444ÛÛÛ ÜÜÜŒŒŒ vvv aaaÛÛÛ · flk µ|p‚œ/...
TRANSCRIPT
![Page 1: 444ÛÛÛ ÜÜÜŒŒŒ VVV AAAÛÛÛ · flK µ|p‚œ/ ppp‚‚‚VVVkkknnnAAAÛÛÛ˙˙˙åååÚÚÚ: Ueno’s book on high dimensional birational geometry (LNM 439, 1975); Reid’s](https://reader033.vdocuments.site/reader033/viewer/2022041715/5e4af2621f35e21ab52688e1/html5/thumbnails/1.jpg)
lll///ªªªzzz���ªªª444ÛÛÛ:::���|||ÜÜÜêêêÆÆÆ���nnn���VVVkkknnnAAAÛÛÛ
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¯K�µ
� VVVkkknnnAAAÛÛÛ���888III888ÜÜÜ={���KKK���êêêqqq}/AAA��� 0 ���êêê444��� k, ~~~XXXµµµk = C.
� 111www���KKK���êêê���=;;;iiiùùù¡¡¡£££Riemannsurfaces¤¤¤¶¶¶• C1
∼= C2 ⇐⇒ K (C1) ∼= K (C2); (kkknnn¼¼¼êêê���ÓÓÓ���)• ººº��� g(C ) = 0, 1, 2, · · ·• ������mmm (Moduli space) Mg .
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¯K�µ
� VVVkkknnnAAAÛÛÛ���888III888ÜÜÜ={���KKK���êêêqqq}/AAA��� 0 ���êêê444��� k, ~~~XXXµµµk = C.
� 111www���KKK���êêê���=;;;iiiùùù¡¡¡£££Riemannsurfaces¤¤¤¶¶¶• C1
∼= C2 ⇐⇒ K (C1) ∼= K (C2); (kkknnn¼¼¼êêê���ÓÓÓ���)• ººº��� g(C ) = 0, 1, 2, · · ·• ������mmm (Moduli space) Mg .
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¯K�µ–�ê¡
� 111www���êêê¡¡¡ (algebraic surfaces)—������������VVVVVVkkknnnAAAÛÛÛ���ÃÃÃÔÔÔ
� ���êêê¡¡¡���ïïïÄÄÄ;;;,,,µµµ¿¿¿���|||ÆÆÆ���!!!���²²²������!!!Mumford!!!Bombieri!!!Beauville!!!���fff!!!���888¡¡¡ÆÆÆ[[[....
� ���êêê¡¡¡���ïïïÄÄÄ���{{{µµµ• VVVkkknnn���ddd (Birational equivalence)• VVVkkknnnØØØCCCþþþ (Birational invariants)• ������mmmnnnØØØ (Moduli spaces)
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¯K�µ–�ê¡
� 111www���êêê¡¡¡ (algebraic surfaces)—������������VVVVVVkkknnnAAAÛÛÛ���ÃÃÃÔÔÔ
� ���êêê¡¡¡���ïïïÄÄÄ;;;,,,µµµ¿¿¿���|||ÆÆÆ���!!!���²²²������!!!Mumford!!!Bombieri!!!Beauville!!!���fff!!!���888¡¡¡ÆÆÆ[[[....
� ���êêê¡¡¡���ïïïÄÄÄ���{{{µµµ• VVVkkknnn���ddd (Birational equivalence)• VVVkkknnnØØØCCCþþþ (Birational invariants)• ������mmmnnnØØØ (Moduli spaces)
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¯K�µ–�ê¡
� 111www���êêê¡¡¡ (algebraic surfaces)—������������VVVVVVkkknnnAAAÛÛÛ���ÃÃÃÔÔÔ
� ���êêê¡¡¡���ïïïÄÄÄ;;;,,,µµµ¿¿¿���|||ÆÆÆ���!!!���²²²������!!!Mumford!!!Bombieri!!!Beauville!!!���fff!!!���888¡¡¡ÆÆÆ[[[....
� ���êêê¡¡¡���ïïïÄÄÄ���{{{µµµ• VVVkkknnn���ddd (Birational equivalence)• VVVkkknnnØØØCCCþþþ (Birational invariants)• ������mmmnnnØØØ (Moduli spaces)
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¯K�µ—Invariants
� lll������¡¡¡���ØØØCCCþþþüüüzzzµµµ
Curves: g–(genus)V
Surfaces
K 2 (canonical volume)
pg (geometric genus)
q (irregularity)
χ(O) (Euler characteristic)
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¯K�µ—p��/
� ppp���VVVkkknnnAAAÛÛÛ���åååÚÚÚ:• Ueno’s book on high dimensionalbirational geometry (LNM 439, 1975);
• Reid’s paper ”Canonical 3-folds” in 1979;
• ÜÜÜ©©©Mori (Annals of Math.,1982);Kawamata, Shokurov, ..., Mori (JAMS,1988)¶¶¶
• ”Minimal Model Program”—Mori’sField’s Medal in 1990.
