The Schottky Problem

Samuel Grushevsky

Princeton University Schottky problem is the following question:

Which principally polarized abelian varieties are Jacobians of curves?

Mg moduli space of curves C of genus g Ag moduli space of g-dimensional abelian varieties (A, Θ) (complex principally polarized) Jac : Mg ,→ Ag Torelli map Jg := Jac(Mg ) locus of Jacobians (Recall that A is a projective variety with a group structure; Θ is an ample divisor on A with h0(A, Θ) = 1; Jac(C) = Picg−1(C) ' Pic0(C))

Schottky problem.

Describe/characterize Jg ⊂ Ag .

What is the Schottky problem? Mg moduli space of curves C of genus g Ag moduli space of g-dimensional abelian varieties (A, Θ) (complex principally polarized) Jac : Mg ,→ Ag Torelli map Jg := Jac(Mg ) locus of Jacobians (Recall that A is a projective variety with a group structure; Θ is an ample divisor on A with h0(A, Θ) = 1; Jac(C) = Picg−1(C) ' Pic0(C))

Schottky problem.

Describe/characterize Jg ⊂ Ag .

What is the Schottky problem?

Schottky problem is the following question:

Which principally polarized abelian varieties are Jacobians of curves? Mg moduli space of curves C of genus g Ag moduli space of g-dimensional abelian varieties (A, Θ) (complex principally polarized) Jac : Mg ,→ Ag Torelli map Jg := Jac(Mg ) locus of Jacobians (Recall that A is a projective variety with a group structure; Θ is an ample divisor on A with h0(A, Θ) = 1; Jac(C) = Picg−1(C) ' Pic0(C))

Schottky problem.

Describe/characterize Jg ⊂ Ag .

What is the Schottky problem?

Schottky problem is the following question (over C):

Which principally polarized abelian varieties are Jacobians of curves? (Recall that A is a projective variety with a group structure; Θ is an ample divisor on A with h0(A, Θ) = 1; Jac(C) = Picg−1(C) ' Pic0(C))

Schottky problem.

Describe/characterize Jg ⊂ Ag .

What is the Schottky problem?

Schottky problem is the following question (over C):

Which principally polarized abelian varieties are Jacobians of curves?

Describe/characterize Jg ⊂ Ag .

What is the Schottky problem?

Schottky problem is the following question (over C):

Which principally polarized abelian varieties are Jacobians of curves?

Schottky problem is the following question (over C):

Which principally polarized abelian varieties are Jacobians of curves?

Schottky problem.

Describe/characterize Jg ⊂ Ag . Coleman’s conjecture For g suﬃciently large there are ﬁnitely many curves of genus g such that their Jacobians have complex multiplication.

(Super)string scattering amplitudes [D’Hoker-Phong], . . .

Why might we care about the Schottky problem? Coleman’s conjecture For g suﬃciently large there are ﬁnitely many curves of genus g such that their Jacobians have complex multiplication.

Could have applications to problems about curves easily stated in terms of the Jacobian:

(Super)string scattering amplitudes [D’Hoker-Phong], . . .

Why might we care about the Schottky problem?

Relates two important moduli spaces. Lots of beautiful geometry arises in this study. A “good” answer could help relate the geometry of Mg and Ag . Coleman’s conjecture For g suﬃciently large there are ﬁnitely many curves of genus g such that their Jacobians have complex multiplication.

Stronger conjecture (+ Andre-Oort =⇒ Coleman). There do not exist any complex geodesics for the natural metric on Ag that are contained in Jg (and intersect Jg ). [Work on this by M¨oller-Viehweg-Zuo;Hain, Toledo...] (Super)string scattering amplitudes [D’Hoker-Phong], . . .

Why might we care about the Schottky problem?

Relates two important moduli spaces. Lots of beautiful geometry arises in this study. A “good” answer could help relate the geometry of Mg and Ag . Could have applications to problems about curves easily stated in terms of the Jacobian: Stronger conjecture (+ Andre-Oort =⇒ Coleman). There do not exist any complex geodesics for the natural metric on Ag that are contained in Jg (and intersect Jg ). [Work on this by M¨oller-Viehweg-Zuo;Hain, Toledo...] (Super)string scattering amplitudes [D’Hoker-Phong], . . .

Why might we care about the Schottky problem?

Relates two important moduli spaces. Lots of beautiful geometry arises in this study. A “good” answer could help relate the geometry of Mg and Ag . Could have applications to problems about curves easily stated in terms of the Jacobian: Coleman’s conjecture For g suﬃciently large there are ﬁnitely many curves of genus g such that their Jacobians have complex multiplication. (Super)string scattering amplitudes [D’Hoker-Phong], . . .

Why might we care about the Schottky problem?

Relates two important moduli spaces. Lots of beautiful geometry arises in this study. A “good” answer could help relate the geometry of Mg and Ag . Could have applications to problems about curves easily stated in terms of the Jacobian: Coleman’s conjecture For g suﬃciently large there are ﬁnitely many curves of genus g such that their Jacobians have complex multiplication.

5 12 +3 = 15 Partial results

(g−3)(g−2) g(g+1) g 3g − 3 + 2 = 2 “weak” solutions (up to extra components)

Dimension counts

g dim Mg dim Ag

1 1 = 1 indecomposable 2 3 = 3 Mg = Ag 3 6 = 6 5 12 +3 = 15 Partial results

(g−3)(g−2) g(g+1) g 3g − 3 + 2 = 2 “weak” solutions (up to extra components)

Dimension counts

g dim Mg dim Ag

1 1 = 1 indecomposable 2 3 = 3 Mg = Ag 3 6 = 6

4 9 +1 = 10 Schottky0s original equation (g−3)(g−2) g(g+1) g 3g − 3 + 2 = 2 “weak” solutions (up to extra components)

Dimension counts

g dim Mg dim Ag

1 1 = 1 indecomposable 2 3 = 3 Mg = Ag 3 6 = 6

4 9 +1 = 10 Schottky0s original equation

5 12 +3 = 15 Partial results Dimension counts

g dim Mg dim Ag

1 1 = 1 indecomposable 2 3 = 3 Mg = Ag 3 6 = 6

4 9 +1 = 10 Schottky0s original equation

5 12 +3 = 15 Partial results

(g−3)(g−2) g(g+1) g 3g − 3 + 2 = 2 “weak” solutions (up to extra components) N Embed Ag into P and write equations for the image of Jg .

Hg := Siegel upper half-space of dimension g t = {τ ∈ Matg×g (C) | τ = τ, Imτ > 0}. g g g Given τ ∈ Hg , have Aτ := C /(τZ + Z ) ∈ Ag .

AB For γ = ∈ Sp(2g, ) let γ ◦ τ := (Cτ + D)−1(Aτ + B). CD Z Claim: Ag = Hg / Sp(2g, Z). Deﬁnition A modular form of weight k with respect to Γ ⊂ Sp(2g, Z) is a function F : Hg → C such that

k F (γ ◦ τ) = det(Cτ + D) F (τ) ∀γ ∈ Γ, ∀τ ∈ Hg

Classical (Riemann-Schottky) approach Hg := Siegel upper half-space of dimension g t = {τ ∈ Matg×g (C) | τ = τ, Imτ > 0}. g g g Given τ ∈ Hg , have Aτ := C /(τZ + Z ) ∈ Ag .

k F (γ ◦ τ) = det(Cτ + D) F (τ) ∀γ ∈ Γ, ∀τ ∈ Hg

Classical (Riemann-Schottky) approach

N Embed Ag into P and write equations for the image of Jg . g g g Given τ ∈ Hg , have Aτ := C /(τZ + Z ) ∈ Ag .

k F (γ ◦ τ) = det(Cτ + D) F (τ) ∀γ ∈ Γ, ∀τ ∈ Hg

Classical (Riemann-Schottky) approach

N Embed Ag into P and write equations for the image of Jg .

Hg := Siegel upper half-space of dimension g t = {τ ∈ Matg×g (C) | τ = τ, Imτ > 0}. AB For γ = ∈ Sp(2g, ) let γ ◦ τ := (Cτ + D)−1(Aτ + B). CD Z Claim: Ag = Hg / Sp(2g, Z). Deﬁnition A modular form of weight k with respect to Γ ⊂ Sp(2g, Z) is a function F : Hg → C such that

k F (γ ◦ τ) = det(Cτ + D) F (τ) ∀γ ∈ Γ, ∀τ ∈ Hg

Classical (Riemann-Schottky) approach

N Embed Ag into P and write equations for the image of Jg .

Hg := Siegel upper half-space of dimension g t = {τ ∈ Matg×g (C) | τ = τ, Imτ > 0}. g g g Given τ ∈ Hg , have Aτ := C /(τZ + Z ) ∈ Ag . Claim: Ag = Hg / Sp(2g, Z). Deﬁnition A modular form of weight k with respect to Γ ⊂ Sp(2g, Z) is a function F : Hg → C such that

k F (γ ◦ τ) = det(Cτ + D) F (τ) ∀γ ∈ Γ, ∀τ ∈ Hg

Classical (Riemann-Schottky) approach

N Embed Ag into P and write equations for the image of Jg .

Hg := Siegel upper half-space of dimension g t = {τ ∈ Matg×g (C) | τ = τ, Imτ > 0}. g g g Given τ ∈ Hg , have Aτ := C /(τZ + Z ) ∈ Ag .

AB For γ = ∈ Sp(2g, ) let γ ◦ τ := (Cτ + D)−1(Aτ + B). CD Z Deﬁnition A modular form of weight k with respect to Γ ⊂ Sp(2g, Z) is a function F : Hg → C such that

k F (γ ◦ τ) = det(Cτ + D) F (τ) ∀γ ∈ Γ, ∀τ ∈ Hg

Classical (Riemann-Schottky) approach

N Embed Ag into P and write equations for the image of Jg .

Hg := Siegel upper half-space of dimension g t = {τ ∈ Matg×g (C) | τ = τ, Imτ > 0}. g g g Given τ ∈ Hg , have Aτ := C /(τZ + Z ) ∈ Ag .

AB For γ = ∈ Sp(2g, ) let γ ◦ τ := (Cτ + D)−1(Aτ + B). CD Z Claim: Ag = Hg / Sp(2g, Z). Classical (Riemann-Schottky) approach

N Embed Ag into P and write equations for the image of Jg .

Hg := Siegel upper half-space of dimension g t = {τ ∈ Matg×g (C) | τ = τ, Imτ > 0}. g g g Given τ ∈ Hg , have Aτ := C /(τZ + Z ) ∈ Ag .

k F (γ ◦ τ) = det(Cτ + D) F (τ) ∀γ ∈ Γ, ∀τ ∈ Hg Deﬁnition 1 g g For ε ∈ 2 Z /Z the theta function of the second order is ε Θ[ε](τ, z) := θ (2τ, 2z). 0

As a function of z, θm(τ, z) is a section of tmΘ n (tm =translate by m) on Aτ , so θm(τ, z) is a section of nΘ.

For n = 2, θm(τ, z) is even/odd in z depending on whether 4ε · δ is even/odd. For m odd θm(τ, 0) ≡ 0.

0 Theta functions of the second order generate H (Aτ , 2Θ).

Deﬁnition 1 g g For ε, δ ∈ n Z /Z the theta function with characteristic ε, δ is

ε X h i θ (τ, z) := exp πi(N + ε, τ(N + ε)) + 2πi(N + ε, z + δ) δ g N∈Z Deﬁnition 1 g g For ε ∈ 2 Z /Z the theta function of the second order is ε Θ[ε](τ, z) := θ (2τ, 2z). 0

As a function of z, θm(τ, z) is a section of tmΘ n (tm =translate by m) on Aτ , so θm(τ, z) is a section of nΘ.

