given by g(t) k[[t]]

49 1-dimensional formal group laws G = Spf k[[t]] a 1-dimensional formal group law over k, k algebraically closed, char(k) = p > 0 multiplication µ : G G G is given by f (x, y) k[[x, y]] G is determined by its height h(g) multiplication by p : G G is given by g(t) k[[t]] h(g) := { if g(t) = 0 h if g(t) = t ph + higher order terms.

50 Supersingular K3 surfaces Fact For a K3 surface X in odd characteristic, X is supersingular h( Br X ) =.

51 Supersingular K3 surfaces Fact For a K3 surface X in odd characteristic, X is supersingular h( Br X ) =. Reminder: want to prove X supersingular K3 X is unirational.

52 Families of torsors X P 1 an elliptic K3 surface with a section.

53 Families of torsors X P 1 an elliptic K3 surface with a section. X P 1 Weierstraß model

54 Families of torsors X P 1 an elliptic K3 surface with a section. X P 1 Weierstraß model A X identity component of Néron-model.

55 Families of torsors X P 1 an elliptic K3 surface with a section. X P 1 Weierstraß model A X identity component of Néron-model. Want to classify families of A-torsors where X A A P 1 = P 1 P 1 S

56 Families of torsors X P 1 an elliptic K3 surface with a section. X P 1 Weierstraß model A X identity component of Néron-model. Want to classify families of A-torsors where X A A P 1 = P 1 P 1 S S is spectrum of a local, complete, and Noetherian k-algebra with residue field k.

57 Families of torsors X P 1 an elliptic K3 surface with a section. X P 1 Weierstraß model A X identity component of Néron-model. Want to classify families of A-torsors where X A A P 1 = P 1 P 1 S S is spectrum of a local, complete, and Noetherian k-algebra with residue field k. RHS is Cartesian.

58 Families of torsors X P 1 an elliptic K3 surface with a section. X P 1 Weierstraß model A X identity component of Néron-model. Want to classify families of A-torsors where X A A P 1 = P 1 P 1 S S is spectrum of a local, complete, and Noetherian k-algebra with residue field k. RHS is Cartesian. together with a degree N-multisection of A P 1 S.

59 Families of torsors classified by Abelian group ker ( H 1 ét (P1 S, A) ) res Hét 1 (P1, A) [N]

60 Families of torsors classified by Abelian group ker ( H 1 ét (P1 S, A) ) res Hét 1 (P1, A) [N] Grothendieck Leray spectral sequence for X f P 1 E i,j 2 := H í et (X, R j f G m ) H i+j ét (X, G m )

61 Families of torsors Proposition X P 1 elliptic K3 with section.

62 Families of torsors Proposition X P 1 elliptic K3 with section. S spectrum of local, complete, and Noetherian k-algebra with residue field k.

63 Families of torsors Proposition X P 1 elliptic K3 with section. S spectrum of local, complete, and Noetherian k-algebra with residue field k. Then, families of A-torsors over S s.th. there exists degree-n multisection are classified by Br X (S) [N].

64 Families of torsors Corollary Non-trivial families of A-torsors over S = Spec k[[t]] can and do exist if and only if h( Br X ) =,

65 Families of torsors Corollary Non-trivial families of A-torsors over S = Spec k[[t]] can and do exist if and only if h( Br X ) =, that is, X is supersingular,

66 Families of torsors Corollary Non-trivial families of A-torsors over S = Spec k[[t]] can and do exist if and only if h( Br X ) =, that is, X is supersingular, and p divides N.

67 Families of torsors Theorem X P 1 supersingular K3 with elliptic fibration with section. Then, there exists a smooth family of supersingular K3 surfaces X X Spec k S = Spec k[[t]] have 0 Pic(X η ) Pic(X ) Z/pZ 0

68 Families of torsors Theorem X P 1 supersingular K3 with elliptic fibration with section. Then, there exists a smooth family of supersingular K3 surfaces have X X Spec k S = Spec k[[t]] 0 Pic(X η ) Pic(X ) Z/pZ 0 In particular: non-trivial moduli.

