.

.. ..

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The Noether Inequality for Algebraic Threefolds

Meng Chen

School of Mathematical Sciences, Fudan University

2018.09.08

. . . . . .

. . . . . .

Back ground

.. P1. Surface geography

• Two famous inequalities

-

6

χ(O)

c21

O ���������

����

�����

Miyaoka-Yau inequality

Noether inequality

• Further classification — the geometry of Mc21,c2 .

Meng Chen (Fudan University) The Noether Inequality for Algebraic Threefolds

. . . . . .

Back ground

.. P1. Surface geography

• Two famous inequalities

-

6

χ(O)

c21

O ���������

����

�����

Miyaoka-Yau inequality

Noether inequality

• Further classification — the geometry of Mc21,c2 .

Meng Chen (Fudan University) The Noether Inequality for Algebraic Threefolds

. . . . . .

Threefold geography

.. P2. The strategy for threefold geography

• X: a minimal 3-fold of general type

• Main birational invariants: K3X, χ(OX), q(X) = h1(OX),

q2 = h2(OX), pg(X) = h3(OX).

• Main task: find optimal relations among above birationalinvariants.

Meng Chen (Fudan University) The Noether Inequality for Algebraic Threefolds

. . . . . .

Threefold geography

.. P2. The strategy for threefold geography

• X: a minimal 3-fold of general type

• Main birational invariants: K3X, χ(OX), q(X) = h1(OX),

q2 = h2(OX), pg(X) = h3(OX).

• Main task: find optimal relations among above birationalinvariants.

Meng Chen (Fudan University) The Noether Inequality for Algebraic Threefolds

. . . . . .

Threefold geography

.. P2. The strategy for threefold geography

• X: a minimal 3-fold of general type

• Main birational invariants: K3X, χ(OX), q(X) = h1(OX),

q2 = h2(OX), pg(X) = h3(OX).

• Main task: find optimal relations among above birationalinvariants.

Meng Chen (Fudan University) The Noether Inequality for Algebraic Threefolds

. . . . . .

Threefold geography

.. P3. The possible Noether type inequality

• The 3-dimensional analogy of Miyaoka-Yau inequality:Neither “K3

X ≤?χ(OX)” is possible, as −∞ < χ(OX) < +∞,nor is “K3

X ≤?pg(X)” possible.

• Seek for the Noether type inequality: K3X ≥ a pg(X)− b,

a, b ∈ Q>0.

-

6

pg(X)

K3X

O ����

�����

The Noether inequality

Meng Chen (Fudan University) The Noether Inequality for Algebraic Threefolds

. . . . . .

Threefold geography

.. P3. The possible Noether type inequality

• The 3-dimensional analogy of Miyaoka-Yau inequality:Neither “K3

X ≤?χ(OX)” is possible, as −∞ < χ(OX) < +∞,nor is “K3

X ≤?pg(X)” possible.

• Seek for the Noether type inequality: K3X ≥ a pg(X)− b,

a, b ∈ Q>0.

-

6

pg(X)

K3X

O ����

�����

The Noether inequality

Meng Chen (Fudan University) The Noether Inequality for Algebraic Threefolds

. . . . . .

Threefold geography

.. P4. The simplest case: Gorenstein minimal 3-folds

• X minimal, the Cartier index rX ≥ 1.

X is Gorenstein ⇐⇒ rX = 1

{smooth minimal 3-folds} ⊂ {Grenstein minimal 3-folds}

⊂ {arbitrary minimal 3-folds}

Meng Chen (Fudan University) The Noether Inequality for Algebraic Threefolds

. . . . . .

Threefold geography

.. P5. The Noether inequality for Gorenstein minimal 3-folds

• Kobayashi (1992): an infinite series of examples ofcanonically polarized 3-folds satisfying K3

X = 43pg(X)− 10

3 .

• M. Chen (2004): K3X ≥ 4

3pg(X)− 103 for canonically

polarized 3-folds.

• Catanese-Chen-Zhang (2006): K3X ≥ 4

3pg(X)− 103 for

smooth minimal 3-folds of general type.