For n = 2, θm(τ, z) is even/odd in z depending on whether 4ε · δ is even/odd. For m odd θm(τ, 0) ≡ 0.

0 Theta functions of the second order generate H (Aτ , 2Θ).

Deﬁnition 1 g g For ε, δ ∈ n Z /Z (or m = τε + δ ∈ Aτ [n]) the theta function with characteristic ε, δ or m is

X h i θm(τ, z) := exp πi(N + ε, τ(N + ε)) + 2πi(N + ε, z + δ) g N∈Z Deﬁnition 1 g g For ε ∈ 2 Z /Z the theta function of the second order is ε Θ[ε](τ, z) := θ (2τ, 2z). 0

As a function of z, θm(τ, z) is a section of tmΘ n (tm =translate by m) on Aτ , so θm(τ, z) is a section of nΘ.

For n = 2, θm(τ, z) is even/odd in z depending on whether 4ε · δ is even/odd. For m odd θm(τ, 0) ≡ 0.

0 Theta functions of the second order generate H (Aτ , 2Θ).

Deﬁnition 1 g g For ε, δ ∈ n Z /Z (or m = τε + δ ∈ Aτ [n]) the theta function with characteristic ε, δ or m is

ε θ (τ, z) := θ (τ, z) := (exponential factor) θ(τ, z + m) m δ Deﬁnition 1 g g For ε ∈ 2 Z /Z the theta function of the second order is ε Θ[ε](τ, z) := θ (2τ, 2z). 0

For n = 2, θm(τ, z) is even/odd in z depending on whether 4ε · δ is even/odd. For m odd θm(τ, 0) ≡ 0.

0 Theta functions of the second order generate H (Aτ , 2Θ).

Deﬁnition 1 g g For ε, δ ∈ n Z /Z (or m = τε + δ ∈ Aτ [n]) the theta function with characteristic ε, δ or m is

ε θ (τ, z) := θ (τ, z) := (exponential factor) θ(τ, z + m) m δ

As a function of z, θm(τ, z) is a section of tmΘ n (tm =translate by m) on Aτ , so θm(τ, z) is a section of nΘ. Deﬁnition 1 g g For ε ∈ 2 Z /Z the theta function of the second order is ε Θ[ε](τ, z) := θ (2τ, 2z). 0

0 Theta functions of the second order generate H (Aτ , 2Θ).

Deﬁnition 1 g g For ε, δ ∈ n Z /Z (or m = τε + δ ∈ Aτ [n]) the theta function with characteristic ε, δ or m is

ε θ (τ, z) := θ (τ, z) := (exponential factor) θ(τ, z + m) m δ

As a function of z, θm(τ, z) is a section of tmΘ n (tm =translate by m) on Aτ , so θm(τ, z) is a section of nΘ.

For n = 2, θm(τ, z) is even/odd in z depending on whether 4ε · δ is even/odd. For m odd θm(τ, 0) ≡ 0. 0 Theta functions of the second order generate H (Aτ , 2Θ).

Deﬁnition 1 g g For ε, δ ∈ n Z /Z (or m = τε + δ ∈ Aτ [n]) the theta function with characteristic ε, δ or m is

ε θ (τ, z) := θ (τ, z) := (exponential factor) θ(τ, z + m) m δ

As a function of z, θm(τ, z) is a section of tmΘ n (tm =translate by m) on Aτ , so θm(τ, z) is a section of nΘ.

For n = 2, θm(τ, z) is even/odd in z depending on whether 4ε · δ is even/odd. For m odd θm(τ, 0) ≡ 0. Deﬁnition 1 g g For ε ∈ 2 Z /Z the theta function of the second order is ε Θ[ε](τ, z) := θ (2τ, 2z). 0 Deﬁnition 1 g g For ε, δ ∈ n Z /Z (or m = τε + δ ∈ Aτ [n]) the theta function with characteristic ε, δ or m is

ε θ (τ, z) := θ (τ, z) := (exponential factor) θ(τ, z + m) m δ

As a function of z, θm(τ, z) is a section of tmΘ n (tm =translate by m) on Aτ , so θm(τ, z) is a section of nΘ.

0 Theta functions of the second order generate H (Aτ , 2Θ). Theorem (Igusa, Mumford, Salvati Manni) For any n ≥ 2 theta constants embed n2g −1 Ag (2n, 4n) := Hg /Γ(2n, 4n) ,→ P ε τ 7→ θ (τ) δ 1 g g all ε,δ∈ n Z /Z

Classical Riemann-Schottky problem Write the deﬁning equations for

2g −1 Th(Jg (2, 4)) ⊂ Th(Ag (2, 4)) ⊂ P .

Theta constants of the second order Θ[ε](τ, 0) are modular forms of weight 1/2 for Γ(2, 4).

Theta constants of the second order deﬁne a generically 2g −1 injective Th : Ag (2, 4) → P (conjecturally an embedding).

Theta constants θm(τ, 0) are modular forms of weight 1/2 for a certain ﬁnite index normal subgroup Γ(2n, 4n) ⊂ Sp(2g, Z). Classical Riemann-Schottky problem Write the deﬁning equations for

2g −1 Th(Jg (2, 4)) ⊂ Th(Ag (2, 4)) ⊂ P .

Theorem (Igusa, Mumford, Salvati Manni) For any n ≥ 2 theta constants embed n2g −1 Ag (2n, 4n) := Hg /Γ(2n, 4n) ,→ P ε τ 7→ θ (τ) δ 1 g g all ε,δ∈ n Z /Z Theta constants of the second order deﬁne a generically 2g −1 injective Th : Ag (2, 4) → P (conjecturally an embedding).

2g −1 Th(Jg (2, 4)) ⊂ Th(Ag (2, 4)) ⊂ P .

Theta constants of the second order deﬁne a generically 2g −1 injective Th : Ag (2, 4) → P (conjecturally an embedding).

2g −1 Th(Jg (2, 4)) ⊂ Th(Ag (2, 4)) ⊂ P .

Theta constants θm(τ, 0) are modular forms of weight 1/2 for a certain finite index normal subgroup Γ(2n, 4n) ⊂ Sp(2g, Z). Theta constants of the second order Θ[ε](τ, 0) are modular forms of weight 1/2 for Γ(2, 4). Theorem (Igusa, Mumford, Salvati Manni) For any n ≥ 2 theta constants embed n2g −1 Ag (2n, 4n) := Hg /Γ(2n, 4n) ,→ P ε τ 7→ θ (τ) δ 1 g g all ε,δ∈ n Z /Z Theta constants of the second order define a generically 2g −1 injective Th : Ag (2, 4) → P (conjecturally an embedding). Theta constants θm(τ, 0) are modular forms of weight 1/2 for a certain finite index normal subgroup Γ(2n, 4n) ⊂ Sp(2g, Z). Theta constants of the second order Θ[ε](τ, 0) are modular forms of weight 1/2 for Γ(2, 4). Theorem (Igusa, Mumford, Salvati Manni) For any n ≥ 2 theta constants embed n2g −1 Ag (2n, 4n) := Hg /Γ(2n, 4n) ,→ P ε τ 7→ θ (τ) δ 1 g g all ε,δ∈ n Z /Z Theta constants of the second order define a generically 2g −1 injective Th : Ag (2, 4) → P (conjecturally an embedding). Classical Riemann-Schottky problem Write the defining equations for

2g −1 Th(Jg (2, 4)) ⊂ Th(Ag (2, 4)) ⊂ P . 4 208896 = 16· 13056

Theorem (Schottky, Igusa)

The deﬁning equation for J4 ⊂ A4 is

g deg Th(Jg (2, 4)) deg Th(Ag (2, 4))

1 1 = 1 2 1 = 1 3 16 = 16 Theorem (Schottky, Igusa)

The deﬁning equation for J4 ⊂ A4 is

g deg Th(Jg (2, 4)) deg Th(Ag (2, 4))

1 1 = 1 2 1 = 1 3 16 = 16 4 208896 = 16· 13056 g deg Th(Jg (2, 4)) deg Th(Ag (2, 4))

1 1 = 1 2 1 = 1 3 16 = 16 4 208896 = 16· 13056

Theorem (Schottky, Igusa)

The deﬁning equation for J4 ⊂ A4 is

 2 X ε X ε F (τ) := 24 θ (τ)16 −  θ (τ)8 . 4 δ  δ  1 g g 1 g g ε,δ∈ 2 Z /Z ε,δ∈ 2 Z /Z g deg Th(Jg (2, 4)) deg Th(Ag (2, 4))

1 1 = 1 2 1 = 1 3 16 = 16 4 208896 = 16· 13056

Theorem (Schottky, Igusa)

The deﬁning equation for J4 ⊂ A4 is  2 4 X 16 X 8 F4 := 2 θm (τ) −  θm(τ) m∈A[2] m∈A[2] g deg Th(Jg (2, 4)) deg Th(Ag (2, 4))

1 1 = 1 2 1 = 1 3 16 = 16 4 208896 = 16· 13056

Theorem (Schottky, Igusa)

The deﬁning equation for J4 ⊂ A4 is  2 4 X 16 X 8 F4 := 2 θm (τ) −  θm(τ) m∈A[2] m∈A[2]

Exercise. Check if there exist A4-geodesics lying in M4. g deg Th(Jg (2, 4)) deg Th(Ag (2, 4))

1 1 = 1 2 1 = 1 3 16 = 16 4 208896 = 16· 13056

Theorem (Schottky, Igusa)

The deﬁning equation for J4 ⊂ A4 is  2 4 X 16 X 8 F4 := 2 θm (τ) −  θm(τ) m∈A[2] m∈A[2]

Open Problem

Construct all geodesics for the metric on A4 contained in M4. are dual, thus . . . expectation values of . . . are equal . . . , so θ + ' θE ×E , and thus D16 8 8 2 g X 16 X 8 Fg := 2 θm − θm

vanishes on Jg for any g (this is true for g ≤ 4).

In fact the zero locus of F5 on M5 is the divisor of trigonal curves.

Physics conjecture (Belavin, Knizhnik, D’Hoker-Phong, ...)

The . . . SO(32) . . . type . . . superstring theory . . . and . . . E8 × E8 theory . . .

Theorem (G.-Salvati Manni) This conjecture is false for any g ≥ 5.

In terms of lattice theta functions,

X 16 X 8 θ (τ) = θ + (τ), θ (τ) = θE (τ). m D16 m 8 In fact the zero locus of F5 on M5 is the divisor of trigonal curves.

are dual, thus . . . expectation values of . . . are equal . . . , so θ + ' θE ×E , and thus D16 8 8 2 g X 16 X 8 Fg := 2 θm − θm

vanishes on Jg for any g (this is true for g ≤ 4).

Theorem (G.-Salvati Manni) This conjecture is false for any g ≥ 5.

In terms of lattice theta functions,

X 16 X 8 θ (τ) = θ + (τ), θ (τ) = θE (τ). m D16 m 8

Physics conjecture (Belavin, Knizhnik, D’Hoker-Phong, ...)

The . . . SO(32) . . . type . . . superstring theory . . . and . . . E8 × E8 theory . . . In fact the zero locus of F5 on M5 is the divisor of trigonal curves.

so θ + ' θE ×E , and thus D16 8 8 2 g X 16 X 8 Fg := 2 θm − θm

vanishes on Jg for any g (this is true for g ≤ 4).