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n�q�lÑØCþ
� nnn���MMP V nnn���444������...þþþ���kkk///ªªªàààÛÛÛ:::000(Terminal singularities)§§§ÚÚÚ¡¡¡ QFTÛÛÛ:::"""
� ªªªàààÛÛÛ:::©©©üüüaaaµµµ(I)!!!(II)"""aaa...(I)—ÌÌÌ���ûûûÛÛÛ::: C3/µr .aaa...(II)—���¡¡¡ûûûÛÛÛ:::.
� (I) ...ÛÛÛ:::µµµε ∈ µr ,
ε · (x , y , z) := (εa1x , εa2y , εa3z).
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n�q�lÑØCþ
� nnn���MMP V nnn���444������...þþþ���kkk///ªªªàààÛÛÛ:::000(Terminal singularities)§§§ÚÚÚ¡¡¡ QFTÛÛÛ:::"""
� ªªªàààÛÛÛ:::©©©üüüaaaµµµ(I)!!!(II)"""aaa...(I)—ÌÌÌ���ûûûÛÛÛ::: C3/µr .aaa...(II)—���¡¡¡ûûûÛÛÛ:::.
� (I) ...ÛÛÛ:::µµµε ∈ µr ,
ε · (x , y , z) := (εa1x , εa2y , εa3z).
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n�q�lÑØCþ
� nnn���MMP V nnn���444������...þþþ���kkk///ªªªàààÛÛÛ:::000(Terminal singularities)§§§ÚÚÚ¡¡¡ QFTÛÛÛ:::"""
� ªªªàààÛÛÛ:::©©©üüüaaaµµµ(I)!!!(II)"""aaa...(I)—ÌÌÌ���ûûûÛÛÛ::: C3/µr .aaa...(II)—���¡¡¡ûûûÛÛÛ:::.
� (I) ...ÛÛÛ:::µµµε ∈ µr ,
ε · (x , y , z) := (εa1x , εa2y , εa3z).
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n�q�lÑØCþ
� (I) ...ÛÛÛ:::V (a1, a2, a3) ≡ (1,−1, b),0 < b < r , b coprime with r .
Q =1
r(1,−1, b) := (b, r).
2× (b, r) LLL«««üüü���dddaaa...ÛÛÛ:::"""
� Reid V ØØØ777úúú%%%(II)...ÛÛÛ:::���
� ïïïÄÄÄ������
V := {Projective 3-folds with QFT singularities}.
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n�q�lÑØCþ
� (I) ...ÛÛÛ:::V (a1, a2, a3) ≡ (1,−1, b),0 < b < r , b coprime with r .
Q =1
r(1,−1, b) := (b, r).
2× (b, r) LLL«««üüü���dddaaa...ÛÛÛ:::"""
� Reid V ØØØ777úúú%%%(II)...ÛÛÛ:::���
� ïïïÄÄÄ������
V := {Projective 3-folds with QFT singularities}.
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n�q�lÑØCþ
� (I) ...ÛÛÛ:::V (a1, a2, a3) ≡ (1,−1, b),0 < b < r , b coprime with r .
Q =1
r(1,−1, b) := (b, r).
2× (b, r) LLL«««üüü���dddaaa...ÛÛÛ:::"""
� Reid V ØØØ777úúú%%%(II)...ÛÛÛ:::���
� ïïïÄÄÄ������
V := {Projective 3-folds with QFT singularities}.
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n�q�lÑØCþ
� Kodaira’s classification
V = V−∞ ∪ V0 ∪ V1 ∪ V2 ∪ V3.
� V−∞ ���¹¹¹ Fano 3-folds¶¶¶
� V0 ���))) Calabi-Yau 3-folds!!!Abelian3-folds ������¶¶¶
� V3 ���)))������...nnn���qqq"""
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n�q�lÑØCþ
� Kodaira’s classification
V = V−∞ ∪ V0 ∪ V1 ∪ V2 ∪ V3.
� V−∞ ���¹¹¹ Fano 3-folds¶¶¶
� V0 ���))) Calabi-Yau 3-folds!!!Abelian3-folds ������¶¶¶
� V3 ���)))������...nnn���qqq"""
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n�q�lÑØCþ
� Kodaira’s classification
V = V−∞ ∪ V0 ∪ V1 ∪ V2 ∪ V3.
� V−∞ ���¹¹¹ Fano 3-folds¶¶¶
� V0 ���))) Calabi-Yau 3-folds!!!Abelian3-folds ������¶¶¶
� V3 ���)))������...nnn���qqq"""
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n�q�lÑØCþ
� Kodaira’s classification
V = V−∞ ∪ V0 ∪ V1 ∪ V2 ∪ V3.