• Chen-Chen (2015): K3X ≥ 4

3pg(X)− 103 for Gorenstein

minimal 3-folds of general type.

Meng Chen (Fudan University) The Noether Inequality for Algebraic Threefolds

. . . . . .

Threefold geography

.. P5. The Noether inequality for Gorenstein minimal 3-folds

• Kobayashi (1992): an infinite series of examples ofcanonically polarized 3-folds satisfying K3

X = 43pg(X)− 10

3 .

• M. Chen (2004): K3X ≥ 4

3pg(X)− 103 for canonically

polarized 3-folds.

• Catanese-Chen-Zhang (2006): K3X ≥ 4

3pg(X)− 103 for

smooth minimal 3-folds of general type.

• Chen-Chen (2015): K3X ≥ 4

3pg(X)− 103 for Gorenstein

minimal 3-folds of general type.

Meng Chen (Fudan University) The Noether Inequality for Algebraic Threefolds

. . . . . .

Threefold geography

.. P5. The Noether inequality for Gorenstein minimal 3-folds

• Kobayashi (1992): an infinite series of examples ofcanonically polarized 3-folds satisfying K3

X = 43pg(X)− 10

3 .

• M. Chen (2004): K3X ≥ 4

3pg(X)− 103 for canonically

polarized 3-folds.

• Catanese-Chen-Zhang (2006): K3X ≥ 4

3pg(X)− 103 for

smooth minimal 3-folds of general type.

• Chen-Chen (2015): K3X ≥ 4

3pg(X)− 103 for Gorenstein

minimal 3-folds of general type.

Meng Chen (Fudan University) The Noether Inequality for Algebraic Threefolds

. . . . . .

Threefold geography

.. P5. The Noether inequality for Gorenstein minimal 3-folds

• Kobayashi (1992): an infinite series of examples ofcanonically polarized 3-folds satisfying K3

X = 43pg(X)− 10

3 .

• M. Chen (2004): K3X ≥ 4

3pg(X)− 103 for canonically

polarized 3-folds.

• Catanese-Chen-Zhang (2006): K3X ≥ 4

3pg(X)− 103 for

smooth minimal 3-folds of general type.

• Chen-Chen (2015): K3X ≥ 4

3pg(X)− 103 for Gorenstein

minimal 3-folds of general type.

Meng Chen (Fudan University) The Noether Inequality for Algebraic Threefolds

. . . . . .

The main statement

.. P6. General case: non-Gorenstein minimal 3-folds

• May always assume pg(X) ≥ 2, since K3X > 0. So φ|KX| is

non-trivial.

• Set up for φ1 = φ|KX|.

X

X′

Σ′

Γ-

? ?

@@@

@@R- - - - - - - - - - --

f

sπ

φ|KX|

g

Meng Chen (Fudan University) The Noether Inequality for Algebraic Threefolds

. . . . . .

The main statement

.. P6. General case: non-Gorenstein minimal 3-folds

• May always assume pg(X) ≥ 2, since K3X > 0. So φ|KX| is

non-trivial.

• Set up for φ1 = φ|KX|.

X

X′

Σ′

Γ-

? ?

@@@

@@R- - - - - - - - - - --

f

sπ

φ|KX|

g

Meng Chen (Fudan University) The Noether Inequality for Algebraic Threefolds

. . . . . .

The main statement

.. P7. Notations

• |KX′ | = |M|+ Z, S a generic irreducible element of |M|.

• The canonical dimension dX = dim(Γ).

• π∗(KX) ∼ M + E′, E′ an effective Q-divisor.

• When dX ≥ 2, S ∼ M,

K3X ≥ (π∗(KX)|S · M|S) + (π∗(KX)|S · E′|S).

• When dX = 1, M ≡ pS = pF, F a general fiber of f,p ≥ pg(X)− 1.

Meng Chen (Fudan University) The Noether Inequality for Algebraic Threefolds

. . . . . .

The main statement

.. P7. Notations

• |KX′ | = |M|+ Z, S a generic irreducible element of |M|.

• The canonical dimension dX = dim(Γ).

• π∗(KX) ∼ M + E′, E′ an effective Q-divisor.