Theorem (G.-Salvati Manni) This conjecture is false for any g ≥ 5.

In terms of lattice theta functions,

X 16 X 8 θ (τ) = θ + (τ), θ (τ) = θE (τ). m D16 m 8

Physics conjecture (Belavin, Knizhnik, D’Hoker-Phong, ...)

The . . . SO(32) . . . type . . . superstring theory . . . and . . . E8 × E8 theory . . . are dual, thus . . . expectation values of . . . are equal . . . , In fact the zero locus of F5 on M5 is the divisor of trigonal curves.

2 g X 16 X 8 Fg := 2 θm − θm

vanishes on Jg for any g (this is true for g ≤ 4).

Theorem (G.-Salvati Manni) This conjecture is false for any g ≥ 5.

In terms of lattice theta functions,

X 16 X 8 θ (τ) = θ + (τ), θ (τ) = θE (τ). m D16 m 8

Physics conjecture (Belavin, Knizhnik, D’Hoker-Phong, ...)

The . . . SO(32) . . . type . . . superstring theory . . . and . . . E8 × E8 theory . . . are dual, thus . . . expectation values of . . . are equal . . . , so θ + ' θE ×E , and thus D16 8 8 In fact the zero locus of F5 on M5 is the divisor of trigonal curves.

Theorem (G.-Salvati Manni) This conjecture is false for any g ≥ 5.

In terms of lattice theta functions,

X 16 X 8 θ (τ) = θ + (τ), θ (τ) = θE (τ). m D16 m 8

Physics conjecture (Belavin, Knizhnik, D’Hoker-Phong, ...)

The . . . SO(32) . . . type . . . superstring theory . . . and . . . E8 × E8 theory . . . are dual, thus . . . expectation values of . . . are equal . . . , so θ + ' θE ×E , and thus D16 8 8 2 g X 16 X 8 Fg := 2 θm − θm vanishes on Jg for any g (this is true for g ≤ 4). In fact the zero locus of F5 on M5 is the divisor of trigonal curves.

In terms of lattice theta functions,

X 16 X 8 θ (τ) = θ + (τ), θ (τ) = θE (τ). m D16 m 8

Physics conjecture (Belavin, Knizhnik, D’Hoker-Phong, ...)

Theorem (G.-Salvati Manni) This conjecture is false for any g ≥ 5. In terms of lattice theta functions,

X 16 X 8 θ (τ) = θ + (τ), θ (τ) = θE (τ). m D16 m 8

Physics conjecture (Belavin, Knizhnik, D’Hoker-Phong, ...)

Theorem (G.-Salvati Manni) This conjecture is false for any g ≥ 5. In fact the zero locus of F5 on M5 is the divisor of trigonal curves. These are the top self-intersection numbers of λ1/2 on Mg and Ag times the degree of Ag (2, 4) → Ag . (G., using Faber’s algorithm) Corollary

Th(Jg (2, 4)) ⊂ Th(Ag (2, 4)) is not a complete intersection for g = 5, 6, 7. (previously proven by Faber)

Challenge

Write at least one modular form vanishing on J5.

g deg Th(Jg (2, 4)) deg Th(Ag (2, 4))

1 1 1 2 1 1 3 16 16 4 208896 13056 5 282654670848 1234714624 6 23303354757572198400 25653961176383488 7 87534047502300588892024209408 197972857997555419746140160 Corollary

Th(Jg (2, 4)) ⊂ Th(Ag (2, 4)) is not a complete intersection for g = 5, 6, 7. (previously proven by Faber)

Challenge

Write at least one modular form vanishing on J5.

g deg Th(Jg (2, 4)) deg Th(Ag (2, 4))

1 1 1 2 1 1 3 16 16 4 208896 13056 5 282654670848 1234714624 6 23303354757572198400 25653961176383488 7 87534047502300588892024209408 197972857997555419746140160

These are the top self-intersection numbers of λ1/2 on Mg and Ag times the degree of Ag (2, 4) → Ag . (G., using Faber’s algorithm) Challenge

Write at least one modular form vanishing on J5.

g deg Th(Jg (2, 4)) deg Th(Ag (2, 4))

1 1 1 2 1 1 3 16 16 4 208896 13056 5 282654670848 1234714624 6 23303354757572198400 25653961176383488 7 87534047502300588892024209408 197972857997555419746140160

These are the top self-intersection numbers of λ1/2 on Mg and Ag times the degree of Ag (2, 4) → Ag . (G., using Faber’s algorithm) Corollary

Th(Jg (2, 4)) ⊂ Th(Ag (2, 4)) is not a complete intersection for g = 5, 6, 7. (previously proven by Faber) g deg Th(Jg (2, 4)) deg Th(Ag (2, 4))

1 1 1 2 1 1 3 16 16 4 208896 13056 5 282654670848 1234714624 6 23303354757572198400 25653961176383488 7 87534047502300588892024209408 197972857997555419746140160

These are the top self-intersection numbers of λ1/2 on Mg and Ag times the degree of Ag (2, 4) → Ag . (G., using Faber’s algorithm) Corollary

Th(Jg (2, 4)) ⊂ Th(Ag (2, 4)) is not a complete intersection for g = 5, 6, 7. (previously proven by Faber)

Challenge

Write at least one modular form vanishing on J5. g deg Th(Jg (2, 4)) deg Th(Ag (2, 4))

1 1 1 2 1 1 3 16 16 4 208896 13056 5 282654670848 1234714624 6 23303354757572198400 25653961176383488 7 87534047502300588892024209408 197972857997555419746140160

These are the top self-intersection numbers of λ1/2 on Mg and Ag times the degree of Ag (2, 4) → Ag . (G., using Faber’s algorithm) Corollary

Th(Jg (2, 4)) ⊂ Th(Ag (2, 4)) is not a complete intersection for g = 5, 6, 7. (previously proven by Faber)

Challenge

Write at least one (nice/invariant) modular form vanishing on J5. Theorem (Mumford, Poor) For any g there exist sets of characteristics 1 2g 2g S1,..., SN ⊂ 2 Z /Z such that τ ∈ Ag is the period matrix of a hyperelliptic Jacobian (τ ∈ Hypg ) if and only if for some 1 ≤ i ≤ N

∀m {θm(τ) = 0 ⇐⇒ m ∈ Si }

The hyperelliptic Schottky problem The hyperelliptic Schottky problem

Theorem (Mumford, Poor) For any g there exist sets of characteristics 1 2g 2g S1,..., SN ⊂ 2 Z /Z such that τ ∈ Ag is the period matrix of a hyperelliptic Jacobian (τ ∈ Hypg ) if and only if for some 1 ≤ i ≤ N

∀m {θm(τ) = 0 ⇐⇒ m ∈ Si } Deﬁnition

The Prym variety for an ´etaledouble cover C˜ → C of C ∈ Mg (given by a point η ∈ Jac(C)[2] \{0}) is

Prym(C, η) := Ker0(Jac(C˜) → Jac(C))

Denote Pg ⊂ Ag the Prym locus. Theorem (Schottky-Jung, Farkas-Rauch proportionality) Let τ be the period matrix of C and let π be the period matrix of the Prym (for the simplest choice of η). Then

ε 0 ε 0 ε 1 θ (π)2 = const θ (τ) · θ (τ) ∀ε, δ ∈ g−1/ g−1 δ 0 δ 1 δ 2Z Z

Using this allows us to get some equations for Th(Jg (2, 4)) from equations for Th(Ag−1(2, 4)).

Schottky-Jung approach (H. Farkas-Rauch) Denote Pg ⊂ Ag the Prym locus. Theorem (Schottky-Jung, Farkas-Rauch proportionality) Let τ be the period matrix of C and let π be the period matrix of the Prym (for the simplest choice of η). Then

ε 0 ε 0 ε 1 θ (π)2 = const θ (τ) · θ (τ) ∀ε, δ ∈ g−1/ g−1 δ 0 δ 1 δ 2Z Z

Using this allows us to get some equations for Th(Jg (2, 4)) from equations for Th(Ag−1(2, 4)).

Schottky-Jung approach (H. Farkas-Rauch)

Deﬁnition

The Prym variety for an ´etaledouble cover C˜ → C of C ∈ Mg (given by a point η ∈ Jac(C)[2] \{0}) is

Prym(C, η) := Ker0(Jac(C˜) → Jac(C)) Denote Pg ⊂ Ag the Prym locus. Theorem (Schottky-Jung, Farkas-Rauch proportionality) Let τ be the period matrix of C and let π be the period matrix of the Prym (for the simplest choice of η). Then

ε 0 ε 0 ε 1 θ (π)2 = const θ (τ) · θ (τ) ∀ε, δ ∈ g−1/ g−1 δ 0 δ 1 δ 2Z Z

Using this allows us to get some equations for Th(Jg (2, 4)) from equations for Th(Ag−1(2, 4)).

Schottky-Jung approach (H. Farkas-Rauch)

Deﬁnition

The Prym variety for an ´etaledouble cover C˜ → C of C ∈ Mg (given by a point η ∈ Jac(C)[2] \{0}) is

Prym(C, η) := Ker0(Jac(C˜) → Jac(C))∈ Ag−1 Theorem (Schottky-Jung, Farkas-Rauch proportionality) Let τ be the period matrix of C and let π be the period matrix of the Prym (for the simplest choice of η). Then

ε 0 ε 0 ε 1 θ (π)2 = const θ (τ) · θ (τ) ∀ε, δ ∈ g−1/ g−1 δ 0 δ 1 δ 2Z Z

Using this allows us to get some equations for Th(Jg (2, 4)) from equations for Th(Ag−1(2, 4)).

Schottky-Jung approach (H. Farkas-Rauch)

Deﬁnition

The Prym variety for an ´etaledouble cover C˜ → C of C ∈ Mg (given by a point η ∈ Jac(C)[2] \{0}) is

Prym(C, η) := Ker0(Jac(C˜) → Jac(C)) ∈ Ag−1

Denote Pg ⊂ Ag the Prym locus. Using this allows us to get some equations for Th(Jg (2, 4)) from equations for Th(Ag−1(2, 4)).

Schottky-Jung approach (H. Farkas-Rauch)

Deﬁnition

The Prym variety for an ´etaledouble cover C˜ → C of C ∈ Mg (given by a point η ∈ Jac(C)[2] \{0}) is

Prym(C, η) := Ker0(Jac(C˜) → Jac(C)) ∈ Ag−1

ε 0 ε 0 ε 1 θ (π)2 = const θ (τ) · θ (τ) ∀ε, δ ∈ g−1/ g−1 δ 0 δ 1 δ 2Z Z Schottky-Jung approach (H. Farkas-Rauch)

Deﬁnition

The Prym variety for an ´etaledouble cover C˜ → C of C ∈ Mg (given by a point η ∈ Jac(C)[2] \{0}) is

Prym(C, η) := Ker0(Jac(C˜) → Jac(C)) ∈ Ag−1

ε 0 ε 0 ε 1 θ (π)2 = const θ (τ) · θ (τ) ∀ε, δ ∈ g−1/ g−1 δ 0 δ 1 δ 2Z Z

Using this allows us to get some equations for Th(Jg (2, 4)) from equations for Th(Ag−1(2, 4)). Conjecture

J5 is equal to the Schottky-Jung locus in genus 5.

