Post on 22-Jul-2020
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The Noether Inequality for Algebraic Threefolds
Meng Chen
School of Mathematical Sciences, Fudan University
2018.09.08
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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..
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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..
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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..
.. ..
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.
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..
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.
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
. . . . . .
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
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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