• When dX ≥ 2, S ∼ M,

K3X ≥ (π∗(KX)|S · M|S) + (π∗(KX)|S · E′|S).

• When dX = 1, M ≡ pS = pF, F a general fiber of f,p ≥ pg(X)− 1.

Meng Chen (Fudan University) The Noether Inequality for Algebraic Threefolds

. . . . . .

The main statement

.. P7. Notations

• |KX′ | = |M|+ Z, S a generic irreducible element of |M|.

• The canonical dimension dX = dim(Γ).

• π∗(KX) ∼ M + E′, E′ an effective Q-divisor.

• When dX ≥ 2, S ∼ M,

K3X ≥ (π∗(KX)|S · M|S) + (π∗(KX)|S · E′|S).

• When dX = 1, M ≡ pS = pF, F a general fiber of f,p ≥ pg(X)− 1.

Meng Chen (Fudan University) The Noether Inequality for Algebraic Threefolds

. . . . . .

The main statement

.. P7. Notations

• |KX′ | = |M|+ Z, S a generic irreducible element of |M|.

• The canonical dimension dX = dim(Γ).

• π∗(KX) ∼ M + E′, E′ an effective Q-divisor.

• When dX ≥ 2, S ∼ M,

K3X ≥ (π∗(KX)|S · M|S) + (π∗(KX)|S · E′|S).

• When dX = 1, M ≡ pS = pF, F a general fiber of f,p ≥ pg(X)− 1.

Meng Chen (Fudan University) The Noether Inequality for Algebraic Threefolds

. . . . . .

The main statement

.. P7. Notations

• |KX′ | = |M|+ Z, S a generic irreducible element of |M|.

• The canonical dimension dX = dim(Γ).

• π∗(KX) ∼ M + E′, E′ an effective Q-divisor.

• When dX ≥ 2, S ∼ M,

K3X ≥ (π∗(KX)|S · M|S) + (π∗(KX)|S · E′|S).

• When dX = 1, M ≡ pS = pF, F a general fiber of f,p ≥ pg(X)− 1.

Meng Chen (Fudan University) The Noether Inequality for Algebraic Threefolds

. . . . . .

The main statement

.. P8. The main statement.Theorem..

.. ..

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.

Let X be a minimal projective 3-fold of general type. Assume thatone of the following holds:

dX ≥ 2; or

dX = 1 and |KX| is not composed with a rational pencil of(1, 2)-surfaces; or

dX = 1, |KX| is composed with a rational pencil of(1, 2)-surfaces, and |KX| is weakly free.

Then the inequality

K3X ≥ 4

3pg(X)−

10

3

holds.Meng Chen (Fudan University) The Noether Inequality for Algebraic Threefolds

. . . . . .

The main statement

.. P9. Preparation: surfaces admitting a genus 2 fibration

.Proposition..

.. ..

.

.

Let S be a smooth projective surface of general type and T asmooth complete curve. Suppose that f : S → T is a fibration ofwhich the general fiber C is of genus 2 . Assume that pg(S) ≥ 3

and KS ≡ nC + G for some effective integral divisor G on S and apositive integer n. Then

vol(S) ≥ 8

3(n − 1).

Meng Chen (Fudan University) The Noether Inequality for Algebraic Threefolds

. . . . . .

The main statement

.. P10. Preparation: Kawamata extension theorem

.Theorem..

.. ..

.

.

Let V be a smooth variety and D a smooth divisor on V. Assumethat KV + D ∼Q A + B where A is an ample Q-divisor and B is aneffective Q-divisor such that D ⊆ Supp(B). Then the naturalhomomorphism

H0(V,OV(m(KV + D))) → H0(D,OD(mKD))

is surjective for all m ≥ 2.

Meng Chen (Fudan University) The Noether Inequality for Algebraic Threefolds

. . . . . .

The main statement

.. P11. Preparation: comparison result

.Corollary..

.. ..

.

.

