Gromov—Witten invariants

Gromov—Witten invariants

introduction to results of A. Okounkov

Maciej Borodzik

Instytut Matematyki, Uniwersytet Warszawski

The moduli problem

Consider the flat family of smooth n−pointed curves of fixed genus g,

so a typical member is a curve C

with pairway distinct points p₁, . . . , p_n ∈ C. A family can be written as

C F

B

The moduli problem consist of finding a universal

family of curves such that any other family is induced from M by a unique morphism b.

∃!b b^∗

Such a space, if exists, is called a moduli space of pointed curves.

The moduli problem

Consider the flat family of smooth n−pointed curves of fixed genus g, so a typical member is a curve C with pairway distinct points p₁, . . . , p_n ∈ C.

A family can be written as

C F

B

The moduli problem consist of finding a universal

family of curves such that any other family is induced from M by a unique morphism b.

∃!b b^∗

Such a space, if exists, is called a moduli space of pointed curves.

The moduli problem

Consider the flat family of smooth n−pointed curves of fixed genus g, so a typical member is a curve C with pairway distinct points p₁, . . . , p_n ∈ C. A

family can be written as

C F

B The moduli problem consist of finding a universal

family of curves such that any other family is induced from M by a unique morphism b.

∃!b b^∗

Such a space, if exists, is called a moduli space of pointed curves.

The moduli problem

Consider the flat family of smooth n−pointed curves of fixed genus g, so a typical member is a curve C with pairway distinct points p₁, . . . , p_n ∈ C. A

family can be written as

C F

B The moduli problem consist of finding a universal

family of curves such that any other family is induced from M by a unique morphism b.

∃!b b^∗

Such a space, if exists, is called a moduli space of pointed curves.

The moduli problem

Consider the flat family of smooth n−pointed curves of fixed genus g, so a typical member is a curve C with pairway distinct points p₁, . . . , p_n ∈ C. A

family can be written as

C F C

B M

The moduli problem consist of finding a universal family of curves

such that any other family is induced from M by a unique morphism b.

∃!bb^∗

Such a space, if exists, is called a moduli space of pointed curves.

The moduli problem

Consider the flat family of smooth n−pointed curves of fixed genus g, so a typical member is a curve C with pairway distinct points p₁, . . . , p_n ∈ C. A

family can be written as

C F C

B M

The moduli problem consist of finding a universal family of curves such that any other family

is induced from M by a unique morphism b.

∃!bb^∗

Such a space, if exists, is called a moduli space of pointed curves.

The moduli problem

Consider the flat family of smooth n−pointed curves of fixed genus g, so a typical member is a curve C with pairway distinct points p₁, . . . , p_n ∈ C. A

family can be written as

C F C

B M

The moduli problem consist of finding a universal

family of curves such that any other family is induced from M by a unique morphism b.

∃!b b^∗

Such a space, if exists, is called a moduli space of pointed curves.

Moduli space. Abstract proper- ties.

Consider a functor F : Schemes → Set

F (B) is set of all flat families over B (modulo

isomorphisms). If there exists a scheme M, such that F (B) = M or(B, M),

then it is exactly a moduli space.

Problems:

• In order to make it exists, you have to allow singular curves.

• In most cases it does not exists in the category of schemes (need to use stacks)

Moduli space. Abstract proper- ties.

Consider a functor

F : Schemes → Set

F (B) is set of all flat families over B (modulo isomorphisms).

If there exists a scheme M, such that F (B) = M or(B, M),

then it is exactly a moduli space.

Problems:

• In order to make it exists, you have to allow singular curves.

• In most cases it does not exists in the category of schemes (need to use stacks)

Moduli space. Abstract proper- ties.

Consider a functor

F : Schemes → Set

F (B) is set of all flat families over B (modulo

isomorphisms). If there exists a scheme M, such that F (B) = M or(B, M),

then it is exactly a moduli space.

Problems:

• In order to make it exists, you have to allow singular curves.

• In most cases it does not exists in the category of schemes (need to use stacks)

Moduli space. Abstract proper- ties.

Consider a functor

F : Schemes → Set

F (B) is set of all flat families over B (modulo

isomorphisms). If there exists a scheme M, such that F (B) = M or(B, M),

then it is exactly a moduli space.

Problems:

• In order to make it exists, you have to allow singular curves.

• In most cases it does not exists in the category of schemes (need to use stacks)

Moduli space. Abstract proper- ties.

Consider a functor

F : Schemes → Set

F (B) is set of all flat families over B (modulo

isomorphisms). If there exists a scheme M, such that F (B) = M or(B, M),

then it is exactly a moduli space.

Problems:

• In order to make it exists, you have to allow singular curves.

