Title Ascending chain condition for F-pure thresholds

Author(s) 佐藤, 謙太

Citation 代数幾何学シンポジウム記録 (2017), 2017: 69-79

Issue Date 2017

URL http://hdl.handle.net/2433/229092

Right

Type Departmental Bulletin Paper

Textversion publisher

Kyoto University

ASCENDING CHAIN CONDITION FOR F -PURE THRESHOLDS

. 2017

F (F -purethreshold) ( 1.9). F (ACC)

1 20 lct ACC

1 3 Ultraproduct2 4 1 5

1.

F F log canonical, klt

1.1. .k , (X,∆) k X k ∆

R- KX +∆ R-Cartier f : Y −→ XY Y R- ∆Y

∆Y := f∗(KX +∆)−KY

KY Y f KX

1.1. (X,∆) kawamata log terminal ( klt) (resp. log canonical lc))Y f : Y −→ X ∆Y 1 (resp.

1 )

klt lc (X,∆) log resolution g : Z −→ X(X,∆) klt (resp. lc) ∆Z 1 (resp. )

( [KM, Corollary 2.31 (3)]) X (X, 0) kltk 0 klt lc

klt (X,∆)klt ∆

1.2. (X,∆) lc D ̸= 0 X R-Cartier D (X,∆)log canonical threshold lct(X,∆;D)

lct(X,∆;D) := sup{t ∈ R!0 | (X,∆+ tD) lc} ∈ R!0log canonical threshold

1

69

2

1.3. (X,∆) k 0 ̸= a ⊆ R , t ! 0(X,∆, at) klt (resp. lc) f : Y −→ X a · OY = OY (−F )

Y Cartier F Y R ∆Y + tF1 (resp. )

klt lc log resolution a = 0(X,∆, at) klt lc

1.4. (X,∆) k lc a ! R 0 ̸= a ⊆ OX (X,∆)log canonical threshold

lct(X,∆; a) := sup{t ∈ R!0 | (X,∆, at) lc} ∈ R!0.

(X,∆) lc a = 0 lct(X,∆; a) 0 ∆ = 0lct(X, 0; a) lct(X; a)

1.5. (X,∆)

(1) D 0 Cartier lct(X,∆;OX(−D)) = lct(X,∆;D)(2) (X,∆) klt lct(X,∆; a) = sup{t ! 0 | (X,∆, at) klt}.

1.2. .lc, klt, lct

F F fpt

R p > 0 F : R −→ R (r ∈ R F (r) := rp)R- M e ! 0 M F e R F e∗MZ- M = F e∗M m ∈ M F e∗m ∈ F e∗M

R F - FR p > 0 F -

1.6. F - R F -

(1) R(2) R F -

1.7 ([HW02], [Sch08]). ∆ SpecR R- 0 ̸= a ⊆ R, t ∈ R!0(1) (R,∆, at) F e ! 1 , 0 a ∈ a⌈t(pe−1)⌉ R

R −→ F e∗ (R(⌈(pe − 1)∆⌉))r (−→ F e∗ (arp

e

)

R-(2) (R,∆) F c ̸= 0 ∈ R e ! 1 0

a ∈ a⌈t(pe−1)⌉ RR −→ F e∗ (R(⌈(pe − 1)∆⌉))r (−→ F e∗ (carp

e

)

R-

a = R t = 0 (R,∆, at) (R,∆) ∆ = 0 RF R F a = 0 F F

F F R R F

klt lc

70

3

1.8 ([HW02, Theorem 3.3], [Har98]). (X = SpecR,∆) k

(1) (X,∆) F lc(2) (X,∆) F klt(3) dimX = 2, ∆ = 0, p > 5 F klt

klt F mod p([Tak04]) lc F mod p

0 kltF ( ) ([ST14]).

1.9. (R,∆) F 0 ̸= a ! R a (R,∆) F

fpt(R,∆; a) := sup{t ! 0 | (R,∆, at) F }

(R,∆) F a = 0 fpt 0 (R,∆) F ,fpt(R,∆; a) := sup{t ! 0 | (R,∆, at) F } ( [Sch08])

1.10. (R,∆) 1.7 a, b ⊆ R 0 s, t ! 0 (R,∆, atbs)F e > 0, a ∈ a⌈t(pe−1)⌉ b ∈ b⌈s(pe−1)⌉

R −→ F e∗ (R(⌈(pe − 1)∆⌉))r (−→ F e∗ (abrp

e

)

R-

F

1.3. . F

- 1.11 ([Sch10, Theorem 6.3]). (R,∆, atbs) 1.10 R J ⊆ R(R,∆, atbs) F -compatible e ! 0 ϕ ∈ HomR(F e∗ (R(⌈(pe−1)∆⌉)), R) ϕ(F e∗ (Ja⌈t(p

e−1)⌉b⌈s(pe−1)⌉)) ⊆ J

0 F -compatible(R,∆, atbs) (test ideal) τ(R,∆, atbs)

a = 0 b = 0 0 b = R, s = 0 τ(R,∆, atbs)τ(R,∆, at)