Let X be a minimal projective 3-fold of general type and D asemi-ample Weil divisor on X. Let π : W → X be a resolution andS, a semi-ample divisor on W, is assumed to be a smooth surfaceof general type. Assume that λπ∗D − S is Q-effective for somepositive rational number λ. Then π∗(KX + λD)|S − σ∗KS0

isQ-effective on S, where σ : S → S0 is the contraction onto theminimal model S0.

Meng Chen (Fudan University) The Noether Inequality for Algebraic Threefolds

. . . . . .

The main statement

.. P12. (Preparation): weak positivity

Let X be a smooth projective variety and F a torsion-freecoherent sheaf on X. We say that F is weakly positive on X ifthere exists some Zariski open subvariety U ⊆ X such that for everyample invertible sheaf H and every positive integer α, there existssome positive integer β such that (SαβF)∗∗ ⊗Hβ is generated byglobal sections over U, which means that the natural map

H0(X, (SαβF)∗∗ ⊗Hβ)⊗OX → (SαβF)∗∗ ⊗Hβ

is surjective over U. Here (SkF)∗∗ denotes the reflexive hull of thesymmetric product SkF .

Meng Chen (Fudan University) The Noether Inequality for Algebraic Threefolds

. . . . . .

The main statement

.. P13. (Preparation): Viehweg-Campana weak positivity

.Theorem..

.. ..

.

.

Let g : Y → Z be a surjective morphism between smooth projectivevarieties and D a reduced divisor on Y with simple normal crossingsupport. Then g∗OY(m(KY/Z + D)) is torsion free and weaklypositive for every integer m > 0.

Meng Chen (Fudan University) The Noether Inequality for Algebraic Threefolds

. . . . . .

The main statement

.. P14. (Preparation): log canonical threshold

• The log canonical threshold of D with respect to (X,B) isdefined by

lct(X,B;D) = sup{t ≥ 0 | (X,B + tD) is lc}.

• Definition of global log canonical threshold (“glct”) :Let Y be a normal projective variety with at worst kltsingularities such that KY is nef and big. The glct of Y isdefined as the following:

glct(Y) = inf{lct(Y;D) | 0 ≤ D ∼Q KY}

= sup{t ≥ 0 | (Y, tD) is lc for all 0 ≤ D ∼Q KY}.

Meng Chen (Fudan University) The Noether Inequality for Algebraic Threefolds

. . . . . .

The main statement

.. P14. (Preparation): log canonical threshold

• The log canonical threshold of D with respect to (X,B) isdefined by

lct(X,B;D) = sup{t ≥ 0 | (X,B + tD) is lc}.

• Definition of global log canonical threshold (“glct”) :Let Y be a normal projective variety with at worst kltsingularities such that KY is nef and big. The glct of Y isdefined as the following:

glct(Y) = inf{lct(Y;D) | 0 ≤ D ∼Q KY}

= sup{t ≥ 0 | (Y, tD) is lc for all 0 ≤ D ∼Q KY}.

Meng Chen (Fudan University) The Noether Inequality for Algebraic Threefolds

. . . . . .

The main statement

.. P15. The “glct” of (1,2)-surfaces

• (Chen-Chen-Jiang) Let S be a minimal (1, 2)-surface (i.e.K2

S = 1, pg(S) = 2). Then glct(S) ≥ 113 .

• (Janos Kollar) glct(1, 2) ≥ 110 . (optimal)

• Example of Kollar: Consider the pair

S :=(x7y3+y10+z5+t2 = 0

)⊂ P(1, 1, 2, 5) and ∆ := (y = 0).

It is easy to check that S has a unique singular point, at(1:0:0:0), and it has type E8. Thus S is a projective surfacewith Du Val singularities, KS = OS(1) is ample, K2

S = 1 andh0(S,OS(2KS)

)= 4. Furthermore, lct(S,∆) = 1

10 .

Meng Chen (Fudan University) The Noether Inequality for Algebraic Threefolds

. . . . . .

The main statement

.. P15. The “glct” of (1,2)-surfaces

• (Chen-Chen-Jiang) Let S be a minimal (1, 2)-surface (i.e.K2

S = 1, pg(S) = 2). Then glct(S) ≥ 113 .