• In most cases it does not exists in the category of schemes (need to use stacks)

Moduli space. Abstract proper- ties.

Consider a functor

F : Schemes → Set

F (B) is set of all flat families over B (modulo

isomorphisms). If there exists a scheme M, such that F (B) = M or(B, M),

then it is exactly a moduli space.

Problems:

• In order to make it exists, you have to allow singular curves.

• In most cases it does not exists in the category of schemes (need to use stacks)

Moduli space. Abstract proper- ties.

Consider a functor

F : Schemes → Set

F (B) is set of all flat families over B (modulo

isomorphisms). If there exists a scheme M, such that F (B) = M or(B, M),

then it is exactly a moduli space.

Problems:

• In order to make it exists, you have to allow singular curves.

• In most cases it does not exists in the category of schemes (need to use stacks)

Examples

M_0,3 is obviously a point.

Now M_0,4.

P¹ P¹

p3

p2

p1

Blow up the green points. Hence

M_0,4 = P¹ \ {0, 1, ∞}. ¯M_0,4 = P¹.

Examples

Now M_0,4.

P¹ P¹

p3

p2

p1

Blow up the green points. Hence

M_0,4 = P¹ \ {0, 1, ∞}. ¯M_0,4 = P¹.

Examples

Now M_0,4.

P¹ P¹

p3

p2

p1

λ Blow up the green points. Hence

M_0,4 = P¹ \ {0, 1, ∞}. ¯M_0,4 = P¹.

Examples

Now M_0,4.

P¹ P¹

p3

p2

p1

Blow up the green points. Hence

M_0,4 = P¹ \ {0, 1, ∞}. ¯M_0,4 = P¹.

Examples

Now M_0,4.

P¹ P¹

p3

p2

p1

Blow up the green points.

Hence

M_0,4 = P¹ \ {0, 1, ∞}. ¯M_0,4 = P¹.

Examples

Now M_0,4.

P¹ P¹

p3

p2

p1

Blow up the green points. Hence M_0,4 = P¹ \ {0, 1, ∞}. ¯M_0,4 = P¹.

Examples

Now M_0,4.

P¹ P¹

p3

p2

p1

Blow up the green points. Hence M_0,4 = P¹ \ {0, 1, ∞}. ¯M_0,4 = P¹.

Examples

The boundary divisor δ = ¯M_0,4 \ M_0,4 consists of three points.

Each point corresponds to splitting

{p₁, p₂, p₃, p₄} = {p_i1, p_i2} ∪ {p_i3, p_i4}.

Over a point {p₁, p₂} ∪ {p₃, p₄} the fibre of the previous diagram looks like

P¹ P¹

p₁

p₂ p₃ p₄

Important, albeit trivial! All divisors corresponding to different splittings are linearly equivalent.

Examples

The boundary divisor δ = ¯M_0,4 \ M_0,4 consists of three points.

Each point corresponds to splitting

{p₁, p₂, p₃, p₄} = {p_i1, p_i2} ∪ {p_i3, p_i4}.

Over a point {p₁, p₂} ∪ {p₃, p₄} the fibre of the previous diagram looks like

P¹ P¹

p₁

p₂ p₃ p₄

Important, albeit trivial! All divisors corresponding to different splittings are linearly equivalent.

Examples

The boundary divisor δ = ¯M_0,4 \ M_0,4 consists of three points.

Each point corresponds to splitting

{p₁, p₂, p₃, p₄} = {p_i1, p_i2} ∪ {p_i3, p_i4}.

Over a point {p₁, p₂} ∪ {p₃, p₄} the fibre of the previous diagram looks like

P¹ P¹

p₁

p₂ p₃ p₄

Important, albeit trivial! All divisors corresponding to different splittings are linearly equivalent.

Examples

The boundary divisor δ = ¯M_0,4 \ M_0,4 consists of three points.

Each point corresponds to splitting

{p₁, p₂, p₃, p₄} = {p_i1, p_i2} ∪ {p_i3, p_i4}.

Over a point {p₁, p₂} ∪ {p₃, p₄} the fibre of the previous diagram looks like

P¹ P¹

p₁

p₂ p₃ p₄

Important, albeit trivial! All divisors corresponding to different splittings are linearly equivalent.

Examples

The boundary divisor δ = ¯M_0,4 \ M_0,4 consists of three points.

Each point corresponds to splitting

{p₁, p₂, p₃, p₄} = {p_i1, p_i2} ∪ {p_i3, p_i4}.

Over a point {p₁, p₂} ∪ {p₃, p₄} the fibre of the previous diagram looks like

P¹ P¹

p₁

p₂ p₃ p₄

Important, albeit trivial! All divisors corresponding to different splittings are linearly equivalent.