� V−∞ ���¹¹¹ Fano 3-folds¶¶¶
� V0 ���))) Calabi-Yau 3-folds!!!Abelian3-folds ������¶¶¶
� V3 ���)))������...nnn���qqq"""
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n�q�lÑØCþ–Riemann-Roch
� Reid’s Riemann-Roch Formula:∀X ∈ V, ∃! basket of QFT datum
B[X ] = {1
ri(1,−1, bi)|i = 1, ..., s}
such that
χ(OX (mKX )) =1
12m(m− 1)(2m− 1)K 3
X − (2m− 1)χ(OX ) + l(m),
l(m) =∑s
i=1
∑m−1j=0
¯jbi (ri− ¯jbi )2ri
.
� If X is smooth, B[X ] = ∅.
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n�q�lÑØCþ–Riemann-Roch
� Reid’s Riemann-Roch Formula:∀X ∈ V, ∃! basket of QFT datum
B[X ] = {1
ri(1,−1, bi)|i = 1, ..., s}
such that
χ(OX (mKX )) =1
12m(m− 1)(2m− 1)K 3
X − (2m− 1)χ(OX ) + l(m),
l(m) =∑s
i=1
∑m−1j=0
¯jbi (ri− ¯jbi )2ri
.
� If X is smooth, B[X ] = ∅.
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n�q�lÑØCþ
� B[X ]ØØØ´VVVkkknnnØØØCCCþþþ���eee X ���444���§§§KKKPPPReid ���ÛÛÛ:::;;;��� BX .
� ∀ X ∈ V and X minimal,(1) if X is of general type, BX ���VVVkkknnnØØØCCCþþþ£££ÛÛÛ:::ØØØCCCþþþ¤¤¤"""(2) if X is Q-Fano, BX ØØØ´VVVkkknnnØØØCCCþþþ"""
� ¯KKKµµµ===êêêâââ���±±±������///(((½½½VVVkkknnnØØØCCCþþþººº
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n�q�lÑØCþ
� B[X ]ØØØ´VVVkkknnnØØØCCCþþþ���eee X ���444���§§§KKKPPPReid ���ÛÛÛ:::;;;��� BX .
� ∀ X ∈ V and X minimal,(1) if X is of general type, BX ���VVVkkknnnØØØCCCþþþ£££ÛÛÛ:::ØØØCCCþþþ¤¤¤"""(2) if X is Q-Fano, BX ØØØ´VVVkkknnnØØØCCCþþþ"""
� ¯KKKµµµ===êêêâââ���±±±������///(((½½½VVVkkknnnØØØCCCþþþººº
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n�q�lÑØCþ
� B[X ]ØØØ´VVVkkknnnØØØCCCþþþ���eee X ���444���§§§KKKPPPReid ���ÛÛÛ:::;;;��� BX .
� ∀ X ∈ V and X minimal,(1) if X is of general type, BX ���VVVkkknnnØØØCCCþþþ£££ÛÛÛ:::ØØØCCCþþþ¤¤¤"""(2) if X is Q-Fano, BX ØØØ´VVVkkknnnØØØCCCþþþ"""
� ¯KKKµµµ===êêêâââ���±±±������///(((½½½VVVkkknnnØØØCCCþþþººº
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n�q�lÑØCþ
� ∀X ∈ V, X kkkeee���ØØØCCCþþþµµµ• pg(X ) := h3(OX );• q1 := h1(OX );• q2 := h2(OX );• χ(OX );• vol(X );• Pm(X ) := h0(X ,mKX ), m ≥ 2;• BX — X ���444������...þþþ���ÛÛÛ:::;;;"""(General type case)
� ===ØØØCCCþþþ���±±±(((½½½ÙÙÙ¦¦¦ØØØCCCþþþººº
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n�q�lÑØCþ
� ∀X ∈ V, X kkkeee���ØØØCCCþþþµµµ• pg(X ) := h3(OX );• q1 := h1(OX );• q2 := h2(OX );• χ(OX );• vol(X );• Pm(X ) := h0(X ,mKX ), m ≥ 2;• BX — X ���444������...þþþ���ÛÛÛ:::;;;"""(General type case)
� ===ØØØCCCþþþ���±±±(((½½½ÙÙÙ¦¦¦ØØØCCCþþþººº
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Û:;�8Ü
� ���888ÜÜÜ S0 := {ÛÛÛ:::;;;}.
� ½½½ÂÂÂ\\\���ÛÛÛ:::;;; B := {B , χ2, χ}, B ∈ S0,χ2, χ ∈ Z.
� ½½½ÂÂÂ888ÜÜÜ
S := {B|B ���\\\���ÛÛÛ:::;;;}.
� Reid V ���333NNN���µµµ
θ : V −→ S0.
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Û:;�8Ü
� ���888ÜÜÜ S0 := {ÛÛÛ:::;;;}.� ½½½ÂÂÂ\\\���ÛÛÛ:::;;; B := {B , χ2, χ}, B ∈ S0,χ2, χ ∈ Z.