• (Janos Kollar) glct(1, 2) ≥ 110 . (optimal)

• Example of Kollar: Consider the pair

S :=(x7y3+y10+z5+t2 = 0

)⊂ P(1, 1, 2, 5) and ∆ := (y = 0).

It is easy to check that S has a unique singular point, at(1:0:0:0), and it has type E8. Thus S is a projective surfacewith Du Val singularities, KS = OS(1) is ample, K2

S = 1 andh0(S,OS(2KS)

)= 4. Furthermore, lct(S,∆) = 1

10 .

Meng Chen (Fudan University) The Noether Inequality for Algebraic Threefolds

. . . . . .

The main statement

.. P15. The “glct” of (1,2)-surfaces

• (Chen-Chen-Jiang) Let S be a minimal (1, 2)-surface (i.e.K2

S = 1, pg(S) = 2). Then glct(S) ≥ 113 .

• (Janos Kollar) glct(1, 2) ≥ 110 . (optimal)

• Example of Kollar: Consider the pair

S :=(x7y3+y10+z5+t2 = 0

)⊂ P(1, 1, 2, 5) and ∆ := (y = 0).

It is easy to check that S has a unique singular point, at(1:0:0:0), and it has type E8. Thus S is a projective surfacewith Du Val singularities, KS = OS(1) is ample, K2

S = 1 andh0(S,OS(2KS)

)= 4. Furthermore, lct(S,∆) = 1

10 .

Meng Chen (Fudan University) The Noether Inequality for Algebraic Threefolds

. . . . . .

The main statement

.. P16. A pencil of surfaces on a 3-fold over a curve

.Proposition..

.. ..

.

.

Assume that there exists a resolution π : W → X such that Wadmits a fibration structure f : W → Γ onto a smooth curve Γ.Denote by F a general fiber of f and F0 the minimal model of F.Assume that

...1 there exists a π-exceptional prime divisor E0 on W such that(π∗(KX)|F · E0|F) > 0, and

...2 π∗(KX) ∼Q bF + D for some rational number b > 0 and aneffective Q-divisor D on W.

Then b ≤ 2glct(F0)

.

(Application of Kollar’s “connectness lemma”)Meng Chen (Fudan University) The Noether Inequality for Algebraic Threefolds

. . . . . .

The main statement

.. P17. When is |KX| weakly free!

.Corollary..

.. ..

.

.

Let X be a minimal projective 3-fold of general type such that |KX|is composed with a pencil of (1, 2)-surfaces. Assume that one ofthe following holds:

...1 |KX| is composed with an irrational pencil; or

...2 |KX| is composed with a rational pencil and pg(X) ≥ 21; or

...3 X is Gorenstein.

Then there exists a minimal projective 3-fold Y, being birational toX, such that Mov|KY| is base point free.

Meng Chen (Fudan University) The Noether Inequality for Algebraic Threefolds

. . . . . .

Main steps of the proof

.. P18. Proof: The case dX ≥ 2

• dX = 3, Kobayashi V K3X ≥ 2pg(X)− 6.

• dX = 2, f : X′ → Γ, C the general fiber of f.

• dX = 2 and g(C) ≥ 3:

K3X ≥ (π∗KX|S ·S|S) = a(π∗KX|S ·C) ≥ (pg(X)−2)(π∗KX|S ·C).

The comparison inequality implies (π∗KX|S · C) ≥ 2. Hence

K3X ≥ 2pg(X)− 4.

Meng Chen (Fudan University) The Noether Inequality for Algebraic Threefolds

. . . . . .

Main steps of the proof

.. P18. Proof: The case dX ≥ 2

• dX = 3, Kobayashi V K3X ≥ 2pg(X)− 6.

• dX = 2, f : X′ → Γ, C the general fiber of f.

• dX = 2 and g(C) ≥ 3:

K3X ≥ (π∗KX|S ·S|S) = a(π∗KX|S ·C) ≥ (pg(X)−2)(π∗KX|S ·C).

The comparison inequality implies (π∗KX|S · C) ≥ 2. Hence

K3X ≥ 2pg(X)− 4.