The locus of intermediate Jacobians of cubic threefolds is contained in the “big” (if we take just one η) Schottky-Jung locus in genus 5.

For g ≥ 7, Pg−1 ( Ag−1, so may have more equations ⇒ need to solve the Prym Schottky problem if the above is not enough.

Theorem (van Geemen)

The locus Jg is an irreducible component of the Schottky-Jung locus — the locus obtained by taking the ideal of equations deﬁning Th(Ag−1(2, 4)) and applying the proportionality for all double covers η. The locus of intermediate Jacobians of cubic threefolds is contained in the “big” (if we take just one η) Schottky-Jung locus in genus 5.

For g ≥ 7, Pg−1 ( Ag−1, so may have more equations ⇒ need to solve the Prym Schottky problem if the above is not enough.

Theorem (van Geemen)

Conjecture

J5 is equal to the Schottky-Jung locus in genus 5. The locus of intermediate Jacobians of cubic threefolds is contained in the “big” (if we take just one η) Schottky-Jung locus in genus 5.

For g ≥ 7, Pg−1 ( Ag−1, so may have more equations ⇒ need to solve the Prym Schottky problem if the above is not enough.

Theorem (van Geemen / Donagi)

The locus Jg is an irreducible component of the small / big Schottky-Jung locus — the locus obtained by taking the ideal of equations deﬁning Th(Ag−1(2, 4)) and applying the proportionality for all / for just one double cover(s) η.

Conjecture

J5 is equal to the“small” (i.e., if we take all η) Schottky-Jung locus in genus 5. For g ≥ 7, Pg−1 ( Ag−1, so may have more equations ⇒ need to solve the Prym Schottky problem if the above is not enough.

Theorem (van Geemen / Donagi)

Conjecture

J5 is equal to the“small” (i.e., if we take all η) Schottky-Jung locus in genus 5.

The locus of intermediate Jacobians of cubic threefolds is contained in the “big” (if we take just one η) Schottky-Jung locus in genus 5. Theorem (van Geemen / Donagi)

Conjecture

J5 is equal to the“small” (i.e., if we take all η) Schottky-Jung locus in genus 5.

The locus of intermediate Jacobians of cubic threefolds is contained in the “big” (if we take just one η) Schottky-Jung locus in genus 5.

For g ≥ 7, Pg−1 ( Ag−1, so may have more equations ⇒ need to solve the Prym Schottky problem if the above is not enough. We do get the one deﬁning equation for J4.

Get 8 conjectural deﬁning equations for J5 [Accola] that involve lots of combinatorics, unlike the deﬁning equation F4 for J4.

This is so far a “weak” (i.e., up to extra components) solution to the Schottky problem. Boundary degeneration of Pryms is hard [Alexeev-Birkenhake-Hulek]

+ We get explicit algebraic equations for theta constants.

− We do not really know Th(Ag−1(2, 4)) entirely (though we do know many elements of the ideal).

Equations for theta constants: recap This is so far a “weak” (i.e., up to extra components) solution to the Schottky problem. Boundary degeneration of Pryms is hard [Alexeev-Birkenhake-Hulek]

We do get the one deﬁning equation for J4.

Get 8 conjectural deﬁning equations for J5 [Accola] that involve lots of combinatorics, unlike the deﬁning equation F4 for J4.

− We do not really know Th(Ag−1(2, 4)) entirely (though we do know many elements of the ideal).

Equations for theta constants: recap

+ We get explicit algebraic equations for theta constants. This is so far a “weak” (i.e., up to extra components) solution to the Schottky problem. Boundary degeneration of Pryms is hard [Alexeev-Birkenhake-Hulek]

Get 8 conjectural deﬁning equations for J5 [Accola] that involve lots of combinatorics, unlike the deﬁning equation F4 for J4.

− We do not really know Th(Ag−1(2, 4)) entirely (though we do know many elements of the ideal).

Equations for theta constants: recap

+ We get explicit algebraic equations for theta constants.

We do get the one deﬁning equation for J4. This is so far a “weak” (i.e., up to extra components) solution to the Schottky problem. Boundary degeneration of Pryms is hard [Alexeev-Birkenhake-Hulek]

− We do not really know Th(Ag−1(2, 4)) entirely (though we do know many elements of the ideal).

Equations for theta constants: recap

+ We get explicit algebraic equations for theta constants.

We do get the one deﬁning equation for J4.

Get 8 conjectural deﬁning equations for J5 [Accola] that involve lots of combinatorics, unlike the deﬁning equation F4 for J4. This is so far a “weak” (i.e., up to extra components) solution to the Schottky problem. Boundary degeneration of Pryms is hard [Alexeev-Birkenhake-Hulek]

Equations for theta constants: recap

+ We get explicit algebraic equations for theta constants.

We do get the one deﬁning equation for J4.

Get 8 conjectural deﬁning equations for J5 [Accola] that involve lots of combinatorics, unlike the deﬁning equation F4 for J4.

− We do not really know Th(Ag−1(2, 4)) entirely (though we do know many elements of the ideal). Boundary degeneration of Pryms is hard [Alexeev-Birkenhake-Hulek]

Equations for theta constants: recap

+ We get explicit algebraic equations for theta constants.

We do get the one deﬁning equation for J4.

Get 8 conjectural deﬁning equations for J5 [Accola] that involve lots of combinatorics, unlike the deﬁning equation F4 for J4.

+ We get explicit algebraic equations for theta constants.

We do get the one deﬁning equation for J4.

Get 8 conjectural deﬁning equations for J5 [Accola] that involve lots of combinatorics, unlike the deﬁning equation F4 for J4.

− We do not really know Th(Ag−1(2, 4)) entirely (though we do know many elements of the ideal). This is so far a “weak” (i.e., up to extra components) solution to the Schottky problem. Boundary degeneration of Pryms is hard [Alexeev-Birkenhake-Hulek] For C ∈ Hypg have dim(Sing ΘJac(C)) = g − 3. For C ∈ Mg \ Hypg have dim(Sing ΘJac(C)) = g − 4. (By Riemann’s theta singularity theorem) Deﬁnition (Andreotti-Mayer loci) n o Nk := (A, Θ) ∈ Ag | dim Sing Θ ≥ k

Theorem (Andreotti-Mayer)

Hypg is an irreducible component of Ng−3. Jg is an irreducible component of Ng−4.

Theorem (Debarre)

Pg is an irreducible component of Ng−6.

Singularities of the theta divisor approach Deﬁnition (Andreotti-Mayer loci) n o Nk := (A, Θ) ∈ Ag | dim Sing Θ ≥ k

Theorem (Andreotti-Mayer)

Hypg is an irreducible component of Ng−3. Jg is an irreducible component of Ng−4.

Theorem (Debarre)

Pg is an irreducible component of Ng−6.

Singularities of the theta divisor approach

For C ∈ Hypg have dim(Sing ΘJac(C)) = g − 3. For C ∈ Mg \ Hypg have dim(Sing ΘJac(C)) = g − 4. (By Riemann’s theta singularity theorem) Theorem (Andreotti-Mayer)

Hypg is an irreducible component of Ng−3. Jg is an irreducible component of Ng−4.

Theorem (Debarre)

Pg is an irreducible component of Ng−6.

Singularities of the theta divisor approach

For C ∈ Hypg have dim(Sing ΘJac(C)) = g − 3. For C ∈ Mg \ Hypg have dim(Sing ΘJac(C)) = g − 4. (By Riemann’s theta singularity theorem) Deﬁnition (Andreotti-Mayer loci) n o Nk := (A, Θ) ∈ Ag | dim Sing Θ ≥ k Theorem (Debarre)

Pg is an irreducible component of Ng−6.

Singularities of the theta divisor approach

Theorem (Andreotti-Mayer)

Hypg is an irreducible component of Ng−3. Jg is an irreducible component of Ng−4. Singularities of the theta divisor approach

Theorem (Andreotti-Mayer)

Hypg is an irreducible component of Ng−3. Jg is an irreducible component of Ng−4.

Theorem (Debarre)

Pg is an irreducible component of Ng−6. Deﬁnition (Theta-null divisor)    Y ε  θ : = τ | θ (τ) = 0 null δ  1 g g   ε,δ∈ 2 Z /Z even  n even o = (A, Θ) ∈ Ag | A[2] ∩ Θ 6= ∅

N0 is a divisor in Ag [Beauville] 0 N0 = 2N0 ∪ θnull, two irreducible components [Debarre]

N1 ( N0, codimAg N1 ≥ 2 [Mumford]

codimAg N1 ≥ 3 [Ciliberto-van der Geer]

Andreotti-Mayer divisor N0

N0 ( Ag [Andreotti-Mayer] Deﬁnition (Theta-null divisor)    Y ε  θ : = τ | θ (τ) = 0 null δ  1 g g   ε,δ∈ 2 Z /Z even  n even o = (A, Θ) ∈ Ag | A[2] ∩ Θ 6= ∅

0 N0 = 2N0 ∪ θnull, two irreducible components [Debarre]

N1 ( N0, codimAg N1 ≥ 2 [Mumford]

codimAg N1 ≥ 3 [Ciliberto-van der Geer]

Andreotti-Mayer divisor N0

N0 ( Ag [Andreotti-Mayer] N0 is a divisor in Ag [Beauville] Deﬁnition (Theta-null divisor)    Y ε  θ : = τ | θ (τ) = 0 null δ  1 g g   ε,δ∈ 2 Z /Z even  n even o = (A, Θ) ∈ Ag | A[2] ∩ Θ 6= ∅

N1 ( N0, codimAg N1 ≥ 2 [Mumford]

codimAg N1 ≥ 3 [Ciliberto-van der Geer]

Andreotti-Mayer divisor N0

N0 ( Ag [Andreotti-Mayer] N0 is a divisor in Ag [Beauville] 0 N0 = 2N0 ∪ θnull, two irreducible components [Debarre] N1 ( N0, codimAg N1 ≥ 2 [Mumford]

codimAg N1 ≥ 3 [Ciliberto-van der Geer]

Andreotti-Mayer divisor N0

N0 ( Ag [Andreotti-Mayer] N0 is a divisor in Ag [Beauville] 0 N0 = 2N0 ∪ θnull, two irreducible components [Debarre]

Deﬁnition (Theta-null divisor)    Y ε  θ : = τ | θ (τ) = 0 null δ  1 g g   ε,δ∈ 2 Z /Z even  n even o = (A, Θ) ∈ Ag | A[2] ∩ Θ 6= ∅ codimAg N1 ≥ 3 [Ciliberto-van der Geer]

Andreotti-Mayer divisor N0

N0 ( Ag [Andreotti-Mayer] N0 is a divisor in Ag [Beauville] 0 N0 = 2N0 ∪ θnull, two irreducible components [Debarre]

Deﬁnition (Theta-null divisor)    Y ε  θ : = τ | θ (τ) = 0 null δ  1 g g   ε,δ∈ 2 Z /Z even  n even o = (A, Θ) ∈ Ag | A[2] ∩ Θ 6= ∅

N1 ( N0, codimAg N1 ≥ 2 [Mumford] Andreotti-Mayer divisor N0

N0 ( Ag [Andreotti-Mayer] N0 is a divisor in Ag [Beauville] 0 N0 = 2N0 ∪ θnull, two irreducible components [Debarre]

Deﬁnition (Theta-null divisor)    Y ε  θ : = τ | θ (τ) = 0 null δ  1 g g   ε,δ∈ 2 Z /Z even  n even o = (A, Θ) ∈ Ag | A[2] ∩ Θ 6= ∅