Examples

The boundary divisor δ = ¯M_0,4 \ M_0,4 consists of three points.

Each point corresponds to splitting

{p₁, p₂, p₃, p₄} = {p_i1, p_i2} ∪ {p_i3, p_i4}.

Over a point {p₁, p₂} ∪ {p₃, p₄} the fibre of the previous diagram looks like

P¹ P¹

p₁

p₂ p₃ p₄

Important, albeit trivial! All divisors corresponding to different splittings are linearly equivalent.

(Counter) examples

Consider M_0,2. P¹

P^{1 p1}

(Counter) examples

Consider M_0,2.

P¹

P^{1 p1}

(Counter) examples

Consider M_0,2.

P¹

P^{1 p1}

λ

(Counter) examples

Consider M_0,2.

P¹

P^{1 p1}

λ

All curves corresponing to different values of λ 6= p₁ are equivalent.

(Counter) examples

Consider M_0,2.

P¹

P^{1 p1}

λ

Blow up the green point. We get a family of identical curves tending to a different curve.

(Counter) examples

Consider M_0,2.

P¹

P^{1 p1}

λ

The space of curves with such topology is not Hausdorff.

Stable curves

• Problem: a curve P¹ has a continuous (i.e. not zero–dimensional) group of automorphisms fixing given two points p₁, p₂.

• The pointed curve C, {p₁, . . . , p_n} is called stable if it has only finitely many automorphisms.

• Exercise.A smooth curve of genus g is stable if 3g − 3 + n ≥ 0.

• A curve is called nodal (or, misleadingly, cuspidal) if it has only double points as its singularities.

Stable curves

• Problem: a curve P¹ has a continuous (i.e. not zero–dimensional) group of automorphisms fixing given two points p₁, p₂.

• The pointed curve C, {p₁, . . . , p_n} is called stable if it has only finitely many automorphisms.

• Exercise.A smooth curve of genus g is stable if 3g − 3 + n ≥ 0.

• A curve is called nodal (or, misleadingly, cuspidal) if it has only double points as its singularities.

Stable curves

• Problem: a curve P¹ has a continuous (i.e. not zero–dimensional) group of automorphisms fixing given two points p₁, p₂.

• The pointed curve C, {p₁, . . . , p_n} is called stable if it has only finitely many automorphisms.

• Exercise.

A smooth curve of genus g is stable if 3g − 3 + n ≥ 0.

• A curve is called nodal (or, misleadingly, cuspidal) if it has only double points as its singularities.

Stable curves

• Problem: a curve P¹ has a continuous (i.e. not zero–dimensional) group of automorphisms fixing given two points p₁, p₂.

• The pointed curve C, {p₁, . . . , p_n} is called stable if it has only finitely many automorphisms.

• Exercise. A smooth curve of genus g is stable if 3g − 3 + n ≥ 0.

• A curve is called nodal (or, misleadingly, cuspidal) if it has only double points as its singularities.

Stable curves

• Problem: a curve P¹ has a continuous (i.e. not zero–dimensional) group of automorphisms fixing given two points p₁, p₂.

• The pointed curve C, {p₁, . . . , p_n} is called stable if it has only finitely many automorphisms.

• Exercise. A smooth curve of genus g is stable if 3g − 3 + n ≥ 0.

• A curve is called nodal (or, misleadingly, cuspidal) if it has only double points as its singularities.

Geometric point of view.

C smooth of genus g. p₁, . . . , p_n points.

C₀ = C \ {p₁, . . . , p_n}.

• C₀ has only one topological (and differential) model.

• Complex (algebraic) structures on C₀ are in 1—1 correspondence with −1 complete metrics on C₀.

• The space G of such metrics is non–compact.

• Injectivity radius r_i — lenght of the shortest closed non-contractible geodesic.

• For δ > 0 space G_δ is compact (relatively easy).

Geometric point of view.

C smooth of genus g. p₁, . . . , p_n points.

C₀ = C \ {p₁, . . . , p_n}.

• C₀ has only one topological (and differential) model.

• Complex (algebraic) structures on C₀ are in 1—1 correspondence with −1 complete metrics on C₀.

• The space G of such metrics is non–compact.

• Injectivity radius r_i — lenght of the shortest closed non-contractible geodesic.

• For δ > 0 space G_δ is compact (relatively easy).

Geometric point of view.

C smooth of genus g. p₁, . . . , p_n points.

C₀ = C \ {p₁, . . . , p_n}.

stability → χ(C₀) < 0.

• C₀ has only one topological (and differential) model.

• Complex (algebraic) structures on C₀ are in 1—1 correspondence with −1 complete metrics on C₀.

