the noether lefschetz theorem

The Noether–Lefschetz theorem

Lena Ji

Princeton University/University of Michigan

Stanford AG SeminarJune 11, 2021

Lena Ji (Princeton/Michigan) The Noether–Lefschetz theorem June 11, 2021 1 / 25

Outline

1 Introduction

2 Quartic surfaces in P3

3 Proof



Max Noether (1844–1921) Solomon Lefschetz (1884–1972)

Theorem (Noether–Lefschetz Theorem)

For a very general surface Sd ⊂ P3C of degree d ≥ 4, the restriction map

Pic(P3C)→ Pic(Sd) is an isomorphism.



M. Noether (1882): The only curves on a “general” surface Fµ ⊂ P3C of degree

µ ≥ 4 are complete intersections of Fµ with another surface

M. Noether, Zur Grundlegung der Theorie algebraischen Raumcurven (1882).



Theorem (Noether–Lefschetz Theorem)

For a very general surface Sd ⊂ P3C of degree d ≥ 4, the restriction map

Pic(P3C)→ Pic(Sd) is an isomorphism.

Very general means away from a countable union

1882: stated by M. Noether

1920s: proved by Lefschetz using topological methods for complex surfaces

S. Lefschetz. On certain numerical invariants of algebraic varieties with application to abelian varieties (1921), p. 359.

Generalizations?

Replace P3C by X

Replace C by k



X ⊂ PNk smooth subvariety, H ⊂ PN

k very general hypersurface

Pic(X ) // Pic(X ∩ H)

X smooth 3-fold + char k = 0:Moishezon 1960s, Carlson–Green–Griffiths–Harris 1983 (Hodge theory),Griffiths–Harris 1985 (degeneration + monodromy), Joshi 1995...

X ⊂ PNk complete intersection + char k ≥ 0:

Deligne 1960s (l-adic cohomology)

X normal 3-fold + char k = 0: Cl(X )→ Cl(X ∩ H)Ravindra–Srinivas 2008 (infinitesimal methods)

Today’s talk: X normal 3-fold + char k ≥ 0



X ⊂ PNk normal subvariety, H ⊂ PN

k very general hypersurface

Cl(X ) // Cl(X ∩ H)

X smooth 3-fold + char k = 0:Moishezon 1960s, Carlson–Green–Griffiths–Harris 1983 (Hodge theory),Griffiths–Harris 1985 (degeneration + monodromy), Joshi 1995...

X ⊂ PNk complete intersection + char k ≥ 0:

Deligne 1960s (l-adic cohomology)

X normal 3-fold + char k = 0: Cl(X )→ Cl(X ∩ H)Ravindra–Srinivas 2008 (formal completion)

Today: X normal 3-fold + char k = p ≥ 0


Main result

Theorem

Let X ⊂ PNk be a normal 3-fold and H ⊂ PN

k a very general hypersurface of degree4 or ≥ 6. Then Cl(X )→ Cl(X ∩ H) is an isomorphism up to torsion.

No cohomology, Hodge theory, or monodromy

Works for algebraically closed k of infinite transcendence degree and in anycharacteristic

e.g. C, Q(t1, t2, t3, . . . , ), Fp(t1, t2, t3, . . . , )


Quartic surfaces in P3

Set-up

C ⊂ P3K degree 4 elliptic curve of rank ≥ 18

q1, q2, p1, . . . , p15 ∈ C (K ) independent in Pic(C )/〈H〉 where H = OP3 (1)|C

∃! quadric Qj = (fj = 0) such that

Im(Pic(Qj)→ Pic(C )) = 〈2qj ,H − 2qj〉

∃! p16 such thatp1 + · · ·+ p15 + p16 = C ∩ T

where T = (g = 0) ⊂ P3 is a quartic surface

Claim

ρ(f1f2 + λg = 0) = 1 for any λ 6∈ K


Claim: ρ (Tλ := (f1f2 + λg = 0)) = 1 for any λ 6∈ K

Qj = (fj = 0) degree 2, T = (g = 0) degree 4

Proof of claim:

X → P1 pencil spanned by Q1 + Q2 and T

Dλ curve on Tλ

Can find Γ→ P1 and divisor D on X ×P1 Γ that restricts to Dλ

Pic(Q1)×Pic(C) Pic(Q2) ∼= Z, so want to restrict D to Q1 + Q2

But D need not be Q-Cartier along C = Q1 ∩ Q2

(D|Q1 )|C − (D|Q2 )|C is supported over {p1, . . . , p16} = C ∩ T

(D|Q1 )|C − (D|Q2 )|C = a1p1 + · · ·+ a16p16 ∈ 〈H, 2q1, 2q2〉

q1, q2, p1, . . . , p15 ∈ C (K ) were chosen independent in Pic(C )/〈H〉ai ’s are all the same a


Claim: ρ (Tλ := (f1f2 + λg = 0)) = 1 for any λ 6∈ K

Proof of claim (continued):

(D|Q1 )|C − (D|Q2 )|C = a(p1 + · · ·+ p16)

Local class group over pi is Z = 〈Q1pi 〉 = 〈−Q2pi 〉D − aQ1 is Q-Cartier along C

Pic(Q1)×Pic(C) Pic(Q2) ∼= Z is generated by restriction of OP3 (1)

(D − aQ1)|Q1+Q2 comes from a divisor D on P3

D|Tλ∼ Dλ


Set-up

For simplicity: Assume X smooth 3-fold

Set-up

L very ample line bundle such that for a general net in |L|

X // P2

1 all but finitely many fibers are irreducible curves, and

2 generic fiber Cη has genus > dim Alb(X ).

Side lemma: If M is very ample, then L =M≥2 has these properties

Want to show

Pic(X )∼=−→ Pic(T ) (up to torsion) for very general divisor T

Recover statement for Pic(X )→ Pic(X ∩ H) by embedding X


Outline of proof

5 main steps

Want to show

Pic(X )∼=−→ Pic(T ) up to torsion

Injectivity: Fiber X by curves to study Pic(X )

(1) Pic(X )→ Jac(generic fiber)

(2) Jac(generic fiber)→ Jac(very general fiber)

Surjectivity: Degeneration argument

(3) Surjectivity mod torsion for X to reducible member S0 + S1 ∈ |L2|(4) Specialize from very general T ∈ |L2| to S0 + S1

(5) Go back from S0 + S1 to T

(+ε) Degeneration argument for odd degrees


Step 1: Pic(X ) ↪→ Jac(Cη)(K )

X smooth 3-fold, φ : X 99K P2 general net

X ′bir //

φ′

��

X

φ~~

P2

Cη = generic fiber of φ′ is a curve over K = k(P2)

Fix a base point of the net, E ⊂ X ′ corresponding exceptional divisor. Define

Pic(X ′) // Jac(Cη)(K )

D ′ � // (D ′ − deg(D ′|Cη)E )|Cη

Restriction to Pic(X ) ⊂ Pic(X ′) is injective

Can show using assumptions on |L|


Step 2: Jac(Cη)(K ) ↪→ Jac(Cs)(k) for very general s ∈ P2

Jac(Cη) is an abelian variety over K = k(P2)

Spreads out to family of abelian varieties J → U ⊂ P2 (relative Jacobian)

Fact: J isogeny−−−−→ (constant family A× U)× (Q with no constant subfamilies)

A× U → U has only constant sections (no nontrivial map P2 → A)

Q → U has only countably many sections (Lang–Neron theorem)

Specialization of sections for Q → U to very general fiber is injective

Steps 1+2 =⇒ injectivity

If Cs is very general, then Pic(X ) ↪→ Pic(Y ) for Y ⊃ Cs


Constant part of J → U

L ⊂ P2 line corresponds to S ∈ |L|

Jfamily of

abelian varieties��

J |L? _oo

��

P2 ⊃ U L? _oo

Turns out that A = Pic0(X )

Pic0(X ) ⊂ Pic(X ) subgroup of algebraically trivial divisorsPic(X )/Pic0(X ) is a finitely-generated group

S. Lang and A. Neron. Rational points of abelian varieties over function fields (1959), p. 97

Pic0(X )∼=−→ Pic0(S) for general S ∈ |L|


Step 3: Jac(Cη)(K ) ∼= {sections over v gen line pair in U}First step toward surjectivity

Theorem (Graber–Starr 2013)

If J → B is a family of abelian varieties, then sections extend from a very generalpair of incident lines in B.