N1 ( N0, codimAg N1 ≥ 2 [Mumford]

codimAg N1 ≥ 3 [Ciliberto-van der Geer] 3 =: θnull

Theorem (G.-Salvati Manni, Smith-Varley + Debarre) 3 g−1 0 (Jg ∩ θnull) ⊂ θnull ⊂ θnull ⊂ (θnull ∩ N0) ⊂ Sing N0

Conjecture (Beauville, Debarre, ...) indec Ng−3 = Hypg ; Ng−4 \Jg ⊂ θnull within Ag 3 =: θnull

Conjecture (H. Farkas) Theorem (G.-Salvati Manni; Smith-Varley) even J4 ∩ θnull = ∃m ∈ A[2] θ(τ, m) = det ∂z ∂z θ(τ, m) = 0 i,j i j even = ∃m ∈ A[2] ∩ Θ; TCmΘ has rank ≤ 3

Theorem (G.-Salvati Manni, Smith-Varley + Debarre) 3 g−1 0 (Jg ∩ θnull) ⊂ θnull ⊂ θnull ⊂ (θnull ∩ N0) ⊂ Sing N0

Conjecture (Beauville, Debarre, ...) indec Ng−3 = Hypg ; Ng−4 \Jg ⊂ θnull within Ag

Thus interested in Jg ∩ θnull. For genus 4 have N0 = J4 ∪ θnull, so J4 \ θnull = N0 \ θnull. 3 =: θnull

Conjecture (H. Farkas) Theorem (G.-Salvati Manni; Smith-Varley) even J4 ∩ θnull = ∃m ∈ A[2] θ(τ, m) = det ∂z ∂z θ(τ, m) = 0 i,j i j even = ∃m ∈ A[2] ∩ Θ; TCmΘ has rank ≤ 3

Theorem (G.-Salvati Manni, Smith-Varley + Debarre) 3 g−1 0 (Jg ∩ θnull) ⊂ θnull ⊂ θnull ⊂ (θnull ∩ N0) ⊂ Sing N0

Conjecture (Beauville, Debarre, ...) indec Ng−3 = Hypg ; Ng−4 \Jg ⊂ θnull within Ag

Thus interested in Jg ∩ θnull. For genus 4 have N0 = J4 ∪ θnull, so J4 \ θnull = N0 \ θnull. Conjecture (H. Farkas) Theorem (G.-Salvati Manni) even J4 ∩ θnull = ∃m ∈ A[2] θ(τ, m) = det ∂z ∂z θ(τ, m) = 0 i,j i j 3 =: θnull

Theorem (G.-Salvati Manni, Smith-Varley + Debarre) 3 g−1 0 (Jg ∩ θnull) ⊂ θnull ⊂ θnull ⊂ (θnull ∩ N0) ⊂ Sing N0

Conjecture (Beauville, Debarre, ...) indec Ng−3 = Hypg ; Ng−4 \Jg ⊂ θnull within Ag

Thus interested in Jg ∩ θnull. For genus 4 have N0 = J4 ∪ θnull, so J4 \ θnull = N0 \ θnull. Conjecture (H. Farkas) Theorem (G.-Salvati Manni; Smith-Varley) even J4 ∩ θnull = ∃m ∈ A[2] θ(τ, m) = det ∂z ∂z θ(τ, m) = 0 i,j i j even = ∃m ∈ A[2] ∩ Θ; TCmΘ has rank ≤ 3 Theorem (G.-Salvati Manni, Smith-Varley + Debarre) 3 g−1 0 (Jg ∩ θnull) ⊂ θnull ⊂ θnull ⊂ (θnull ∩ N0) ⊂ Sing N0

Conjecture (Beauville, Debarre, ...) indec Ng−3 = Hypg ; Ng−4 \Jg ⊂ θnull within Ag

Thus interested in Jg ∩ θnull. For genus 4 have N0 = J4 ∪ θnull, so J4 \ θnull = N0 \ θnull. Conjecture (H. Farkas) Theorem (G.-Salvati Manni; Smith-Varley) even J4 ∩ θnull = ∃m ∈ A[2] θ(τ, m) = det ∂z ∂z θ(τ, m) = 0 i,j i j even 3 = ∃m ∈ A[2] ∩ Θ; TCmΘ has rank ≤ 3 =: θnull Conjecture (Beauville, Debarre, ...) indec Ng−3 = Hypg ; Ng−4 \Jg ⊂ θnull within Ag

Thus interested in Jg ∩ θnull. For genus 4 have N0 = J4 ∪ θnull, so J4 \ θnull = N0 \ θnull. Conjecture (H. Farkas) Theorem (G.-Salvati Manni; Smith-Varley) even J4 ∩ θnull = ∃m ∈ A[2] θ(τ, m) = det ∂z ∂z θ(τ, m) = 0 i,j i j even 3 = ∃m ∈ A[2] ∩ Θ; TCmΘ has rank ≤ 3 =: θnull

Theorem (G.-Salvati Manni, Smith-Varley + Debarre) 3 g−1 0 (Jg ∩ θnull) ⊂ θnull ⊂ θnull ⊂ (θnull ∩ N0) ⊂ Sing N0 Note that ΘA1×A2 = (ΘA1 × A2) ∪ (A1 × ΘA2 ).

Thus Sing(ΘA1×A2 ) ⊃ ΘA1 × ΘA2 . Conjecture (Arbarello-De Concini) Theorem (Ein-Lazarsfeld)

decomposable Ng−2 = Ag

Conjecture (Ciliberto-van der Geer) (k + 1)(k + 2) codim indec Nk ≥ Ag 2

Question

Is it possible that Nk = Nk+1 for some k?

More questions on Nk Conjecture (Arbarello-De Concini) Theorem (Ein-Lazarsfeld)

decomposable Ng−2 = Ag

Conjecture (Ciliberto-van der Geer) (k + 1)(k + 2) codim indec Nk ≥ Ag 2

Question

Is it possible that Nk = Nk+1 for some k?

More questions on Nk

Note that ΘA1×A2 = (ΘA1 × A2) ∪ (A1 × ΘA2 ).

Thus Sing(ΘA1×A2 ) ⊃ ΘA1 × ΘA2 . Conjecture (Arbarello-De Concini) Theorem (Ein-Lazarsfeld)

decomposable Ng−2 = Ag

Conjecture (Ciliberto-van der Geer) (k + 1)(k + 2) codim indec Nk ≥ Ag 2 More questions on Nk

Note that ΘA1×A2 = (ΘA1 × A2) ∪ (A1 × ΘA2 ).

Thus Sing(ΘA1×A2 ) ⊃ ΘA1 × ΘA2 . Conjecture (Arbarello-De Concini) Theorem (Ein-Lazarsfeld)

decomposable Ng−2 = Ag

Conjecture (Ciliberto-van der Geer) (k + 1)(k + 2) codim indec Nk ≥ Ag 2

Question

Is it possible that Nk = Nk+1 for some k? Theorem (Koll´ar)

For any (A, Θ) ∈ Ag , any z ∈ A we have multz Θ ≤ g.

Theorem (Smith-Varley)

If multz Θ = g, then A = E1 × · · · × Eg .

Conjecture

indec j g+1 k For A ∈ Ag and any z ∈ A, multz Θ ≤ 2 .

The bound holds and is achieved for Jacobians [Riemann] The bound holds and is achieved for Pryms [Mumford, Smith-Varley, Casalaina-Martin]

Thus the conjecture is true for g ≤ 5 (P5 = A5)

Multiplicity of the theta divisor Theorem (Smith-Varley)

If multz Θ = g, then A = E1 × · · · × Eg .

Conjecture

indec j g+1 k For A ∈ Ag and any z ∈ A, multz Θ ≤ 2 .

The bound holds and is achieved for Jacobians [Riemann] The bound holds and is achieved for Pryms [Mumford, Smith-Varley, Casalaina-Martin]

Thus the conjecture is true for g ≤ 5 (P5 = A5)

Multiplicity of the theta divisor

Theorem (Koll´ar)

For any (A, Θ) ∈ Ag , any z ∈ A we have multz Θ ≤ g. Conjecture

indec j g+1 k For A ∈ Ag and any z ∈ A, multz Θ ≤ 2 .

The bound holds and is achieved for Jacobians [Riemann] The bound holds and is achieved for Pryms [Mumford, Smith-Varley, Casalaina-Martin]

Thus the conjecture is true for g ≤ 5 (P5 = A5)

Multiplicity of the theta divisor

Theorem (Koll´ar)

For any (A, Θ) ∈ Ag , any z ∈ A we have multz Θ ≤ g.

Theorem (Smith-Varley)

If multz Θ = g, then A = E1 × · · · × Eg . The bound holds and is achieved for Jacobians [Riemann] The bound holds and is achieved for Pryms [Mumford, Smith-Varley, Casalaina-Martin]

Thus the conjecture is true for g ≤ 5 (P5 = A5)

Multiplicity of the theta divisor

Theorem (Koll´ar)

For any (A, Θ) ∈ Ag , any z ∈ A we have multz Θ ≤ g.

Theorem (Smith-Varley)

If multz Θ = g, then A = E1 × · · · × Eg .

Conjecture

indec j g+1 k For A ∈ Ag and any z ∈ A, multz Θ ≤ 2 . The bound holds and is achieved for Pryms [Mumford, Smith-Varley, Casalaina-Martin]

Thus the conjecture is true for g ≤ 5 (P5 = A5)

Multiplicity of the theta divisor

Theorem (Koll´ar)

For any (A, Θ) ∈ Ag , any z ∈ A we have multz Θ ≤ g.

Theorem (Smith-Varley)

If multz Θ = g, then A = E1 × · · · × Eg .

Conjecture

indec j g+1 k For A ∈ Ag and any z ∈ A, multz Θ ≤ 2 .

The bound holds and is achieved for Jacobians [Riemann] Thus the conjecture is true for g ≤ 5 (P5 = A5)

Multiplicity of the theta divisor

Theorem (Koll´ar)

For any (A, Θ) ∈ Ag , any z ∈ A we have multz Θ ≤ g.

Theorem (Smith-Varley)

If multz Θ = g, then A = E1 × · · · × Eg .

Conjecture

indec j g+1 k For A ∈ Ag and any z ∈ A, multz Θ ≤ 2 .

The bound holds and is achieved for Jacobians [Riemann] The bound holds and is achieved for Pryms [Mumford, Smith-Varley, Casalaina-Martin] Multiplicity of the theta divisor

Theorem (Koll´ar)

For any (A, Θ) ∈ Ag , any z ∈ A we have multz Θ ≤ g.

Theorem (Smith-Varley)

If multz Θ = g, then A = E1 × · · · × Eg .

Conjecture

indec j g+1 k For A ∈ Ag and any z ∈ A, multz Θ ≤ 2 .

The bound holds and is achieved for Jacobians [Riemann] The bound holds and is achieved for Pryms [Mumford, Smith-Varley, Casalaina-Martin]

Thus the conjecture is true for g ≤ 5 (P5 = A5) Geometric solution in genus 4.

Only a weak solution (at least so far) in higher genera.

+ Geometric conditions for an abelian variety to be a Jacobian.

− dim Sing Θτ hard to compute for an explicitly given τ ∈ Hg .

Andreotti-Mayer approach: recap Only a weak solution (at least so far) in higher genera.

Geometric solution in genus 4.

− dim Sing Θτ hard to compute for an explicitly given τ ∈ Hg .