Meng Chen (Fudan University) The Noether Inequality for Algebraic Threefolds

. . . . . .

Main steps of the proof

.. P19. Proof:The case dX = 2 and g(C) = 2

• KS ≡ nC +∆, n ≥ 2(pg(X)− 2), S admits a pencil of genus2. Hence

vol(S) ≥ 8

3(n − 1).

• The extension theorem implies

K3X ≥ 1

4vol(S) ≥ 4

3pg(X)−

10

3.

Meng Chen (Fudan University) The Noether Inequality for Algebraic Threefolds

. . . . . .

Main steps of the proof

.. P19. Proof:The case dX = 2 and g(C) = 2

• KS ≡ nC +∆, n ≥ 2(pg(X)− 2), S admits a pencil of genus2. Hence

vol(S) ≥ 8

3(n − 1).

• The extension theorem implies

K3X ≥ 1

4vol(S) ≥ 4

3pg(X)−

10

3.

Meng Chen (Fudan University) The Noether Inequality for Algebraic Threefolds

. . . . . .

Main steps of the proof

.. P20. Proof: The case dX = 1

• f : X′ → Γ is the induced fibration. The surface theoryimplies: (1) F is a (1, 1)-surface; (2) F is a (1, 2)-surface; (3)F is of other types, i.e. K2 ≥ 2 and pg(F) > 0.

• When F is of Type (3): K2F0

≥ 2, the comparison inequalityimplies

K3X ≥ a(π∗(KX)|F)2 ≥

a3(a + 1)2

K2F0

≥ 2a3(a + 1)2

> 2a − 4 ≥ 2pg(X)− 6

where a ≥ pg(X)− 1.

Meng Chen (Fudan University) The Noether Inequality for Algebraic Threefolds

. . . . . .

Main steps of the proof

.. P20. Proof: The case dX = 1

• f : X′ → Γ is the induced fibration. The surface theoryimplies: (1) F is a (1, 1)-surface; (2) F is a (1, 2)-surface; (3)F is of other types, i.e. K2 ≥ 2 and pg(F) > 0.

• When F is of Type (3): K2F0

≥ 2, the comparison inequalityimplies

K3X ≥ a(π∗(KX)|F)2 ≥

a3(a + 1)2

K2F0

≥ 2a3(a + 1)2

> 2a − 4 ≥ 2pg(X)− 6

where a ≥ pg(X)− 1.

Meng Chen (Fudan University) The Noether Inequality for Algebraic Threefolds

. . . . . .

Main steps of the proof

.. P21. Proof: The case dX = 1 and F a (1, 1)-surface

• The main idea is to study the bicanonical restriction:

H0(2KX′)ν2−→ V2 ⊆ H0(F, 2KF).

dim V2 ≤ P2(F) = 3.

• dim V2 = 3 V φ2 is generically finite of degree 4 VK3

X ≥ 2pg(X)− 4.

• dim V2 = 2 V dimφ2(X) = 2.

4K3X ≥ (π∗(KX) · S2 · S2) = a2(π∗(KX)|S2

· C′)

≥ 2aa + 1

(P2(X)− 2) ≥ 2aa + 1

(12

K3X + 3pg(X)− 5

).

Meng Chen (Fudan University) The Noether Inequality for Algebraic Threefolds

. . . . . .

Main steps of the proof

.. P21. Proof: The case dX = 1 and F a (1, 1)-surface

• The main idea is to study the bicanonical restriction:

H0(2KX′)ν2−→ V2 ⊆ H0(F, 2KF).

dim V2 ≤ P2(F) = 3.

• dim V2 = 3 V φ2 is generically finite of degree 4 VK3

X ≥ 2pg(X)− 4.

• dim V2 = 2 V dimφ2(X) = 2.

4K3X ≥ (π∗(KX) · S2 · S2) = a2(π∗(KX)|S2

· C′)

≥ 2aa + 1

(P2(X)− 2) ≥ 2aa + 1

(12

K3X + 3pg(X)− 5

).

Meng Chen (Fudan University) The Noether Inequality for Algebraic Threefolds

. . . . . .