• The space G of such metrics is non–compact.

• Injectivity radius r_i — lenght of the shortest closed non-contractible geodesic.

• For δ > 0 space G_δ is compact (relatively easy).

Geometric point of view.

C smooth of genus g. p₁, . . . , p_n points.

C₀ = C \ {p₁, . . . , p_n}.

• C₀ has only one topological (and differential) model.

• Complex (algebraic) structures on C₀ are in 1—1 correspondence with −1 complete metrics on C₀.

• The space G of such metrics is non–compact.

• Injectivity radius r_i — lenght of the shortest closed non-contractible geodesic.

• For δ > 0 space G_δ is compact (relatively easy).

Geometric point of view.

C smooth of genus g. p₁, . . . , p_n points.

C₀ = C \ {p₁, . . . , p_n}.

• C₀ has only one topological (and differential) model.

• Complex (algebraic) structures on C₀ are in 1—1 correspondence with −1 complete metrics on C₀.

• The space G of such metrics is non–compact.

• Injectivity radius r_i — lenght of the shortest closed non-contractible geodesic.

• For δ > 0 space G_δ is compact (relatively easy).

Geometric point of view.

C smooth of genus g. p₁, . . . , p_n points.

C₀ = C \ {p₁, . . . , p_n}.

• C₀ has only one topological (and differential) model.

• Complex (algebraic) structures on C₀ are in 1—1 correspondence with −1 complete metrics on C₀.

• The space G of such metrics is non–compact.

• Injectivity radius r_i — lenght of the shortest closed non-contractible geodesic.

• For δ > 0 space G_δ is compact (relatively easy).

Geometric point of view.

C smooth of genus g. p₁, . . . , p_n points.

C₀ = C \ {p₁, . . . , p_n}.

• C₀ has only one topological (and differential) model.

• Complex (algebraic) structures on C₀ are in 1—1 correspondence with −1 complete metrics on C₀.

• The space G of such metrics is non–compact.

• Injectivity radius r_i — lenght of the shortest closed non-contractible geodesic.

• For δ > 0 space G_δ is compact (relatively easy).

Geometric point of view.

C smooth of genus g. p₁, . . . , p_n points.

C₀ = C \ {p₁, . . . , p_n}.

• C₀ has only one topological (and differential) model.

• Complex (algebraic) structures on C₀ are in 1—1 correspondence with −1 complete metrics on C₀.

• The space G of such metrics is non–compact.

• Injectivity radius r_i — lenght of the shortest closed non-contractible geodesic.

• For δ > 0 space G_δ is compact (relatively easy).

Degenerations of −1 metrics.

Consider a sequence of metrics g_n on C₀ with r_i⁽ⁿ⁾ → 0.

• Split C₀ in two parts C₀^thick and C₀^thin.

• C₀^thin — points through which there exists a short non–contractible loop.

• If this is sufficiently short, C₀^thin is the sum of annuli.

• The metrics are convergent on C₀^thick.

• Annulus. Small r_i — large modulus (ratio ^R_r ).

Degenerations of −1 metrics.

Consider a sequence of metrics g_n on C₀ with r_i⁽ⁿ⁾ → 0.

• Split C₀ in two parts C₀^thick and C₀^thin.

• C₀^thin — points through which there exists a short non–contractible loop.

• If this is sufficiently short, C₀^thin is the sum of annuli.

• The metrics are convergent on C₀^thick.

• Annulus. Small r_i — large modulus (ratio ^R_r ).

Degenerations of −1 metrics.

Consider a sequence of metrics g_n on C₀ with r_i⁽ⁿ⁾ → 0.

• Split C₀ in two parts C₀^thick and C₀^thin.

• C₀^thin — points through which there exists a short non–contractible loop.

• If this is sufficiently short, C₀^thin is the sum of annuli.

• The metrics are convergent on C₀^thick.

• Annulus. Small r_i — large modulus (ratio ^R_r ).

Degenerations of −1 metrics.

Consider a sequence of metrics g_n on C₀ with r_i⁽ⁿ⁾ → 0.

• Split C₀ in two parts C₀^thick and C₀^thin.

• C₀^thin — points through which there exists a short non–contractible loop.

• If this is sufficiently short, C₀^thin is the sum of

annuli.

• The metrics are convergent on C₀^thick.

• Annulus. Small r_i — large modulus (ratio ^R_r ).

Degenerations of −1 metrics.

Consider a sequence of metrics g_n on C₀ with r_i⁽ⁿ⁾ → 0.

• Split C₀ in two parts C₀^thick and C₀^thin.

• C₀^thin — points through which there exists a short non–contractible loop.

• If this short is sufficiently short, C₀^thin is the sum of annuli.