Andreotti-Mayer approach: recap

+ Geometric conditions for an abelian variety to be a Jacobian. Only a weak solution (at least so far) in higher genera.

− dim Sing Θτ hard to compute for an explicitly given τ ∈ Hg .

Andreotti-Mayer approach: recap

+ Geometric conditions for an abelian variety to be a Jacobian. Geometric solution in genus 4. Only a weak solution (at least so far) in higher genera.

Andreotti-Mayer approach: recap

+ Geometric conditions for an abelian variety to be a Jacobian. Geometric solution in genus 4.

− dim Sing Θτ hard to compute for an explicitly given τ ∈ Hg . Andreotti-Mayer approach: recap

+ Geometric conditions for an abelian variety to be a Jacobian. Geometric solution in genus 4.

− dim Sing Θτ hard to compute for an explicitly given τ ∈ Hg . Only a weak solution (at least so far) in higher genera. For Jacobians have the Abel-Jacobi curve C ,→ Jac(C).

Theorem (Matsusaka-Ran) Θg−1 If ∃C ⊂ A of “minimal” class (g−1)! , then A = Jac(C) For Pryms the Abel-Prym curve C˜ ,→ Jac(C˜) → Prym(C˜ → C). Theorem (Welters) Θg−1 If ∃C ⊂ A of homology class 2 (g−1)! , then A is a Prym (or a degeneration, technical details, . . . )

Here we start with a curve and solve an easier version of Schottky: Given C ⊂ A, is A = Jac(C)?

Curves of small homology class Theorem (Matsusaka-Ran) Θg−1 If ∃C ⊂ A of “minimal” class (g−1)! , then A = Jac(C) For Pryms the Abel-Prym curve C˜ ,→ Jac(C˜) → Prym(C˜ → C). Theorem (Welters) Θg−1 If ∃C ⊂ A of homology class 2 (g−1)! , then A is a Prym (or a degeneration, technical details, . . . )

Here we start with a curve and solve an easier version of Schottky: Given C ⊂ A, is A = Jac(C)?

Curves of small homology class

For Jacobians have the Abel-Jacobi curve C ,→ Jac(C). For Pryms the Abel-Prym curve C˜ ,→ Jac(C˜) → Prym(C˜ → C). Theorem (Welters) Θg−1 If ∃C ⊂ A of homology class 2 (g−1)! , then A is a Prym (or a degeneration, technical details, . . . )

Here we start with a curve and solve an easier version of Schottky: Given C ⊂ A, is A = Jac(C)?

Curves of small homology class

For Jacobians have the Abel-Jacobi curve C ,→ Jac(C).

Theorem (Matsusaka-Ran) Θg−1 If ∃C ⊂ A of “minimal” class (g−1)! , then A = Jac(C) Theorem (Welters) Θg−1 If ∃C ⊂ A of homology class 2 (g−1)! , then A is a Prym (or a degeneration, technical details, . . . )

Here we start with a curve and solve an easier version of Schottky: Given C ⊂ A, is A = Jac(C)?

Curves of small homology class

For Jacobians have the Abel-Jacobi curve C ,→ Jac(C).

Theorem (Matsusaka-Ran) Θg−1 If ∃C ⊂ A of “minimal” class (g−1)! , then A = Jac(C) For Pryms the Abel-Prym curve C˜ ,→ Jac(C˜) → Prym(C˜ → C). Here we start with a curve and solve an easier version of Schottky: Given C ⊂ A, is A = Jac(C)?

Curves of small homology class

For Jacobians have the Abel-Jacobi curve C ,→ Jac(C).

Here we start with a curve and solve an easier version of Schottky: Given C ⊂ A, is A = Jac(C)? Trisecant formula (Fay, Gunning) 0 ∀p, p1, p2, p3 ∈ C ⊂ Jac(C) = Pic (C) the following are collinear:

Kum(p+p1−p2−p3), Kum(p+p2−p1−p3), Kum(p+p3−p1−p2)(∗)

Geometry of the Kummer variety

Definition The Kummer variety is the image of 2g −1 Kum := |2Θ| : Aτ / ± 1 ,→ P z → Θ[ε](τ, z) 1 g g , all ε∈ 2 Z /Z Theorem (Gunning) indec If for some A ∈ Ag there exist infinitely many p such that (∗) (pi fixed, in general position), then A ∈ Jg . This is a solution to the Schottky problem, already given a curve.

Geometry of the Kummer variety

Deﬁnition The Kummer variety is the image of 2g −1 Kum := |2Θ| : Aτ / ± 1 ,→ P z → Θ[ε](τ, z) 1 g g , all ε∈ 2 Z /Z

Trisecant formula (Fay, Gunning) 0 ∀p, p1, p2, p3 ∈ C ⊂ Jac(C) = Pic (C) the following are collinear:

Kum(p+p1−p2−p3), Kum(p+p2−p1−p3), Kum(p+p3−p1−p2)(∗) This is a solution to the Schottky problem, already given a curve.

Geometry of the Kummer variety

Deﬁnition The Kummer variety is the image of 2g −1 Kum := |2Θ| : Aτ / ± 1 ,→ P z → Θ[ε](τ, z) 1 g g , all ε∈ 2 Z /Z

Trisecant formula (Fay, Gunning) 0 ∀p, p1, p2, p3 ∈ C ⊂ Jac(C) = Pic (C) the following are collinear:

Kum(p+p1−p2−p3), Kum(p+p2−p1−p3), Kum(p+p3−p1−p2)(∗)

Theorem (Gunning) indec If for some A ∈ Ag there exist inﬁnitely many p such that (∗) (pi ﬁxed, in general position), then A ∈ Jg . Geometry of the Kummer variety

Deﬁnition The Kummer variety is the image of 2g −1 Kum := |2Θ| : Aτ / ± 1 ,→ P z → Θ[ε](τ, z) 1 g g , all ε∈ 2 Z /Z

Trisecant formula (Fay, Gunning) 0 ∀p, p1, p2, p3 ∈ C ⊂ Jac(C) = Pic (C) the following are collinear:

Kum(p+p1−p2−p3), Kum(p+p2−p1−p3), Kum(p+p3−p1−p2)(∗)

Theorem (Gunning) indec If for some A ∈ Ag there exist inﬁnitely many p such that (∗) (pi ﬁxed, in general position), then A ∈ Jg . This is a solution to the Schottky problem, already given a curve. “Multi”secant formula (Gunning) For any 1 ≤ k ≤ g and for any p1,..., pk+2, q1,..., qk ∈ C ⊂ Jac(C) the k + 2 points k k+2 X X Kum(2pj + qi − pi ), j = 1 ... k + 2 i=1 i=1 are linearly dependent.

g Note Sym C Jac(C), use k = g above. The converse is Conjecture (Buchstaber-Krichever) Theorem (G., Pareschi-Popa) indec Given A ∈ Ag and p1,..., pg+2 ∈ A in general position, if

2g −1 ∀z ∈ A {Kum(2pi + z)}i=1...g+2 ⊂ P

are linearly dependent, then A ∈ Jg .

“Getting rid” of the points of secancy g Note Sym C Jac(C), use k = g above. The converse is Conjecture (Buchstaber-Krichever) Theorem (G., Pareschi-Popa) indec Given A ∈ Ag and p1,..., pg+2 ∈ A in general position, if

2g −1 ∀z ∈ A {Kum(2pi + z)}i=1...g+2 ⊂ P

are linearly dependent, then A ∈ Jg .

“Getting rid” of the points of secancy

“Multi”secant formula (Gunning) For any 1 ≤ k ≤ g and for any p1,..., pk+2, q1,..., qk ∈ C ⊂ Jac(C) the k + 2 points k k+2 X X Kum(2pj + qi − pi ), j = 1 ... k + 2 i=1 i=1 are linearly dependent. Conjecture (Buchstaber-Krichever) Theorem (G., Pareschi-Popa) indec Given A ∈ Ag and p1,..., pg+2 ∈ A in general position, if

2g −1 ∀z ∈ A {Kum(2pi + z)}i=1...g+2 ⊂ P

are linearly dependent, then A ∈ Jg .

“Getting rid” of the points of secancy

“Multi”secant formula (Gunning) For any 1 ≤ k ≤ g and for any p1,..., pk+2, q1,..., qk ∈ C ⊂ Jac(C) the k + 2 points k k+2 X X Kum(2pj + qi − pi ), j = 1 ... k + 2 i=1 i=1 are linearly dependent.

g Note Sym C Jac(C), use k = g above. The converse is “Getting rid” of the points of secancy

g Note Sym C Jac(C), use k = g above. The converse is Conjecture (Buchstaber-Krichever) Theorem (G., Pareschi-Popa) indec Given A ∈ Ag and p1,..., pg+2 ∈ A in general position, if

2g −1 ∀z ∈ A {Kum(2pi + z)}i=1...g+2 ⊂ P

are linearly dependent, then A ∈ Jg . or

Semidegenerate trisecant: a line tangent to Kum(A) at a point not in A[2] intersecting Kum(A) at another point

Flex line: a line tangent to Kum(A) at a point not in A[2] with multiplicity 3 implies that A is a Jacobian.

No general position assumption: only that the points of secancy are not in A[2], so that Kum(A) is smooth at them. Also true for degenerate trisecants, i.e., the existence of a

Trisecant Conjecture (Welters) Theorem (Krichever) indec If Kum(A) has a trisecant, for A ∈ Ag , then A ∈ Jg . or

Semidegenerate trisecant: a line tangent to Kum(A) at a point not in A[2] intersecting Kum(A) at another point

Flex line: a line tangent to Kum(A) at a point not in A[2] with multiplicity 3 implies that A is a Jacobian.

Also true for degenerate trisecants, i.e., the existence of a

Trisecant Conjecture (Welters) Theorem (Krichever) indec If Kum(A) has a trisecant, for A ∈ Ag , then A ∈ Jg .

No general position assumption: only that the points of secancy are not in A[2], so that Kum(A) is smooth at them. or

Semidegenerate trisecant: a line tangent to Kum(A) at a point not in A[2] intersecting Kum(A) at another point

Flex line: a line tangent to Kum(A) at a point not in A[2] with multiplicity 3 implies that A is a Jacobian.

Trisecant Conjecture (Welters) Theorem (Krichever) indec If Kum(A) has a trisecant, for A ∈ Ag , then A ∈ Jg .

No general position assumption: only that the points of secancy are not in A[2], so that Kum(A) is smooth at them. Also true for degenerate trisecants, i.e., the existence of a or Flex line: a line tangent to Kum(A) at a point not in A[2] with multiplicity 3 implies that A is a Jacobian.

Trisecant Conjecture (Welters) Theorem (Krichever) indec If Kum(A) has a trisecant, for A ∈ Ag , then A ∈ Jg .

No general position assumption: only that the points of secancy are not in A[2], so that Kum(A) is smooth at them. Also true for degenerate trisecants, i.e., the existence of a Semidegenerate trisecant: a line tangent to Kum(A) at a point not in A[2] intersecting Kum(A) at another point or Flex line: a line tangent to Kum(A) at a point not in A[2] with multiplicity 3 Trisecant Conjecture (Welters) Theorem (Krichever) indec If Kum(A) has a trisecant, for A ∈ Ag , then A ∈ Jg .