Main steps of the proof

.. P21. Proof: The case dX = 1 and F a (1, 1)-surface

• The main idea is to study the bicanonical restriction:

H0(2KX′)ν2−→ V2 ⊆ H0(F, 2KF).

dim V2 ≤ P2(F) = 3.

• dim V2 = 3 V φ2 is generically finite of degree 4 VK3

X ≥ 2pg(X)− 4.

• dim V2 = 2 V dimφ2(X) = 2.

4K3X ≥ (π∗(KX) · S2 · S2) = a2(π∗(KX)|S2

· C′)

≥ 2aa + 1

(P2(X)− 2) ≥ 2aa + 1

(12

K3X + 3pg(X)− 5

).

Meng Chen (Fudan University) The Noether Inequality for Algebraic Threefolds

. . . . . .

Main steps of the proof

.. P22. Proof: The case dX = 1 and F a (1, 1)-surface

Thus it follows that

K3X ≥ 6a

3a + 4pg(X)−

10a3a + 4

= 2pg(X)− 6 +8a − 8pg(X) + 24

3a + 4

> 2pg(X)− 6.

Meng Chen (Fudan University) The Noether Inequality for Algebraic Threefolds

. . . . . .

Main steps of the proof

.. P23. Proof: The case dX = 1 and F a (1, 1)-surface

• dim V2 = 1, φ2 and φ1 induce the same fibration. One has

2K3X ≥ a2(π∗(KX)|F)2 ≥

a32(a2 + 2)2

K2F0

= a2 − 4 +12a2 + 16

(a2 + 2)2

> P2(X)− 5

≥ 1

2K3

X + 3pg(X)− 8,

which givesK3

X > 2pg(X)−16

3.

Meng Chen (Fudan University) The Noether Inequality for Algebraic Threefolds

. . . . . .

Main steps of the proof

.. P24. Statement for the case: dX = 1 and F a (1, 2)-surface

.Theorem..

.. ..

.

.

Let X be a minimal projective 3-fold of general type withpg(X) ≥ 4. Assume that dX = 1 and |KX| is composed with apencil of (1, 2)-surfaces. Moreover, assume that Mov|KX| is basepoint free. Then

K3X ≥ 4

3pg(X)−

10

3.

.Corollary..

.. ..

.

.

When dX = 1, F is a (1, 2)-surface and pg(X) ≥ 21, then

K3X ≥ 4

3pg(X)−

10

3.

Meng Chen (Fudan University) The Noether Inequality for Algebraic Threefolds

. . . . . .

Main steps of the proof

.. P25. Proof: The case dX = 1 and F a (1, 2)-surface

Step 0. Overall setting.

W

f′

�����������������

π

��

γ

��@@@

@@@@

@@@@

@@@@

β // Σ

s

��à Xfoo

Φ|KX/Z|//______ Z

Meng Chen (Fudan University) The Noether Inequality for Algebraic Threefolds

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Main steps of the proof

.. P26. Proof: The case dX = 1 and F a (1, 2)-surface

Step 1. Let D = KX/Z and S ∈ Mov|xπ∗(KD)y| be a generalmember. We have S|S ≡ aC where a ≥ h0(D)− 2 ≥ pg(X) + 1 andC is a general fiber of the restricted fibration β|S : S → β(S) withpg(C) = 2.

4K3X + 16 = (π∗(KX + D)2 · π∗(D)) ≥ (π∗(KX + D)2 · S)

≥ (σ∗KS0)2 ≥ 8

3(2a − 3).

Condition: a ≥ pg(X) + 2 V

K3X ≥ 4

3pg(X)−

10

3.

Meng Chen (Fudan University) The Noether Inequality for Algebraic Threefolds

. . . . . .

Main steps of the proof

.. P27. Proof: The case dX = 1 and F a (1, 2)-surface

Step 2. a = pg(X) + 1 V Z is normal.

Sub-step 2.1. We claim that Z = P(1, 1, a).

Sub-step 2.2. Existence of a special resolution W, whichfactors through Fa.

W

π

��

f′

��@@@

@@@@

@@@@

@@g // Fa

��X f // P1

Meng Chen (Fudan University) The Noether Inequality for Algebraic Threefolds

. . . . . .