• The metrics are convergent on C₀^thick.

• Annulus. Small r_i — large modulus (ratio ^R_r ).

Degenerations of −1 metrics.

Consider a sequence of metrics g_n on C₀ with r_i⁽ⁿ⁾ → 0.

• Split C₀ in two parts C₀^thick and C₀^thin.

• C₀^thin — points through which there exists a short non–contractible loop.

• If this short is sufficiently short, C₀^thin is the sum of annuli.

• The metrics are convergent on C₀^thick.

• Annulus. Small r_i — large modulus (ratio ^R_r ).

Degenerations of −1 metrics.

Consider a sequence of metrics g_n on C₀ with r_i⁽ⁿ⁾ → 0.

• Split C₀ in two parts C₀^thick and C₀^thin.

• C₀^thin — points through which there exists a short non–contractible loop.

• If this short is sufficiently short, C₀^thin is the sum of annuli.

• The metrics are convergent on C₀^thick.

• Annulus. Small r_i — large modulus (ratio ^R_r ).

Degeneration of annuli

The shape of an annulus is uniquely determined by its modulus.

Annuli degenerate to two discs with one common point.

Degeneration of annuli

The shape of an annulus is uniquely determined by its modulus.

Annuli degenerate to two discs with one common point.

Degeneration of annuli

The shape of an annulus is uniquely determined by its modulus.

Annuli degenerate to two discs with one common point.

Degeneration of annuli

The shape of an annulus is uniquely determined by its modulus.

Annuli degenerate to two discs with one common point.

Degeneration of annuli

The shape of an annulus is uniquely determined by its modulus.

Annuli degenerate to two discs with one common point.

Degeneration of annuli

The shape of an annulus is uniquely determined by its modulus.

Annuli degenerate to two discs with one common point.

Degeneration of annuli

The shape of an annulus is uniquely determined by its modulus.

Annuli degenerate to two discs with one common point.

Degeneration of annuli

The shape of an annulus is uniquely determined by its modulus.

Annuli degenerate to two discs with one common point.

Structure of ¯ M

.

• M¯ _g,nis a smooth compact orbifold of dimension 3g − 3 + n.

• the orbifold means that localy the space looks like the quotient of C^3g−3+n/G. G finite group.

• the set M_g,n ⊂ ¯M_g,n is open dense.

∆_g,n = ¯M_g,n \ M_g,n is a divisor.

• Consider M_g,I, #I = n. Let g = g₁ + g₂, I = I₁ ∪ I₂.

• Glueing map: ¯M_g1,I1∪p × ¯M_g2,I2∪q → M_g,I. Glue p with q.

Structure of ¯ M

.

• M¯ _g,nis a smooth compact orbifold of dimension 3g − 3 + n.

• the orbifold means that localy the space looks like the quotient of C^3g−3+n/G. G finite group.

• the set M_g,n ⊂ ¯M_g,n is open dense.

∆_g,n = ¯M_g,n \ M_g,n is a divisor.

• Consider M_g,I, #I = n. Let g = g₁ + g₂, I = I₁ ∪ I₂.

• Glueing map: ¯M_g1,I1∪p × ¯M_g2,I2∪q → M_g,I. Glue p with q.

Structure of ¯ M

.

• M¯ _g,nis a smooth compact orbifold of dimension 3g − 3 + n.

• the orbifold means that localy the space looks like the quotient of C^3g−3+n/G. G finite group.

• the set M_g,n ⊂ ¯M_g,n is open dense.

∆_g,n = ¯M_g,n \ M_g,n is a divisor.

• Consider M_g,I, #I = n. Let g = g₁ + g₂, I = I₁ ∪ I₂.

• Glueing map: ¯M_g1,I1∪p × ¯M_g2,I2∪q → M_g,I. Glue p with q.

Structure of ¯ M

.

• M¯ _g,nis a smooth compact orbifold of dimension 3g − 3 + n.

• the orbifold means that localy the space looks like the quotient of C^3g−3+n/G. G finite group.

• the orbifold is the most basic non–trivial example of a stack.

• the set M_g,n ⊂ ¯M_g,n is open dense.

∆_g,n = ¯M_g,n \ M_g,n is a divisor.

• Consider M_g,I, #I = n. Let g = g₁ + g₂, I = I₁ ∪ I₂.

• Glueing map: ¯M_g1,I1∪p × ¯M_g2,I2∪q → M_g,I. Glue p with q.

Structure of ¯ M

.

• M¯ _g,nis a smooth compact orbifold of dimension 3g − 3 + n.

• the orbifold means that localy the space looks like the quotient of C^3g−3+n/G. G finite group.