69 Families of torsors Theorem X P 1 supersingular K3 with elliptic fibration with section. Then, there exists a smooth family of supersingular K3 surfaces have X X Spec k S = Spec k[[t]] 0 Pic(X η ) Pic(X ) Z/pZ 0 In particular: non-trivial moduli. have inseparable isogenies X η X k η X η

70 Families of torsors Theorem X P 1 supersingular K3 with elliptic fibration with section. Then, there exists a smooth family of supersingular K3 surfaces have X X Spec k S = Spec k[[t]] 0 Pic(X η ) Pic(X ) Z/pZ 0 In particular: non-trivial moduli. have inseparable isogenies X η X k η X η In particular: X unirational X η unirational.

71 Families of torsors upshot X P 1 supersingular K3 with elliptic fibration with section.

72 Families of torsors upshot X P 1 supersingular K3 with elliptic fibration with section. Then, there exists a non-trivial deformation of X X Spec k[[t]],

73 Families of torsors upshot X P 1 supersingular K3 with elliptic fibration with section. Then, there exists a non-trivial deformation of X X Spec k[[t]], such that X unirational X η unirational

74 Artin invariant Proposition Definition (Artin) Let X be a supersingular K3 surface in odd characteristic.

75 Artin invariant Proposition Definition (Artin) Let X be a supersingular K3 surface in odd characteristic. Then, disc NS(X ) = p 2σ 0

76 Artin invariant Proposition Definition (Artin) Let X be a supersingular K3 surface in odd characteristic. Then, disc NS(X ) = p 2σ 0 for some integer σ 0 1 σ 0 10,

77 Artin invariant Proposition Definition (Artin) Let X be a supersingular K3 surface in odd characteristic. Then, disc NS(X ) = p 2σ 0 for some integer σ 0 1 σ 0 10, called the Artin invariant.

78 Artin invariant Theorem (Rudakov Shafarevich) Let X be a supersingular K3 surface in odd characteristic.

79 Artin invariant Theorem (Rudakov Shafarevich) Let X be a supersingular K3 surface in odd characteristic. Then, NS (X ) is determined up to isometry by σ 0.

80 Artin invariant Theorem (Rudakov Shafarevich) Let X be a supersingular K3 surface in odd characteristic. Then, NS (X ) is determined up to isometry by σ 0. Definition This lattice is called the supersingular K3 lattice of Artin invariant σ 0.

81 Artin invariant Proposition Let X be a supersingular K3 surface in odd characteristic. There exists an elliptic fibration X P 1, possibly without section.

82 Artin invariant Proposition Let X be a supersingular K3 surface in odd characteristic. There exists an elliptic fibration X P 1, possibly without section. If σ 0 9, then there exists an elliptic fibration X P 1 with section.

83 Supersingular K3 crystals X supersingular K3 surface over k

84 Supersingular K3 crystals X supersingular K3 surface over k H := H 2 cris (X /W ) free W (k)-module of rank 22

85 Supersingular K3 crystals X supersingular K3 surface over k H := Hcris 2 (X /W ) free W (k)-module of rank 22 Frobenius ϕ : H H

86 Supersingular K3 crystals X supersingular K3 surface over k H := Hcris 2 (X /W ) free W (k)-module of rank 22 Frobenius ϕ : H H Poincaré duality induces perfect pairing H H W (k)

87 Supersingular K3 crystals X supersingular K3 surface over k H := Hcris 2 (X /W ) free W (k)-module of rank 22 Frobenius ϕ : H H Poincaré duality induces perfect pairing H H W (k) Tate module T H := {x H x = ϕ(x)} H

88 Supersingular K3 crystals X supersingular K3 surface over k H := Hcris 2 (X /W ) free W (k)-module of rank 22 Frobenius ϕ : H H Poincaré duality induces perfect pairing H H W (k) Tate module T H := {x H x = ϕ(x)} H is free Z p -module of rank 22 with disc T H = p 2σ 0.