No general position assumption: only that the points of secancy are not in A[2], so that Kum(A) is smooth at them. Also true for degenerate trisecants, i.e., the existence of a Semidegenerate trisecant: a line tangent to Kum(A) at a point not in A[2] intersecting Kum(A) at another point or Flex line: a line tangent to Kum(A) at a point not in A[2] with multiplicity 3 implies that A is a Jacobian. Theorem (Fay, Beauville-Debarre)

For any p, p1, p2, p3 ∈ C˜ → Prym(C˜ → C) the points

Kum(p + p + p + p ), Kum(p + p − p − p ), 1 2 3 1 2 3 (∗∗) Kum(p + p2 − p1 − p3), Kum(p + p3 − p1 − p2) 2g −1 lie on a 2-plane in P .

Example (Beauville-Debarre)

There exists A ∈ Ag \ Pg such that Kum(A) has a quadrisecant.

Theorem (G.-Krichever) indec For A ∈ Ag and p, p1, p2, p3 ∈ A, if (∗∗), and (∗∗) also holds for −p, p1, p2, p3, then A ∈ Pg . (Characterization by a symmetric pair of quadrisecants)

Kummer images of Prym varieties Example (Beauville-Debarre)

There exists A ∈ Ag \ Pg such that Kum(A) has a quadrisecant.

Theorem (G.-Krichever) indec For A ∈ Ag and p, p1, p2, p3 ∈ A, if (∗∗), and (∗∗) also holds for −p, p1, p2, p3, then A ∈ Pg . (Characterization by a symmetric pair of quadrisecants)

Kummer images of Prym varieties

Theorem (Fay, Beauville-Debarre)

For any p, p1, p2, p3 ∈ C˜ → Prym(C˜ → C) the points

Kum(p + p + p + p ), Kum(p + p − p − p ), 1 2 3 1 2 3 (∗∗) Kum(p + p2 − p1 − p3), Kum(p + p3 − p1 − p2) 2g −1 lie on a 2-plane in P . Example (Beauville-Debarre)

There exists A ∈ Ag \ Pg such that Kum(A) has a quadrisecant.

Theorem (G.-Krichever) indec For A ∈ Ag and p, p1, p2, p3 ∈ A, if (∗∗), and (∗∗) also holds for −p, p1, p2, p3, then A ∈ Pg . (Characterization by a symmetric pair of quadrisecants)

Kummer images of Prym varieties

Theorem (Fay, Beauville-Debarre)

For any p, p1, p2, p3 ∈ C˜ → Prym(C˜ → C) the points

Kum(p + p + p + p ), Kum(p + p − p − p ), 1 2 3 1 2 3 (∗∗) Kum(p + p2 − p1 − p3), Kum(p + p3 − p1 − p2) 2g −1 lie on a 2-plane in P . Theorem (Debarre) indec If for some A ∈ Ag there exist inﬁnitely many p such that (∗∗) (pi ﬁxed and in general position), then A ∈ Pg . Theorem (G.-Krichever) indec For A ∈ Ag and p, p1, p2, p3 ∈ A, if (∗∗), and (∗∗) also holds for −p, p1, p2, p3, then A ∈ Pg . (Characterization by a symmetric pair of quadrisecants)

Kummer images of Prym varieties

Theorem (Fay, Beauville-Debarre)

For any p, p1, p2, p3 ∈ C˜ → Prym(C˜ → C) the points

Example (Beauville-Debarre)

There exists A ∈ Ag \ Pg such that Kum(A) has a quadrisecant. Theorem (G.-Krichever) indec For A ∈ Ag and p, p1, p2, p3 ∈ A, if (∗∗), and (∗∗) also holds for −p, p1, p2, p3, then A ∈ Pg . (Characterization by a symmetric pair of quadrisecants)

Kummer images of Prym varieties

Theorem (Fay, Beauville-Debarre)

For any p, p1, p2, p3 ∈ C˜ → Prym(C˜ → C) the points

Example (Beauville-Debarre)

There exists A ∈ Ag \ Pg such that Kum(A) has a quadrisecant. So what would be the Prym analog of the trisecant conjecture? Kummer images of Prym varieties

Theorem (Fay, Beauville-Debarre)

For any p, p1, p2, p3 ∈ C˜ → Prym(C˜ → C) the points

Kum(p + p + p + p ), Kum(p + p − p − p ), 1 2 3 1 2 3 (∗∗) Kum(p + p2 − p1 − p3), Kum(p + p3 − p1 − p2) 2g −1 lie on a 2-plane in P .

Example (Beauville-Debarre)

There exists A ∈ Ag \ Pg such that Kum(A) has a quadrisecant.

Theorem (G.-Krichever) indec For A ∈ Ag and p, p1, p2, p3 ∈ A, if (∗∗), and (∗∗) also holds for −p, p1, p2, p3, then A ∈ Pg . (Characterization by a symmetric pair of quadrisecants) Finite amount of data involved, no curves or inﬁnitesimal structure.

Challenge Use these characterizations to approach Coleman’s conjecture, or solve the Torelli problem for Pryms (period map generically injective — conjecturally the non-injectivity is due only to the tetragonal construction), or . . .

+ A “strong” solution to Schottky and Prym-Schottky (no extra components).

− The points of the tri(quadri)secancy enter in the equations, i.e., we do not directly get algebraic equations for theta constants.

Secants of the Kummer variety: recap Challenge Use these characterizations to approach Coleman’s conjecture, or solve the Torelli problem for Pryms (period map generically injective — conjecturally the non-injectivity is due only to the tetragonal construction), or . . .

Finite amount of data involved, no curves or inﬁnitesimal structure.

− The points of the tri(quadri)secancy enter in the equations, i.e., we do not directly get algebraic equations for theta constants.

Secants of the Kummer variety: recap

+ A “strong” solution to Schottky and Prym-Schottky (no extra components). Challenge Use these characterizations to approach Coleman’s conjecture, or solve the Torelli problem for Pryms (period map generically injective — conjecturally the non-injectivity is due only to the tetragonal construction), or . . .

− The points of the tri(quadri)secancy enter in the equations, i.e., we do not directly get algebraic equations for theta constants.

Secants of the Kummer variety: recap

+ A “strong” solution to Schottky and Prym-Schottky (no extra components). Finite amount of data involved, no curves or inﬁnitesimal structure. Challenge Use these characterizations to approach Coleman’s conjecture, or solve the Torelli problem for Pryms (period map generically injective — conjecturally the non-injectivity is due only to the tetragonal construction), or . . .

Secants of the Kummer variety: recap

+ A “strong” solution to Schottky and Prym-Schottky (no extra components). Finite amount of data involved, no curves or inﬁnitesimal structure.

− The points of the tri(quadri)secancy enter in the equations, i.e., we do not directly get algebraic equations for theta constants. Secants of the Kummer variety: recap

+ A “strong” solution to Schottky and Prym-Schottky (no extra components). Finite amount of data involved, no curves or inﬁnitesimal structure.

− The points of the tri(quadri)secancy enter in the equations, i.e., we do not directly get algebraic equations for theta constants.

Can characterize Hypg by the value of their Seshadri constant, if the Γ00 conjecture holds [Debarre, Lazarsfeld]

Theorem (Buser, Sarnak) The upper bound for the length of the shortest period for Jacobians is (much) less than the upper bound for the length of the shortest period for abelian varieties. , or does it?

Can characterize Hypg by the value of their Seshadri constant, if the Γ00 conjecture holds [Debarre, Lazarsfeld]

+ Gives a way to tell that some abelian varieties are not Jacobians. − Does not possibly give a way to show that a given abelian variety is a Jacobian

Theorem (Buser, Sarnak) The upper bound for the length of the shortest period for Jacobians is (much) less than the upper bound for the length of the shortest period for abelian varieties.

Theorem (Lazarsfeld, also work by Bauer, Nakamaye) The Seshadri constant for a generic Jacobian is much smaller than for a generic abelian variety. , or does it?

Can characterize Hypg by the value of their Seshadri constant, if the Γ00 conjecture holds [Debarre, Lazarsfeld]

Theorem (Buser, Sarnak) The upper bound for the length of the shortest period for Jacobians is (much) less than the upper bound for the length of the shortest period for abelian varieties.

Theorem (Lazarsfeld, also work by Bauer, Nakamaye) The Seshadri constant for a generic Jacobian is much smaller than for a generic abelian variety. + Gives a way to tell that some abelian varieties are not Jacobians. − Does not possibly give a way to show that a given abelian variety is a Jacobian Theorem (Buser, Sarnak) The upper bound for the length of the shortest period for Jacobians is (much) less than the upper bound for the length of the shortest period for abelian varieties.

Can characterize Hypg by the value of their Seshadri constant, if the Γ00 conjecture holds [Debarre, Lazarsfeld] Theorem (set-theoretically: Welters, scheme-theoretically: Izadi)

For g ≥ 5 on Jac(C) we have Bs(Γ00) = C − C.

Conjecture (van Geemen-van der Geer) indec For A ∈ Ag if Bs(Γ00) 6= {0}, then A ∈ Jg .

Holds for g = 4. [Izadi] Holds for a generic Prym for g ≥ 8. [Izadi] Holds for a generic abelian variety for g = 5 or g ≥ 14. [Beauville-Debarre-Donagi-van der Geer]

Γ00 conjecture

Deﬁnition

0 Γ00 = {f ∈ H (A, 2Θ) | mult0f ≥ 4} Conjecture (van Geemen-van der Geer) indec For A ∈ Ag if Bs(Γ00) 6= {0}, then A ∈ Jg .

Holds for g = 4. [Izadi] Holds for a generic Prym for g ≥ 8. [Izadi] Holds for a generic abelian variety for g = 5 or g ≥ 14. [Beauville-Debarre-Donagi-van der Geer]

Γ00 conjecture

Deﬁnition

0 Γ00 = {f ∈ H (A, 2Θ) | mult0f ≥ 4}

Theorem (set-theoretically: Welters, scheme-theoretically: Izadi)

For g ≥ 5 on Jac(C) we have Bs(Γ00) = C − C. Holds for g = 4. [Izadi] Holds for a generic Prym for g ≥ 8. [Izadi] Holds for a generic abelian variety for g = 5 or g ≥ 14. [Beauville-Debarre-Donagi-van der Geer]

Γ00 conjecture

Deﬁnition

0 Γ00 = {f ∈ H (A, 2Θ) | mult0f ≥ 4}

Theorem (set-theoretically: Welters, scheme-theoretically: Izadi)

For g ≥ 5 on Jac(C) we have Bs(Γ00) = C − C.

Conjecture (van Geemen-van der Geer) indec For A ∈ Ag if Bs(Γ00) 6= {0}, then A ∈ Jg . Holds for a generic Prym for g ≥ 8. [Izadi] Holds for a generic abelian variety for g = 5 or g ≥ 14. [Beauville-Debarre-Donagi-van der Geer]

Γ00 conjecture

Deﬁnition

0 Γ00 = {f ∈ H (A, 2Θ) | mult0f ≥ 4}

Theorem (set-theoretically: Welters, scheme-theoretically: Izadi)

For g ≥ 5 on Jac(C) we have Bs(Γ00) = C − C.

Conjecture (van Geemen-van der Geer) indec For A ∈ Ag if Bs(Γ00) 6= {0}, then A ∈ Jg .

Holds for g = 4. [Izadi] Holds for a generic abelian variety for g = 5 or g ≥ 14. [Beauville-Debarre-Donagi-van der Geer]

Γ00 conjecture

Deﬁnition

0 Γ00 = {f ∈ H (A, 2Θ) | mult0f ≥ 4}

Theorem (set-theoretically: Welters, scheme-theoretically: Izadi)

For g ≥ 5 on Jac(C) we have Bs(Γ00) = C − C.