Split J into (constant family)× (family with no constant subfamilies)Simpler proof when the base is U ⊂ P2

Over very general line L have {sections of J |L → L} ↪→ Js(k)

=⇒ Surjectivity for reducible S0 + S1 ∈ |L2|: for S0,S1 ∈ |L| very general

Pic(X ) // // Pic(S0)×Pic(S0∩S1) Pic(S1)


Step 4: From v general T ∈ |L2| to reducible S0 + S1

Let T ∈ |L2| be very general, DT ∈ Pic(T )

X = BlBs |Λ|(X )→ |Λ| = P1 pencil containing S0 + S1 and T

Chow variety argument: after a base change Γ→ |Λ| can find a divisor D onX ×|Λ| Γ that restricts to DT on T

Look at restriction of D to S0 + S1, want to apply Step 3

Problem: Singularities of X ×|Λ| Γ (blew up singular curve Bs |Λ|)D may not be Q-Cartier over S0 ∩ S1 ∩ T = {p1, . . . , pm}Singularities of X ×|Λ| Γ over {pi} locally look like

OX×|Λ|Γ,pi∼= k[[y1, y2, y3, t]]/(y1y2 − try3)


Step 4 cont.: From v general T ∈ |L2| to reducible S0 + S1

If D not Q-Cartier along S0 ∩ S1 then (D|S0 )|S0∩S1 and (D|S1 )|S0∩S1 may differ

(D|S0 )|S0∩S1 − (D|S1 )|S0∩S1 is supported on {p1, . . . , pm}Conjecture of Kollar (proved by Voisin when k uncountable):If T is very general, then p1, . . . , pm are linearly independent in Pic(S0 ∩ S1).

=⇒ (D|S0 )|S0∩S1 − (D|S1 )|S0∩S1 = a(p1 + · · ·+ pm)

Replace D by D − aS0: this is Q-Cartier along S0 ∩ S1

Step 3 =⇒ can find DX ∈ Pic(X ) such that DX |S0+S1 ∼ (D − aS0)|S0+S1


Step 5: Surjectivity: from reducible to back to v general

Have DX ∈ Pic(X ) such that DX |S0+S1 ∼ (D − aS0)|S0+S1

On X ×|Λ| Γ: Want D − aS0 ∼ ψ∗DX , so that DX |T ∼ DT

ψ∗DX − (D − aS0) is a divisor on X ×|Λ| Γ that is trivial on S0 + S1

For smooth families, specialization of Neron–Severi groups is injective up totorsion. Argument only needs divisors to be Q-Cartier

A multiple of DX |T − DT is in Pic0(T )

Pic0(X ) ∼= Pic0(T )

Pic(X )→ Pic(T ) is surjective up to torsion


Other degrees?

Step 5 works for even multiples of LSpecialized from T ∈ |L2| to (member of |L|)+(member of |L|)Recall set-up: L =Mm for m ≥ 2

Can specialize to (member of |M4|)+(member of |M≥2|)Get result for very general members of |M≥6|

Conclusion

Pic(X )/torsion

// // Pic(T ) for very general T ∈ |M4| or |M≥6|


Generalizations

Singular varieties?

Steps 1–4 work if X is normal, replacing

Pic with ClPic0 with Cl0 = subgroup of algebraically trivial Weil divisors

Step 5 in char 0: OK after base changing to a resolution of singularities of X

Step 5 in char p > 0: OK after base changing to a purely inseparablealteration (purely inseparable morphism + partial resolution)

Torsion?

Expect no prime-to-p-torsion

Examples with p-torsion?? (Please let me know!)


Higher dimensions

If dimX = n ≥ 4

Steps 1–2 show Cl(X )→ Pic(C ) is injective for a very general completeintersection curve in |L|Steps 3–5 show Cl(X )→ Cl(T ) is surjective up to torsion for a very generalcomplete intersection surface in |L2| ∩ |L| ∩ · · · ∩ |L|

By factoring Cl(X )→ Cl(T ) through a divisor in |L| = |M≥2|, get

Theorem

Let X ⊂ PNk be a normal variety of dimension ≥ 4 and H ⊂ PN

k a very generalhypersurface of degree ≥ 2. Then Cl(X )→ Cl(X ∩ H) is an isomorphism up totorsion.


Thank you!


the noether lefschetz theorem

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