Main steps of the proof

.. P28. Proof: The case dX = 1 and F a (1, 2)-surface

Sub-step 2.2. Continued....Denote g∗(σ0) = B which is an effective Cartier divisor on W.We can write

π∗KX + 2F ∼ S + E′′ ∼ aF + B + E′′

for an effective Q-divisor E′′ and

KW = π∗KX + Eπ

for an effective π-exceptional Q-divisor Eπ.

Meng Chen (Fudan University) The Noether Inequality for Algebraic Threefolds

. . . . . .

Main steps of the proof

.. P29. Proof: The case dX = 1 and F a (1, 2)-surface

Sub-step 2.3. Two distinguished components in B and E′′.The following statements hold:

...1 there exists a unique π-exceptional prime divisor E0 on W suchthat E0 dominates Fa. Moreover, (E0 · C) = 1 andcoeffE0E′′ = coeffE0Eπ = 1, where C is a general fiber of g;

...2 there exists a unique prime divisor D0 in B such that(D0 ·E0 ·F) = 1, coeffD0B = 1, and (π∗(KX) · (B−D0) ·F) = 0.

Meng Chen (Fudan University) The Noether Inequality for Algebraic Threefolds

. . . . . .

Main steps of the proof

.. P30. Proof: The case dX = 1 and F a (1, 2)-surface

Sub-step 2.4. “Pseudo-effectivity” of (3π∗KX − (a − 6)F)|D0 .By Viehweg, there is a resolutions ψ′ : Σ′ → Fa and a

resolution W′ of W ×Fa Σ′ giving the commutative diagram:

W′

π′

��

g′ // Σ′

ψ′

��W g // Fa

such that every g′-exceptional divisor is π′-exceptional. Wemay assume that E′

0 is smooth by taking further modification,where E′

0 is the strict transform of E0 on W′.

Meng Chen (Fudan University) The Noether Inequality for Algebraic Threefolds

. . . . . .

Main steps of the proof

.. P31. Proof: The case dX = 1 and F a (1, 2)-surface

.Claim..

.. ..

.

.

For any integer m > 0, there exists an integer c > 0 and aneffective divisor

Dm ∼ cmKW/Fa + cmE0 + cg∗A

such that E0 ⊆ Supp(Dm).

.Corollary..

.. ..

.

.((3π∗KX − (a − 6)F) · D0 · π∗KX) ≥ 0.

Meng Chen (Fudan University) The Noether Inequality for Algebraic Threefolds

. . . . . .

Main steps of the proof

.. P32. Proof: The case dX = 1 and F a (1, 2)-surface

Sub-step 2.5. The main inequality for Step 2.

((3π∗KX − (a − 6)F) · B · π∗KX) ≥ 0.

Since (F ·B · π∗KX) = (F · S · π∗KX) = (C · π∗KX|F) = 1, hence

(π∗K2X · B) ≥ a − 6

3(F · B · π∗KX) ≥

a3− 2.

Finally, we have

K3X = (π∗K2

X · (π∗KX + 2F))− 2

≥ (π∗K2X · (aF + B))− 2

≥ 4a3

− 4 =4

3pg(X)−

8

3.

Meng Chen (Fudan University) The Noether Inequality for Algebraic Threefolds

. . . . . .

Main steps of the proof

.. P33. Summary and open question

Let X be a minimal 3-fold of general type. Then

K3X ≥

11680 , pg(X) = 0;

175 , pg(X) = 1;

13 , pg(X) = 2;

1, pg(X) = 3;

2, pg(X) = 4;

43pg − 14

3 , 5 ≤ pg(X) ≤ 20;

43pg − 10

3 , pg(X) ≥ 21.

Conjecture. K3X ≥ 4

3pg(X)− 103 holds.

Meng Chen (Fudan University) The Noether Inequality for Algebraic Threefolds

. . . . . .

Main steps of the proof

.. P34

.

.. ..

.

.Thank you very much!

Meng Chen (Fudan University) The Noether Inequality for Algebraic Threefolds