• the group G is Aut(C, {p₁, . . . , p_n}) generically trivial.

• the set M_g,n ⊂ ¯M_g,n is open dense.

∆_g,n = ¯M_g,n \ M_g,n is a divisor.

• Consider M_g,I, #I = n. Let g = g₁ + g₂, I = I₁ ∪ I₂.

• Glueing map: ¯M_g1,I1∪p × ¯M_g2,I2∪q → M_g,I. Glue p with q.

Structure of ¯ M

.

• M¯ _g,nis a smooth compact orbifold of dimension 3g − 3 + n.

• the orbifold means that localy the space looks like the quotient of C^3g−3+n/G. G finite group.

• for g = 0, G is always trivial, so ¯M_g,n is a smooth manifold.

• the set M_g,n ⊂ ¯M_g,n is open dense.

∆_g,n = ¯M_g,n \ M_g,n is a divisor.

• Consider M_g,I, #I = n. Let g = g₁ + g₂, I = I₁ ∪ I₂.

• Glueing map: ¯M_g1,I1∪p × ¯M_g2,I2∪q → M_g,I. Glue p with q.

Structure of ¯ M

.

• M¯ _g,nis a smooth compact orbifold of dimension 3g − 3 + n.

• the orbifold means that localy the space looks like the quotient of C^3g−3+n/G. G finite group.

• the set M_g,n ⊂ ¯M_g,n is open dense.

∆_g,n = ¯M_g,n \ M_g,n is a divisor.

• Consider M_g,I, #I = n. Let g = g₁ + g₂, I = I₁ ∪ I₂.

• Glueing map: ¯M_g1,I1∪p × ¯M_g2,I2∪q → M_g,I. Glue p with q.

Structure of ¯ M

.

• M¯ _g,nis a smooth compact orbifold of dimension 3g − 3 + n.

• the orbifold means that localy the space looks like the quotient of C^3g−3+n/G. G finite group.

• the set M_g,n ⊂ ¯M_g,n is open dense.

∆_g,n = ¯M_g,n \ M_g,n is a divisor. for g = 0, n = 4, ∆_0,4 consists of three points.

• Consider M_g,I, #I = n. Let g = g₁ + g₂, I = I₁ ∪ I₂.

• Glueing map: ¯M_g1,I1∪p × ¯M_g2,I2∪q → M_g,I. Glue p with q.

Structure of ¯ M

.

• M¯ _g,nis a smooth compact orbifold of dimension 3g − 3 + n.

• the orbifold means that localy the space looks like the quotient of C^3g−3+n/G. G finite group.

• the set M_g,n ⊂ ¯M_g,n is open dense.

∆_g,n = ¯M_g,n \ M_g,n is a divisor.

• Consider M_g,I, #I = n. Let g = g₁ + g₂, I = I₁ ∪ I₂.

• Glueing map: ¯M_g1,I1∪p × ¯M_g2,I2∪q → M_g,I. Glue p with q.

Structure of ¯ M

.

• M¯ _g,nis a smooth compact orbifold of dimension 3g − 3 + n.

• the orbifold means that localy the space looks like the quotient of C^3g−3+n/G. G finite group.

• the set M_g,n ⊂ ¯M_g,n is open dense.

∆_g,n = ¯M_g,n \ M_g,n is a divisor.

• Consider M_g,I, #I = n. Let g = g₁ + g₂, I = I₁ ∪ I₂.

• Glueing map: ¯M_g1,I1∪p × ¯M_g2,I2∪q → M_g,I. Glue p with q.

Forgetting map.

• The map M_g,n → M_g,n−1

• prolonges to boundary ¯M_g,n → ¯M_g,n−1.

• For M_0,5 → M_0,4 we get

3

1 2

4

5

Forgetting map.

• The map M_g,n → M_g,n−1

• prolonges to boundary ¯M_g,n → ¯M_g,n−1.

• For M_0,5 → M_0,4 we get

3

1 2

4

5

Forgetting map.

• The map M_g,n → M_g,n−1

• prolonges to boundary ¯M_g,n → ¯M_g,n−1.

• For M_0,5 → M_0,4 we get

3

1 2

4

5

Forgetting map.

• The map M_g,n → M_g,n−1

• prolonges to boundary ¯M_g,n → ¯M_g,n−1.

• For M_0,5 → M_0,4 we get

3

1 2

4

5

Forgetting map.

• The map M_g,n → M_g,n−1

• prolonges to boundary ¯M_g,n → ¯M_g,n−1.

• For M_0,5 → M_0,4 we get

3

1 2

4

5 1

2 3

4

Forgetting map.

• The map M_g,n → M_g,n−1

• prolonges to boundary ¯M_g,n → ¯M_g,n−1.