89 Moduli of supersingular K3 crystals Theorem (Ogus) Let N be a supersingular K3 lattice with disc N = p 2σ 0.

90 Moduli of supersingular K3 crystals Theorem (Ogus) Let N be a supersingular K3 lattice with disc N = p 2σ 0. There exists a moduli space M N for supersingular K3 crystals (H, ϕ,, ) with N T H.

91 Moduli of supersingular K3 crystals Theorem (Ogus) Let N be a supersingular K3 lattice with disc N = p 2σ 0. There exists a moduli space M N for supersingular K3 crystals (H, ϕ,, ) with N T H. M N is irreducible, smooth and projective over F p.

92 Ogus period map N a supersingular K3 lattice.

93 Ogus period map N a supersingular K3 lattice. S N M N moduli space of supersingular K3 surfaces X with N NS(X ). moduli space of supersingular K3 crystals H with N T H.

94 Ogus period map N a supersingular K3 lattice. S N M N moduli space of supersingular K3 surfaces X with N NS(X ). moduli space of supersingular K3 crystals H with N T H. Ogus period map π N : S N M N

95 Ogus crystalline Torelli theorem Theorem (Ogus) In characteristic p 5, there exists a period map π N : S N M N.

96 Ogus crystalline Torelli theorem Theorem (Ogus) In characteristic p 5, there exists a period map π N : S N M N. π N is étale, but not of finite type

97 Ogus crystalline Torelli theorem Theorem (Ogus) In characteristic p 5, there exists a period map π N : S N M N. π N is étale, but not of finite type for X, Y supersingular K3 surfaces X = Y Hcris(X 2 /W ) = Hcris(Y 2 /W ). (RHS: isomorphism of supersingular K3 crystals)

98 Families of torsors, II Theorem N, N + supersingular K3 lattices in odd characteristic with σ 0 (N + ) = σ 0 (N) + 1.

99 Families of torsors, II Theorem N, N + supersingular K3 lattices in odd characteristic with σ 0 (N + ) = σ 0 (N) + 1. Then, there exists a P 1 -bundle structure M N+ M N.

100 Families of torsors, II Theorem N, N + supersingular K3 lattices in odd characteristic with σ 0 (N + ) = σ 0 (N) + 1. Then, there exists a P 1 -bundle structure M N+ M N. Corollary M N is an iterated P 1 -bundle over F p 2.

101 Conclusion In odd characteristic, Given supersingular K3 X with σ 0 9, there exists deformation X Spec k[[t]]

102 Conclusion In odd characteristic, Given supersingular K3 X with σ 0 9, there exists deformation X Spec k[[t]] σ 0 (X η ) = σ 0 (X ) + 1.

103 Conclusion In odd characteristic, Given supersingular K3 X with σ 0 9, there exists deformation X Spec k[[t]] σ 0 (X η ) = σ 0 (X ) + 1. X and X η are related by purely inseparable isogeny.

104 Conclusion In odd characteristic, Given supersingular K3 X with σ 0 9, there exists deformation X Spec k[[t]] σ 0 (X η ) = σ 0 (X ) + 1. X and X η are related by purely inseparable isogeny. These deformations induce a P 1 -bundle structure M N+ M N.

105 Conclusion In odd characteristic, Given supersingular K3 X with σ 0 9, there exists deformation X Spec k[[t]] σ 0 (X η ) = σ 0 (X ) + 1. X and X η are related by purely inseparable isogeny. These deformations induce a P 1 -bundle structure M N+ M N. There does exist a unirational K3 surface (Shioda).

106 Conclusion Theorem Let X, Y be supersingular K3 surfaces in characteristic p 5. Then, there exist purely inseparably isogenies X Y X

107 Conclusion Theorem Let X, Y be supersingular K3 surfaces in characteristic p 5. Then, there exist purely inseparably isogenies X Y X Theorem Supersingular K3 surfaces in characteristic p 5 are unirational.

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