1.12. (R,∆, at) F τ(R,∆, at) = R (R,∆) Ffpt(R,∆; a) = sup{t ! 0 | τ(R,∆, at) = R}

R F R KR dualizing complex( [ST17, 2.2 ]) R Q- ∆ KR +∆ (R,∆) log Q-Gorenstein

( dualizing complex )R (R, 0) log Q-Gorenstein

1.13 ([ST14]). (R,∆, at) 1.7 (R,∆) log Q-Gorenstein

(1) s > t τ(R,∆, as) ⊆ τ(R,∆, at).(2) ε > 0 t < s < t+ ε τ(R,∆, as) = τ(R,∆, at).(3) s < t τ(R,∆, as) " τ(R,∆, at) t ∈ Q

1.14. (3) t (R,∆; a) F - (F -jumping number)1.12 (R,∆) F log Q-Gorenstein fpt(R,∆; a) (R,∆; a) F

71

4

R k k[[x1, . . . , xd]] ∆ = 0

F∗R ∼= ⊗λR · F∗(xλ),

λ = (λ1, . . . ,λd) 0 " λi " p − 1 λ ∈ Nd xλ := xλ11 . . . xλdd ∈ RR Tr : F∗R −→ R F∗(xp−11 . . . x

p−1d )

1.15. R = k[[x1, . . . , xd]], a, b ⊆ R 0 s, t ∈ R!0(1) ([HT04]) a l t ! l τ(R, atbs) = aτ(R, at−1bs).(2) ([ST14, Lemma 4.4 (b)])Tr(F∗(τ(R, atbs)) = τ(R, at/pas/p).(3) ([Tak06]) τ(R, (a+ b)t) =

∑u,v∈R!0,u+v=t τ(R, a

ubv).

2.

2.1. 0 . log canonical thresholdlct ACC (I,") (ACC) I(xn)n∈N n xn = xn+1

2.1 ([dFEM10]). 0 k n ! 1 LCTregnACC

LCTregn := {lct(X; a) | X n , a ! O}.

Shokurov ([Sho92]) ,

Hacon, McKernan, Xu [HMX14]ACC 0 lct ACC

lct ACC([HMX14])

2.2. . Blickle, Mustaţă, Smith 2.1

2.2 ([BMS09]). k n ! 1 FPTregn ACC

FPTregn := {fpt(R; a) | X = SpecR k n , a ! R}.

fpt lct 02

2.3 ([Sat17, Theorem 1.2]).

fpt

1 ([Sat17, Main Theorem]). k n ! 1 R = k[[x1, . . . , xn]]FPT(R) ACC

FPT(R) := {fpt(R; a) | a ! R}.

2.4. [Sat17] R[Sat17]

3. Ultraproduct

ultraproduct

72

5

3.1. Ultraproduct .

3.1. N P(N) U ⊆ P(N) non-principalultrafilter

(1) A ⊆ N A ̸∈ U.(2) A ∈ U, A ⊆ B ⊆ N B ∈ U.(3) A,B ∈ U A ∩B ∈ U.(4) A ⊆ N A ∈ U N \A ∈ U.

Zorn non-principal ultrafilter U

3.2. {Tm}m∈N∏

m Tm ∼U(am)m ∼U (bm)m ⇔ {m ∈ N | am = bm} ∈ U.

{Tm}m ultraproduct ulimm Tm

ulimm Tm := (∏

m

Tm)/ ∼U .

(am)m ∈∏

Tm ulim am ∈ ulimm Tm m ∈ Nfm : Tm −→ Sm ulimm fm : ulimm Tm −→ ulimm Sm ulimm am (−→

ulimm fm(am) T m Tm := T ulimm Tm∗T T ultrapower

- 3.3 ([Gol98, Theorem 5.6.1]). w = ulimm am ∈ ∗R supm am < ∞ infm am > −∞w0 ∈ R

ε > 0 {m ∈ N | |am − w0| < ε} ∈ U.w0 w shadow sh(w)

3.2. Ultraproduct . ultraproduct

3.4. m Tm ker(∏

m Tm −→ ulimm Tm)∏

m Tmulimm Tm

∏m Tm m Tm (resp.

, ) ulimTm .

m Rm Mm Rm ulimm Mm ulimm Rmam ⊆ Rm ulimm am ulimm Rm (Rm,mm, km)

(ulimm Rm, ulimm mm, ulimm km)

3.5. ultraproduct R = k[[x]](∗R, ∗m, ∗k) ulimm(xm) ∈ ∩m(∗m)m ∩m(∗m)m ̸= 0 Krull

∗R

3.6. (R,m, k) catapower ∗R

R# :=∗R/(∩m(∗mm))

∗m m# (R#,m#, ∗k)

R#

3.7 ([Scho10, 8.1.19]). (R,m, k) R̂ k −→ R̂

R# ∼= R̂ ⊗̂k(∗k).R#

73

6

ulimm fm ∈ ∗R [fm]m ∈ R# ulimm am ⊆ ∗R [am]m ⊆ R#

3.8. R = k[[x1, . . . , xd]] R# = (∗k)[[x1, . . . , xd]] λ = (λ1, . . . ,λd) ∈ Ndxλ := xλ11 · · ·x

λdd fm =

∑λ∈Nd am,λx

λ ∈ R [fm]m =∑

λ∈Nd [am,λ]mxλ ∈ R#

R# R

3.9. (R,m, k) am, bm ⊆ R [am]m ⊆ [bm]m ⊆ R#n

{m ∈ N | am ⊆ bm +mn} ∈ U.