• For M_0,5 → M_0,4 we get

3

1

5 2

4

Forgetting map.

• The map M_g,n → M_g,n−1

• prolonges to boundary ¯M_g,n → ¯M_g,n−1.

• For M_0,5 → M_0,4 we get

3

1

5

1 2 3

4 2

4

Boundary relations in ¯ M

.

• Consider a forgetting map π = π_{i,j,k,l} : ¯M_0,n → ¯M_0,4

• A, B — partition of [1, . . . , n].

• D(A|B) image of the glueing map M¯ _0,A∪p × ¯M_0,B∪q → ¯M_0,n.

• Then π⁻¹({i, j} ∪ {k, l}) is the sum of all

D(A|B) for partitions {i, j} ⊂ A, {k, l} ⊂ B.

• The divisors {i, j} ∪ {k, l}, {i, k} ∪ {j, l} and {i, l} ∪ {j, k} are linearly equivalent.

Boundary relations in ¯ M

.

• Consider a forgetting map π = π_{i,j,k,l} : ¯M_0,n → ¯M_0,4

• A, B — partition of [1, . . . , n].

• D(A|B) image of the glueing map M¯ _0,A∪p × ¯M_0,B∪q → ¯M_0,n.

• Then π⁻¹({i, j} ∪ {k, l}) is the sum of all

D(A|B) for partitions {i, j} ⊂ A, {k, l} ⊂ B.

• The divisors {i, j} ∪ {k, l}, {i, k} ∪ {j, l} and {i, l} ∪ {j, k} are linearly equivalent.

Boundary relations in ¯ M

.

• Consider a forgetting map π = π_{i,j,k,l} : ¯M_0,n → ¯M_0,4

• A, B — partition of [1, . . . , n].

• D(A|B) image of the glueing map M¯ _0,A∪p × ¯M_0,B∪q → ¯M_0,n.

• Then π⁻¹({i, j} ∪ {k, l}) is the sum of all

D(A|B) for partitions {i, j} ⊂ A, {k, l} ⊂ B.

• The divisors {i, j} ∪ {k, l}, {i, k} ∪ {j, l} and {i, l} ∪ {j, k} are linearly equivalent.

Boundary relations in ¯ M

.

• Consider a forgetting map π = π_{i,j,k,l} : ¯M_0,n → ¯M_0,4

• A, B — partition of [1, . . . , n].

• D(A|B) image of the glueing map M¯ _0,A∪p × ¯M_0,B∪q → ¯M_0,n.

• Then π⁻¹({i, j} ∪ {k, l}) is the sum of all

D(A|B) for partitions {i, j} ⊂ A, {k, l} ⊂ B.

• The divisors {i, j} ∪ {k, l}, {i, k} ∪ {j, l} and {i, l} ∪ {j, k} are linearly equivalent.

Boundary relations in ¯ M

.

• Consider a forgetting map π = π_{i,j,k,l} : ¯M_0,n → ¯M_0,4

• A, B — partition of [1, . . . , n].

• D(A|B) image of the glueing map M¯ _0,A∪p × ¯M_0,B∪q → ¯M_0,n.

• Then π⁻¹({i, j} ∪ {k, l}) is the sum of all

D(A|B) for partitions {i, j} ⊂ A, {k, l} ⊂ B.

• The divisors {i, j} ∪ {k, l}, {i, k} ∪ {j, l} and {i, l} ∪ {j, k} are linearly equivalent.

Boundary relations in ¯ M

.

X

i,j∈A k,l∈B

D(A|B) ∼ X

i,k∈A j,l∈B

D(A|B).

Kontsewich formula N_d = X

d1+d2=d d1,d2>0

N_d1N_d2·



d²₁d²₂ 3d − 4 3d₁ − 2



− d³₁d₂ 3d − 4 3d₁ − 1



. N_d number of plane rational curves going through 3d − 1 points.

Boundary relations in ¯ M

.

X

i,j∈A k,l∈B

D(A|B) ∼ X

i,k∈A j,l∈B

D(A|B).

Kontsewich formula N_d = X

d1+d2=d d1,d2>0

N_d1N_d2·



d²₁d²₂ 3d − 4 3d₁ − 2



− d³₁d₂ 3d − 4 3d₁ − 1



. N_d number of plane rational curves going through

3d − 1 points.

Boundary relations in ¯ M

.

X

i,j∈A k,l∈B

D(A|B) ∼ X

i,k∈A j,l∈B

D(A|B).

Kontsewich formula N_d = X

d1+d2=d d1,d2>0

N_d1N_d2·



d²₁d²₂ 3d − 4 3d₁ − 2



− d³₁d₂ 3d − 4 3d₁ − 1



. N_d number of plane rational curves going through 3d − 1 points.