� ½½½ÂÂÂ888ÜÜÜ
S := {B|B ���\\\���ÛÛÛ:::;;;}.
� Reid V ���333NNN���µµµ
θ : V −→ S0.
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Û:;�8Ü
� ���888ÜÜÜ S0 := {ÛÛÛ:::;;;}.� ½½½ÂÂÂ\\\���ÛÛÛ:::;;; B := {B , χ2, χ}, B ∈ S0,χ2, χ ∈ Z.
� ½½½ÂÂÂ888ÜÜÜ
S := {B|B ���\\\���ÛÛÛ:::;;;}.
� Reid V ���333NNN���µµµ
θ : V −→ S0.
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Û:;�8Ü
� ���888ÜÜÜ S0 := {ÛÛÛ:::;;;}.� ½½½ÂÂÂ\\\���ÛÛÛ:::;;; B := {B , χ2, χ}, B ∈ S0,χ2, χ ∈ Z.
� ½½½ÂÂÂ888ÜÜÜ
S := {B|B ���\\\���ÛÛÛ:::;;;}.
� Reid V ���333NNN���µµµ
θ : V −→ S0.
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S þ�|ÜêÆ
� ∀ B ∈ S0, PPP
B = {(bi , ri )|0 < bi < ri , bi coprime to ri , i = 1, · · · , s}.
� Set σ(B) :=∑
i bi and σ′(B) :=∑
ib2i
ri.
� Let n > 1 be an integer. For each i , setli := xnbiri y and define
∆ni := libin −
1
2(l2i + li)ri , (���KKK���êêê)
Define ∆n(B) =∑t
i=1 ∆ni .
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S þ�|ÜêÆ
� ∀ B ∈ S0, PPP
B = {(bi , ri )|0 < bi < ri , bi coprime to ri , i = 1, · · · , s}.
� Set σ(B) :=∑
i bi and σ′(B) :=∑
ib2i
ri.
� Let n > 1 be an integer. For each i , setli := xnbiri y and define
∆ni := libin −
1
2(l2i + li)ri , (���KKK���êêê)
Define ∆n(B) =∑t
i=1 ∆ni .
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S þ�|ÜêÆ
� ∀ B ∈ S0, PPP
B = {(bi , ri )|0 < bi < ri , bi coprime to ri , i = 1, · · · , s}.
� Set σ(B) :=∑
i bi and σ′(B) :=∑
ib2i
ri.
� Let n > 1 be an integer. For each i , setli := xnbiri y and define
∆ni := libin −
1
2(l2i + li)ri , (���KKK���êêê)
Define ∆n(B) =∑t
i=1 ∆ni .
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S þ�|ÜêÆ
� ???���\\\���ÛÛÛ:::;;; B = {B , χ2, χ} ∈ S.
� vol(B) := −σ + σ′ + 6χ + 2χ2.
� χ2(B) = χ2.
� χ3(B) := −σ(B) + 10χ + 5χ2.
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S þ�|ÜêÆ
� ???���\\\���ÛÛÛ:::;;; B = {B , χ2, χ} ∈ S.
� vol(B) := −σ + σ′ + 6χ + 2χ2.
� χ2(B) = χ2.
� χ3(B) := −σ(B) + 10χ + 5χ2.
������§§§EEE������ÆÆÆ nnn���VVVkkknnnAAAÛÛÛ
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S þ�|ÜêÆ
� ???���\\\���ÛÛÛ:::;;; B = {B , χ2, χ} ∈ S.
� vol(B) := −σ + σ′ + 6χ + 2χ2.
� χ2(B) = χ2.
� χ3(B) := −σ(B) + 10χ + 5χ2.
������§§§EEE������ÆÆÆ nnn���VVVkkknnnAAAÛÛÛ
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S þ�|ÜêÆ
� ???���\\\���ÛÛÛ:::;;; B = {B , χ2, χ} ∈ S.
� vol(B) := −σ + σ′ + 6χ + 2χ2.
� χ2(B) = χ2.
� χ3(B) := −σ(B) + 10χ + 5χ2.
������§§§EEE������ÆÆÆ nnn���VVVkkknnnAAAÛÛÛ
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S þ�|ÜêÆ
� ééém ≥ 4, χm(B) 888BBB½½½ÂÂÂXXXeeeµµµ
χm+1(B)−χm(B) :=m2
2(vol(B)−σ′(B))− 2χ+
m
2σ(B) + ∆m(B).
� ��� B = BX (===AAAÛÛÛÛÛÛ:::;;;¤¤¤, χ = χ(OX ),χ2 = χ(OX (2KX )) ééé X ∈ V, KKK B ���¤¤¤kkkØØØCCCþþþ��� X ���ØØØCCCþþþ������������"""(Riemann-Roch Formula)
������§§§EEE������ÆÆÆ nnn���VVVkkknnnAAAÛÛÛ
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S þ�|ÜêÆ
� ééém ≥ 4, χm(B) 888BBB½½½ÂÂÂXXXeeeµµµ
χm+1(B)−χm(B) :=m2
2(vol(B)−σ′(B))− 2χ+
m
2σ(B) + ∆m(B).