Conjecture (van Geemen-van der Geer) indec For A ∈ Ag if Bs(Γ00) 6= {0}, then A ∈ Jg .

Holds for g = 4. [Izadi] Holds for a generic Prym for g ≥ 8. [Izadi] Γ00 conjecture

Deﬁnition

0 Γ00 = {f ∈ H (A, 2Θ) | mult0f ≥ 4}

Theorem (set-theoretically: Welters, scheme-theoretically: Izadi)

For g ≥ 5 on Jac(C) we have Bs(Γ00) = C − C.

Conjecture (van Geemen-van der Geer) indec For A ∈ Ag if Bs(Γ00) 6= {0}, then A ∈ Jg .

Holds for g = 4. [Izadi] Holds for a generic Prym for g ≥ 8. [Izadi] Holds for a generic abelian variety for g = 5 or g ≥ 14. [Beauville-Debarre-Donagi-van der Geer] m Kum(z) ∈ Kum(0), ∂ ∂ Kum(0) zi zj linear span m P Kum(z) = cKum(0) + cij ∂zi ∂zj Kum(0) (†)

for some c, cij ∈ C

Thus z ∈ Bs(Γ00)

Similar to a semidegenerate trisecant tangent at Kum(0).

For p, q ∈ C ⊂ Jac(C) in fact rk(cij ) = 1 [∼Frobenius]

Theorem (G.) indec For A ∈ Ag if (†) holds with rk(cij ) = 1, then A ∈ Jg .

Idea (Mu~noz-Porras)

If Γ00 conjecture holds, then Jg =small Schottky-Jung locus (methods to prove this by degenerating to the boundary).

Even functions Θ[ε](τ, z) generate H0(A, 2Θ). m Kum(z) ∈ Kum(0), ∂ ∂ Kum(0) zi zj linear span m P Kum(z) = cKum(0) + cij ∂zi ∂zj Kum(0) (†)

for some c, cij ∈ C Similar to a semidegenerate trisecant tangent at Kum(0).

For p, q ∈ C ⊂ Jac(C) in fact rk(cij ) = 1 [∼Frobenius]

Theorem (G.) indec For A ∈ Ag if (†) holds with rk(cij ) = 1, then A ∈ Jg .

Idea (Mu~noz-Porras)

If Γ00 conjecture holds, then Jg =small Schottky-Jung locus (methods to prove this by degenerating to the boundary).

Even functions Θ[ε](τ, z) generate H0(A, 2Θ).

Thus z ∈ Bs(Γ00) m P Kum(z) = cKum(0) + cij ∂zi ∂zj Kum(0) (†)

for some c, cij ∈ C Similar to a semidegenerate trisecant tangent at Kum(0).

For p, q ∈ C ⊂ Jac(C) in fact rk(cij ) = 1 [∼Frobenius]

Theorem (G.) indec For A ∈ Ag if (†) holds with rk(cij ) = 1, then A ∈ Jg .

Idea (Mu~noz-Porras)

If Γ00 conjecture holds, then Jg =small Schottky-Jung locus (methods to prove this by degenerating to the boundary).

Even functions Θ[ε](τ, z) generate H0(A, 2Θ).

Thus z ∈ Bs(Γ00) m Kum(z) ∈ Kum(0), ∂ ∂ Kum(0) zi zj linear span Similar to a semidegenerate trisecant tangent at Kum(0).

For p, q ∈ C ⊂ Jac(C) in fact rk(cij ) = 1 [∼Frobenius]

Theorem (G.) indec For A ∈ Ag if (†) holds with rk(cij ) = 1, then A ∈ Jg .

Idea (Mu~noz-Porras)

If Γ00 conjecture holds, then Jg =small Schottky-Jung locus (methods to prove this by degenerating to the boundary).

Even functions Θ[ε](τ, z) generate H0(A, 2Θ).

Thus z ∈ Bs(Γ00) m Kum(z) ∈ Kum(0), ∂ ∂ Kum(0) zi zj linear span m P Kum(z) = cKum(0) + cij ∂zi ∂zj Kum(0) (†) for some c, cij ∈ C For p, q ∈ C ⊂ Jac(C) in fact rk(cij ) = 1 [∼Frobenius]

Theorem (G.) indec For A ∈ Ag if (†) holds with rk(cij ) = 1, then A ∈ Jg .

Idea (Mu~noz-Porras)

If Γ00 conjecture holds, then Jg =small Schottky-Jung locus (methods to prove this by degenerating to the boundary).

Even functions Θ[ε](τ, z) generate H0(A, 2Θ).

Thus z ∈ Bs(Γ00) m Kum(z) ∈ Kum(0), ∂ ∂ Kum(0) zi zj linear span m P Kum(z) = cKum(0) + cij ∂zi ∂zj Kum(0) (†) for some c, cij ∈ C Similar to a semidegenerate trisecant tangent at Kum(0). Theorem (G.) indec For A ∈ Ag if (†) holds with rk(cij ) = 1, then A ∈ Jg .

Idea (Mu~noz-Porras)

If Γ00 conjecture holds, then Jg =small Schottky-Jung locus (methods to prove this by degenerating to the boundary).

Even functions Θ[ε](τ, z) generate H0(A, 2Θ).

Thus z ∈ Bs(Γ00) m Kum(z) ∈ Kum(0), ∂ ∂ Kum(0) zi zj linear span m P Kum(z) = cKum(0) + cij ∂zi ∂zj Kum(0) (†) for some c, cij ∈ C Similar to a semidegenerate trisecant tangent at Kum(0).

For p, q ∈ C ⊂ Jac(C) in fact rk(cij ) = 1 [∼Frobenius] Idea (Mu~noz-Porras)

If Γ00 conjecture holds, then Jg =small Schottky-Jung locus (methods to prove this by degenerating to the boundary).

Even functions Θ[ε](τ, z) generate H0(A, 2Θ).

Thus z ∈ Bs(Γ00) m Kum(z) ∈ Kum(0), ∂ ∂ Kum(0) zi zj linear span m P Kum(z) = cKum(0) + cij ∂zi ∂zj Kum(0) (†)

for some c, cij ∈ C Similar to a semidegenerate trisecant tangent at Kum(0).

For p, q ∈ C ⊂ Jac(C) in fact rk(cij ) = 1 [∼Frobenius]

Theorem (G.) indec For A ∈ Ag if (†) holds with rk(cij ) = 1, then A ∈ Jg . Even functions Θ[ε](τ, z) generate H0(A, 2Θ).

Thus z ∈ Bs(Γ00) m Kum(z) ∈ Kum(0), ∂ ∂ Kum(0) zi zj linear span m P Kum(z) = cKum(0) + cij ∂zi ∂zj Kum(0) (†)

for some c, cij ∈ C Similar to a semidegenerate trisecant tangent at Kum(0).

For p, q ∈ C ⊂ Jac(C) in fact rk(cij ) = 1 [∼Frobenius]

Theorem (G.) indec For A ∈ Ag if (†) holds with rk(cij ) = 1, then A ∈ Jg .

Idea (Mu~noz-Porras)

If Γ00 conjecture holds, then Jg =small Schottky-Jung locus (methods to prove this by degenerating to the boundary). For the results on Schottky’s form and theta divisors:

Theta functions are not a spectator sport. . .

--- Lipman Bers

For the characterization of Pryms by pairs of quadrisecants: Use integrable systems.

. . . and neither are integrable systems.

Proofs Theta functions are not a spectator sport. . .

--- Lipman Bers

For the characterization of Pryms by pairs of quadrisecants: Use integrable systems.

. . . and neither are integrable systems.

Proofs

For the results on Schottky’s form and theta divisors: For the characterization of Pryms by pairs of quadrisecants: Use integrable systems.

. . . and neither are integrable systems.

Proofs

For the results on Schottky’s form and theta divisors:

Theta functions are not a spectator sport. . .

--- Lipman Bers Use integrable systems.

. . . and neither are integrable systems.

Proofs

For the results on Schottky’s form and theta divisors:

Theta functions are not a spectator sport. . .

--- Lipman Bers

For the characterization of Pryms by pairs of quadrisecants: . . . and neither are integrable systems.

Proofs

For the results on Schottky’s form and theta divisors:

Theta functions are not a spectator sport. . .

--- Lipman Bers

For the characterization of Pryms by pairs of quadrisecants: Use integrable systems. . . . and neither are integrable systems.

Proofs

For the results on Schottky’s form and theta divisors:

Theta functions are not a spectator sport. . .

--- Lipman Bers

For the characterization of Pryms by pairs of quadrisecants: Use integrable systems. Dubrovin, Krichever, Novikov, Arbarello, De Concini, Shiota, Mulase, Marini, Mu~nozPorras, Plaza Martin, ... Proofs

For the results on Schottky’s form and theta divisors:

Theta functions are not a spectator sport. . .

--- Lipman Bers

For the characterization of Pryms by pairs of quadrisecants: Use integrable systems.

. . . and neither are integrable systems. KP hierarchy of PDEs is the hierarchy of the conditions for the existence of n-jets of a family of flex lines, for each n ∈ N. The existence of an n-jet of a family of flexes for any n gives a formal one-dimensional family of flexes. Such a formal family comes from an actual geometric family. Thus if the KP hierarchy is satisfied by the theta function, the abelian variety is a Jacobian. Shiota proved that the KP equation suffices to recover the hierarchy. Krichever showed that the obstruction for extending one flex to a family of flexes vanishes in Taylor series. For Pryms, G.-Krichever needed a new hierarchy, etc.

Kadomtsev-Petviashvili (KP) equation is the condition for the existence of a 1-jet of a family of degenerate trisecants (flex lines of the Kummer). The existence of an n-jet of a family of flexes for any n gives a formal one-dimensional family of flexes. Such a formal family comes from an actual geometric family. Thus if the KP hierarchy is satisfied by the theta function, the abelian variety is a Jacobian. Shiota proved that the KP equation suffices to recover the hierarchy. Krichever showed that the obstruction for extending one flex to a family of flexes vanishes in Taylor series. For Pryms, G.-Krichever needed a new hierarchy, etc.

Kadomtsev-Petviashvili (KP) equation is the condition for the existence of a 1-jet of a family of degenerate trisecants (flex lines of the Kummer). KP hierarchy of PDEs is the hierarchy of the conditions for the existence of n-jets of a family of flex lines, for each n ∈ N. The existence of an n-jet of a family of flexes for any n gives a formal one-dimensional family of flexes. Such a formal family comes from an actual geometric family. Thus if the KP hierarchy is satisfied by the theta function, the abelian variety is a Jacobian. Shiota proved that the KP equation suffices to recover the hierarchy. Krichever showed that the obstruction for extending one flex to a family of flexes vanishes in Taylor series. Kadomtsev-Petviashvili (KP) equation is the condition for the existence of a 1-jet of a family of degenerate trisecants (flex lines of the Kummer). KP hierarchy of PDEs is the hierarchy of the conditions for the existence of n-jets of a family of flex lines, for each n ∈ N. The existence of an n-jet of a family of flexes for any n gives a formal one-dimensional family of flexes. Such a formal family comes from an actual geometric family. Thus if the KP hierarchy is satisfied by the theta function, the abelian variety is a Jacobian. Shiota proved that the KP equation suffices to recover the hierarchy. Krichever showed that the obstruction for extending one flex to a family of flexes vanishes in Taylor series. For Pryms, G.-Krichever needed a new hierarchy, etc.