Tautological bundles over ¯ M

• L_i has the fiber T_p^∗_iC.

• Λ_g,n — Hodge line bundle. Fiber H⁰(C, ω_C).

• Λ_g,n = π_∗ω_C/M relative dualising sheaf.

• ψ_i = c₁(L_i), c(Λ_g,n) = 1 + λ₁ + λ₂ + · · · + λ_g.

• π : ¯M_g,n+1 → ¯M_g,n. ψ_g,n+1,i = π^∗(ψ_g,n,i) + [D].

• D ' ¯M_{0,{pi}∪{pn+1}∪{p}} × ¯M_{g,[n]\{pi}∪{q}}.

Tautological bundles over ¯ M

• L_i has the fiber T_p^∗_iC.

• Λ_g,n — Hodge line bundle. Fiber H⁰(C, ω_C).

• Λ_g,n = π_∗ω_C/M relative dualising sheaf.

• ψ_i = c₁(L_i), c(Λ_g,n) = 1 + λ₁ + λ₂ + · · · + λ_g.

• π : ¯M_g,n+1 → ¯M_g,n. ψ_g,n+1,i = π^∗(ψ_g,n,i) + [D].

• D ' ¯M_{0,{pi}∪{pn+1}∪{p}} × ¯M_{g,[n]\{pi}∪{q}}.

Tautological bundles over ¯ M

• L_i has the fiber T_p^∗_iC.

• Λ_g,n — Hodge line bundle. Fiber H⁰(C, ω_C).

• Λ_g,n = π_∗ω_C/M relative dualising sheaf.

• ψ_i = c₁(L_i), c(Λ_g,n) = 1 + λ₁ + λ₂ + · · · + λ_g.

• π : ¯M_g,n+1 → ¯M_g,n. ψ_g,n+1,i = π^∗(ψ_g,n,i) + [D].

• D ' ¯M_{0,{pi}∪{pn+1}∪{p}} × ¯M_{g,[n]\{pi}∪{q}}.

Tautological bundles over ¯ M

• L_i has the fiber T_p^∗_iC.

• Λ_g,n — Hodge line bundle. Fiber H⁰(C, ω_C).

• Λ_g,n = π_∗ω_C/M relative dualising sheaf.

• ψ_i = c₁(L_i), c(Λ_g,n) = 1 + λ₁ + λ₂ + · · · + λ_g.

• π : ¯M_g,n+1 → ¯M_g,n. ψ_g,n+1,i = π^∗(ψ_g,n,i) + [D].

• D ' ¯M_{0,{pi}∪{pn+1}∪{p}} × ¯M_{g,[n]\{pi}∪{q}}.

Tautological bundles over ¯ M

• L_i has the fiber T_p^∗_iC.

• Λ_g,n — Hodge line bundle. Fiber H⁰(C, ω_C).

• Λ_g,n = π_∗ω_C/M relative dualising sheaf.

• ψ_i = c₁(L_i), c(Λ_g,n) = 1 + λ₁ + λ₂ + · · · + λ_g.

• π : ¯M_g,n+1 → ¯M_g,n. ψ_g,n+1,i = π^∗(ψ_g,n,i) + [D].

• D ' ¯M_{0,{pi}∪{pn+1}∪{p}} × ¯M_{g,[n]\{pi}∪{q}}.

Tautological bundles over ¯ M

• L_i has the fiber T_p^∗_iC.

• Λ_g,n — Hodge line bundle. Fiber H⁰(C, ω_C).

• Λ_g,n = π_∗ω_C/M relative dualising sheaf.

• ψ_i = c₁(L_i), c(Λ_g,n) = 1 + λ₁ + λ₂ + · · · + λ_g.

• π : ¯M_g,n+1 → ¯M_g,n. ψ_g,n+1,i = π^∗(ψ_g,n,i) + [D].

• D ' ¯M_{0,{pi}∪{pn+1}∪{p}} × ¯M_{g,[n]\{pi}∪{q}}.

Intersections on ¯ M

hτ_k1τ_k2 . . . τ_kni = Z

M¯ g,n

ψ₁^k1ψ₂^k2 . . . ψ_n^kn.

• defined for k₁ + · · · + k_n = 3g − 3 + n, otherwise 0.

• the pull-back property of ψ_i gives string equation and dilaton equation

hτ₀

n

Y

i=1

τ_kii_g =

n

X

j=1

hτ_kj−1 Y

i6=j

τ_kii_g.

hτ₁

n

Y

i=1

τ_kii_g = (2g − 2 + n)h

n

Y

i=1

τ_kii_g.