� ��� B = BX (===AAAÛÛÛÛÛÛ:::;;;¤¤¤, χ = χ(OX ),χ2 = χ(OX (2KX )) ééé X ∈ V, KKK B ���¤¤¤kkkØØØCCCþþþ��� X ���ØØØCCCþþþ������������"""(Riemann-Roch Formula)
������§§§EEE������ÆÆÆ nnn���VVVkkknnnAAAÛÛÛ
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S þ�|ÜêÆ—@Ø S
� ���
B = {(b1, r1), (b2, r2), · · · , (bk , rk)},
¡¡¡
B ′ := {(b1 + b2, r1 + r2), (b3, r3), · · · , (bk , rk)}
´ B ���///@@@ØØØ000, PPP��� B � B ′.
� “�”½½½Â S0 þþþ��� SSS'''XXX"""
� XXXJJJ b1r2 − b2r1 = 1, ···���¡¡¡ B � B ′ ´������ÄÄÄ���@@@ØØØ£££a prime packing¤¤¤.
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S þ�|ÜêÆ—@Ø S
� ���
B = {(b1, r1), (b2, r2), · · · , (bk , rk)},
¡¡¡
B ′ := {(b1 + b2, r1 + r2), (b3, r3), · · · , (bk , rk)}
´ B ���///@@@ØØØ000, PPP��� B � B ′.
� “�”½½½Â S0 þþþ��� SSS'''XXX"""
� XXXJJJ b1r2 − b2r1 = 1, ···���¡¡¡ B � B ′ ´������ÄÄÄ���@@@ØØØ£££a prime packing¤¤¤.
������§§§EEE������ÆÆÆ nnn���VVVkkknnnAAAÛÛÛ
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S þ�|ÜêÆ—@Ø S
� ���
B = {(b1, r1), (b2, r2), · · · , (bk , rk)},
¡¡¡
B ′ := {(b1 + b2, r1 + r2), (b3, r3), · · · , (bk , rk)}
´ B ���///@@@ØØØ000, PPP��� B � B ′.
� “�”½½½Â S0 þþþ��� SSS'''XXX"""
� XXXJJJ b1r2 − b2r1 = 1, ···���¡¡¡ B � B ′ ´������ÄÄÄ���@@@ØØØ£££a prime packing¤¤¤.
������§§§EEE������ÆÆÆ nnn���VVVkkknnnAAAÛÛÛ
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S þ�|ÜêÆ
� “�” ���±±±ggg,,,///½½½ÂÂÂ���888ÜÜÜ Sþþþ"""===
{B , χ2, χ} � {B ′, χ2, χ},
XXXJJJ B � B ′.
� ···���III���ïïïÄÄÄ ����ØØØCCCþþþ������NNN'''XXX"""
������§§§EEE������ÆÆÆ nnn���VVVkkknnnAAAÛÛÛ
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S þ�|ÜêÆ
� “�” ���±±±ggg,,,///½½½ÂÂÂ���888ÜÜÜ Sþþþ"""===
{B , χ2, χ} � {B ′, χ2, χ},
XXXJJJ B � B ′.
� ···���III���ïïïÄÄÄ ����ØØØCCCþþþ������NNN'''XXX"""
������§§§EEE������ÆÆÆ nnn���VVVkkknnnAAAÛÛÛ
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S þ�|ÜêÆ
� @@@ØØØ SSS���555���µµµ
Lemma
Assume B = (B , χ2, χ) � (B ′, χ2, χ) = B′. Then:1. σ(B) = σ(B ′), σ′(B) ≥ σ′(B ′);
2. For all n > 1, ∆n(B) ≥ ∆n(B ′);
3. vol(B) ≥ vol(B′);
4. χm(B) ≥ χm(B′) for all m ≥ 2.
� eee������888IIIµµµXXXÛÛÛAAA^ùùù���555���ººº
������§§§EEE������ÆÆÆ nnn���VVVkkknnnAAAÛÛÛ
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S þ�|ÜêÆ
� @@@ØØØ SSS���555���µµµ
Lemma
Assume B = (B , χ2, χ) � (B ′, χ2, χ) = B′. Then:1. σ(B) = σ(B ′), σ′(B) ≥ σ′(B ′);
2. For all n > 1, ∆n(B) ≥ ∆n(B ′);
3. vol(B) ≥ vol(B′);
4. χm(B) ≥ χm(B′) for all m ≥ 2.
� eee������888IIIµµµXXXÛÛÛAAA^ùùù���555���ººº
������§§§EEE������ÆÆÆ nnn���VVVkkknnnAAAÛÛÛ
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S þ�|ÜêÆ–;�S�
Set S0 := {1n |n ≥ 2} and
S5 := S0 ∪ {2
5};
S6 := S5;
Sn := Sn−1 ∪ {bn| 0 < b <
n
2, b coprime to n}
for all n ≥ 5. Each set Sn gives a division of theinterval (0, 1
2]. Write (0, 12] :=
⋃i
[ω(n)i+1, ω
(n)i ]. Let
ω(n)i+1 = qi+1
pi+1and ω
(n)i = qi
piwith qi+1 coprime to pi+1
and qi coprime to pi .
• qipi+1 − piqi+1 = 1
for all n and i ������§§§EEE������ÆÆÆ nnn���VVVkkknnnAAAÛÛÛ
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S þ�|ÜêÆ–�fB(·)
Let B = {(bi , ri)|r = 1, · · · , t}. For eachBi = (bi , ri) ∈ B, if bi
ri∈ S (n), then we set
B(n)i := {(bi , ri)}. If bi
ri6∈ S (n), then ql+1
pl+1< bi
ri< ql
pl
with qsps
= ω(n)s for some n and s = l , l + 1. In
this situation, we can unpack (bi , ri) to
B(n)i := {(riql − bipl)× (ql+1, pl+1), (−riql+1 +
bipl+1)× (ql , pl)}.
B(n)(B) :=∑i
B(n)i .
Then B(n)(B) � B for all n.
������§§§EEE������ÆÆÆ nnn���VVVkkknnnAAAÛÛÛ
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S þ�|ÜêÆ–;�S�
• B(n−1)(B) = B(n−1)(B(n)(B)) � B(n)(B)
for all n ≥ 1.
B(0)(B) = · · · = B(4)(B) � B(5)(B) � ... � B(n)(B) � ... � B.
The step B(n−1)(B) � B(n)(B) can beachieved by εn = ∆n(B(n−1))−∆n(B) successiveprime packings. Note that εn ≥ 0.
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S þ�|ÜêÆ–;�S�
� Set Bn := B(n)(B). ···���������;;;���SSS���
B0 = B4 � B5 � · · ·Bn � · · · � B .
� Set Bn := {Bn, χ2, χ}. ···���������\\\���ÛÛÛ:::;;;���;;;���SSS���µµµ
B0 = B4 � B5 � · · ·Bn � · · · � B.
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S þ�|ÜêÆ–;�S�
� Set Bn := B(n)(B). ···���������;;;���SSS���
B0 = B4 � B5 � · · ·Bn � · · · � B .
� Set Bn := {Bn, χ2, χ}. ···���������\\\���ÛÛÛ:::;;;���;;;���SSS���µµµ
B0 = B4 � B5 � · · ·Bn � · · · � B.
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S þ�|ÜêÆ–;�S�
� Set Bn := B(n)(B). ···���������;;;���SSS���
B0 = B4 � B5 � · · ·Bn � · · · � B .
� Set Bn := {Bn, χ2, χ}. ···���������\\\���ÛÛÛ:::;;;���;;;���SSS���µµµ
B0 = B4 � B5 � · · ·Bn � · · · � B.
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S þ�|ÜêÆ–;�S�
� Let B (0) = {n01,2 × (1, 2), · · · , n0
1,r × (1, r)}.
σ(B) = σ(B(0)) =∑
n01,r ,
∆3(B) = ∆3(B(0)) = n01,2
∆4(B) = ∆4(B(0)) = 2n01,2 + n0
1,3
B(0)
n0
1,2 = 5χ+ 6χ2 − 4χ3 + χ4
n01,3 = 4χ+ 2χ2 + 2χ3 − 3χ4 + χ5
n01,4 = χ− 3χ2 + χ3 + 2χ4 − χ5 −
∑r≥5 n
01,r
n01,r , r ≥ 5.
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S þ�|ÜêÆ–;�S�
� Let B (0) = {n01,2 × (1, 2), · · · , n0
1,r × (1, r)}.
σ(B) = σ(B(0)) =∑
n01,r ,
∆3(B) = ∆3(B(0)) = n01,2
∆4(B) = ∆4(B(0)) = 2n01,2 + n0
1,3
B(0)
n0
1,2 = 5χ+ 6χ2 − 4χ3 + χ4
n01,3 = 4χ+ 2χ2 + 2χ3 − 3χ4 + χ5
n01,4 = χ− 3χ2 + χ3 + 2χ4 − χ5 −
∑r≥5 n
01,r
n01,r , r ≥ 5.
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S þ�|ÜêÆ–;�S�
ε5 := ∆5(B(0))−∆5(B) = 4n01,2 + 2n0
1,3 + n01,4 −∆5(B)
= 2χ− χ3 + 2χ5 − χ6 − σ5
σ5 :=∑r≥5
n01,r .
B(5) = {n51,2×(1, 2), n5
2,5×(2, 5), n51,3×(1, 3), n5
1,4×(1, 4), n51,5×(1, 5), · · · }
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S þ�|ÜêÆ–;�S�
ε5 := ∆5(B(0))−∆5(B) = 4n01,2 + 2n0
1,3 + n01,4 −∆5(B)
= 2χ− χ3 + 2χ5 − χ6 − σ5
σ5 :=∑r≥5
n01,r .
B(5) = {n51,2×(1, 2), n5
2,5×(2, 5), n51,3×(1, 3), n5
1,4×(1, 4), n51,5×(1, 5), · · · }
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S þ�|ÜêÆ–;�S�
ε5 := ∆5(B(0))−∆5(B) = 4n01,2 + 2n0
1,3 + n01,4 −∆5(B)
= 2χ− χ3 + 2χ5 − χ6 − σ5
σ5 :=∑r≥5
n01,r .
B(5) = {n51,2×(1, 2), n5
2,5×(2, 5), n51,3×(1, 3), n5
1,4×(1, 4), n51,5×(1, 5), · · · }
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S þ�|ÜêÆ–Bn�L«
B(5)
n5
1,2 = 3χ+ 6χ2 − 3χ3 + χ4 − 2χ5 + χ6 + σ5,n5
2,5 = 2χ− χ3 + 2χ5 − χ6 − σ5
n51,3 = 2χ+ 2χ2 + 3χ3 − 3χ4 − χ5 + χ6 + σ5,
n51,4 = χ− 3χ2 + χ3 + 2χ4 − χ5 − σ5
n51,r = n0
1,r , r ≥ 5
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S þ�|ÜêÆ–Bn�L«
B(7)
n71,2 = 2χ+ 7χ2 − 2χ3 + χ4 − 2χ5 − χ7 + χ8 + 3σ5 − 2n0
1,5 − n01,6 + η
n73,7 = χ− χ2 − χ3 + χ6 + χ7 − χ8 − 2σ5 + 2n0
1,5 + n01,6 − η
n72,5 = χ+ χ2 + 2χ5 − 2χ6 − χ7 + χ8 + σ5 − 2n0
1,5 − n01,6 + η
n71,3 = 2χ+ 2χ2 + 3χ3 − 3χ4 − χ5 + χ6 + σ5 − η
n72,7 = η
n71,4 = χ− 3χ2 + χ3 + 2χ4 − χ5 − σ5 − η
n71,r = n0
1,r , r ≥ 5
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S þ�|ÜêÆ–Bn�L«
ε8 = −2χ2 − χ3 − χ4 + χ5 + χ6 + χ8 − χ9 − 3σ5
+3n01,5 + 2n0
1,6 + n01,7;
ε9 = −2χ2 − 2χ3 + χ4 + χ5 − χ7 + χ8 + χ9 − χ10 − 3σ5 + η+2n0
1,5 + 2n01,6 + 2n0
1,7 + n01,8;
ε10 = −5χ2 − χ3 + 2χ6 + χ10 − χ11 − 6σ5 − η+5n0
1,5 + 4n01,6 + 3n0
1,7 + 2n01,8 + n0
1,9;ε12 = −χ− 5χ2 − 3χ3 + 2χ5 + χ6 − χ7 + χ8 + χ12 − χ13 − 8σ5 + η
+7n01,5 + 5n0
1,6 + 5n01,7 + 4n0
1,8 + 3n01,9 + 2n0
1,10 + n01,11.
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S þ�|ÜêÆ–Bn�L«
B(12)
n121,2 = 2χ− 2χ3 + χ4 − 2χ5 − χ7 + χ8 + n0
1,5 + η − ζ − αn12
5,11 = αn12
4,9 = ζ − αn12
3,7 = 2χ+ 2χ3 − 2χ5 + 2χ7 − 2χ8 − χ12 + χ13 − 2η − ζ + n01,5
n125,12 = −χ− 3χ3 + 2χ5 + χ6 − χ7 + χ8 + χ12 − χ13 + η − n0
1,5
n122,5 = 2χ+ 4χ3 + χ4 − χ5 − 4χ6 − χ8 + χ9 − χ12 + χ13
n123,8 = −χ3 − χ4 + χ5 + χ6 + χ8 − χ9 − β
n124,11 = β
n121,3 = 2χ+ 5χ3 − 2χ4 − 2χ5 − 2χ6 − χ8 + χ9 − χ10 + χ11 + 2n0
1,5 − βn12
3,10 = −χ3 + 2χ6 + χ10 − χ11 − n01,5 − η
n122,7 = −χ+ 2χ3 − χ4 − 2χ6 + χ7 − χ9 − χ10 + χ12 + 2n0
1,5 + 2η + ζ + α+ βn12
3,11 = χ− χ3 + χ4 − χ7 + χ9 + χ11 − χ12 − n01,5 − ζ − α− β
n121,4 = 4χ3 − 2χ5 + 2χ7 − χ8 − 2χ9 + χ10 − χ11 + χ12 + n0
1,5 − 2η + 2ζ + α+ βn12
2,9 = −2χ3 + χ4 + χ5 − χ7 + χ8 + χ9 − χ10 − n01,5 + η − ζ
n121,5 = 2χ3 − χ4 − χ5 + χ7 − χ8 − χ9 + χ10 + 2n0
1,5 − η + ζ
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'�Ø�ªµεn ≥ 0
� ε10 + ε12 ≥ 0 V (***):
2χ5 + 3χ6 +χ8 +χ10 +χ12 ≥ χ+ 10χ2 + 4χ3 +χ7 +χ11 +χ13.
� If χm ≤ 1 for all m ≤ 12, Inequality (***)implies χ ≤ 8.
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'�Ø�ªµεn ≥ 0
� ε10 + ε12 ≥ 0 V (***):
2χ5 + 3χ6 +χ8 +χ10 +χ12 ≥ χ+ 10χ2 + 4χ3 +χ7 +χ11 +χ13.
� If χm ≤ 1 for all m ≤ 12, Inequality (***)implies χ ≤ 8.
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� Question: ∀X ∈ V3, X minimal, when isΦm := Φ|mKX | birational?
� Known results:• If BX = ∅, then Φm is birational for m ≥ 5.Wilson(m ≥ 25, 1980); Benveniste (m ≥ 8,1984¤¤¤¶¶¶Matsuki (m = 7, 1986); M.Chen(m = 6, 1998); Chen-Chen-Zhang(m = 5, 2007).• If Pk ≥ 2, then Φm is birational form ≥ 11k + 5. Kollar (1986)
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� Question: ∀X ∈ V3, X minimal, when isΦm := Φ|mKX | birational?
� Known results:• If BX = ∅, then Φm is birational for m ≥ 5.Wilson(m ≥ 25, 1980); Benveniste (m ≥ 8,1984¤¤¤¶¶¶Matsuki (m = 7, 1986); M.Chen(m = 6, 1998); Chen-Chen-Zhang(m = 5, 2007).• If Pk ≥ 2, then Φm is birational form ≥ 11k + 5. Kollar (1986)
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� Chen-Chen (2007–2011)• If Pk ≤ 1 for 2 ≤ k ≤ 12 and BX 6= ∅,
(∗ ∗ ∗)V χ ≤ 8, P13 ≤ 6;
V B has finite possibilities.
• Φm is birational for m ≥ 73.• Chen-Chen (2015): Φm is birational form ≥ 61 and vol(X ) ≥ 1
1680.
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� Chen-Chen (2008): Q-Fano nnn���qqq���KKK;;;���NNNÈÈÈ������ZZZeee...“1/330”.
� Chen-Chen-Chen (2011): Iano-FletcherXXX���ßßß������yyy²²²—���¤¤¤���ªªªàààûûûÛÛÛ:::���\\\������������nnn���qqq���������©©©aaa"""
� Chen-Jiang £££2015¤¤¤µµµQ-Fano nnn���qqq������;;;���AAAÛÛÛ—Φ−m is birational for m ≥ 39.
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� Chen-Chen (2008): Q-Fano nnn���qqq���KKK;;;���NNNÈÈÈ������ZZZeee...“1/330”.
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� Chen-Jiang £££2015¤¤¤µµµQ-Fano nnn���qqq������;;;���AAAÛÛÛ—Φ−m is birational for m ≥ 39.
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� Chen-Chen-Chen (2011): Iano-FletcherXXX���ßßß������yyy²²²—���¤¤¤���ªªªàààûûûÛÛÛ:::���\\\������������nnn���qqq���������©©©aaa"""
� Chen-Jiang £££2015¤¤¤µµµQ-Fano nnn���qqq������;;;���AAAÛÛÛ—Φ−m is birational for m ≥ 39.
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úm¯K
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úm¯K
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J. A. Chen, M. Chen. An OptimalBoundedness on Weak Q-Fano 3-Folds,Adv. Math. (219) 2008, 2086–2104
J. A. Chen, M. Chen. Explicit BirationalGeometry of Threefolds of General Type, I,Ann. Sci. Ec Norm. Sup. (43) 2010,365–394
J. A. Chen, M. Chen. Explicit BirationalGeometry of Threefolds of General Type,II, J. of Differential Geometry 86 (2010),237–271
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J. J. Chen, J. A. Chen, M. Chen.Quasi-smooth Weighted CompleteIntersections, J. of Algebraic Geometry 20(2011), no. 2, 239–262J. A. Chen, M. Chen. Explicit birationalgeometry of 3-folds and 4-folds of generaltype (Part III), Compositio Mathematica,Doi: 10.1112/S0010437X14007817 (42pages). ArXiv: 1302.0374M. Chen, C. Jiang, On the anti-canonicalgeometry of Q-Fano 3-folds, (to appear)Journal of Differential Geometry (45pages). ArXiv: 1408.6349
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