3.3. 2.

2 ([BMS09], [Sat17, Theorem 1.5]). R F {am}m R

sh(ulimm fpt(R; am)) = fpt(R#; a∞).

2 1.13 (3)

3.10. R, {am}m 2 limm−→∞ fpt(R; am)

1

4.

k d ! 1R k[[x1, . . . , xd]]

4.1. A⇒ . A

4.1. a ⊆ R t > 0 M ! 1 (at,M) Afpt(R; a+mM ) < t

{am}m∈N ({am}tm,M) A m (atm,M)A

4.2. 0 ̸= a ⊆ R t > 0 .(1) b ⊇ a fpt(R; b) ! fpt(R; a). t " fpt(R; a) M

(at,M) A(2) fpt(R; a) = limM−→∞ fpt(R; a+mM ). fpt(R; a) < t M

(at,M) A

4.3. M > 0 FPT(R;⊇ mM ) ACC

FPT(R;⊇ mM ) := {fpt(R; b) | mM ⊆ b ⊆ m}.

Proof. [Sat17, Proposition 4.1] M N FPT(R;⊇mM ) ⊆ (1/N)Z ACC . #

4.4. A{am}m∈N {fpt(R; am)}

t := limm−→∞ fpt(R; am) M > 0 ({am}tm,M)A m fpt(R; am+mM ) < t

, 4.2 (1) limm fpt(R; a+mM ) = t Prop4.3

74

7

A Mcatapower

4.5. t > 0 R {am}m a∞ := [am]m ⊆ R#M > 0 (at∞,M) A ({am}tm,M) A

A catapower

(1) 4.2 B B ⇒ A . ( 4.11).(2) 4.3 C C ⇒ B . ( 4.16).(3) 4.4 D D ⇒ C 4.20(4) 4.5 a∞ = [am]m D m D

4.22

D ⇒ C a∞ D ⇒ m D( 4.21 4.23). [Sat17] D C

D CA B

[Sat17, 5 ]

4.2. B ⇒ A. A B B, a ⊆ R fpt(R; a) t (−→ τ(R, at)

R!0 −→ P(R) A(t, s) (−→ τ(R, atms) (R!0)2 −→ P(R)

4.6. a, b ⊆ R (R; a, b) F (F -pure region) FPR(R; a, b) ⊆(R!0)2

FPR(R; a, b) := Cl({(t, s) ∈ (R!0)2 | (R, atbs) F })= Cl({(t, s) ∈ (R!0)2 | τ(R, atbs) = R}),

Cl

4.7 ([Pér13, Example 5.3]). R := F3[[x, y]], a = (xy), b = (x+ y) FPR(R; a, b)1 ( [Pér13, Example 5.3] )

Figure 1 Figure 2

R0 := C[x, y], a = (xy), b = (x + y) log canonical region LCR(R; a, b) :={(t, s) | (R0, atbs) lc} X = SpecR0 (X, a, b) log resolution

LCR(R0; a, b) ⊆ R2 24.8. log canonical region LCR(R0; a, b) log canonical polytopeLCT(R0; a, b) LCR rational polytope

loc canonical log resolutionF polytope fractal

75

8

4.9. a ⊆ R M > 0 (at,M) Bl ⊆ R2 −M (t, 0) l ∩ FPR(R; a,m) = ∅.

4.10. 4.7 R0 = C[x, y], a = (xy), b = (x+ y) t > 1,t −2 l LCR(R; a, b) ∩ l = ∅

R = F3[[x, y]], a = (xy), b = (x+ y) 1t > 1 , (t, 0)

M = −3 FPR(R; a, b)n ! 0 y = −x+ 1

Pn := (3n − 13n

,1

3n) ∈ R2

[Pér13, Example 5.3] Pn FQ ∈ R2 n Pn ≺ Q Q ̸∈ FPR(R; a, b)

P = (x, y), Q = (x′, y′)

P ≺ Q def⇔ x < x′ y < y′

t > 1 (t, 0) −3 l l ∩R2!0 Qn Pn ≺ Q l ∩ FPR(R; a, b) = ∅

4.11. a ⊆ R t > fpt(R; a) M > 0 (at,M)B , (at,M) A

Proof. ε > 0 lε ⊆ R2 (t − ε, 0) −M BFPR(R; a,m) ε > 0 lε ∩ FPR(R; a,m) = ∅

1.15 (3)

τ(R, (a+mM )t−ε) =∑

u,v∈R!0,u+v=t−ετ(R, aumMv)

u + v = t − ε u, v (u,Mv) ∈ R2!0 lε(u,Mv) ̸∈ FPR(R; a,m). τ(R, aumMv) ⊆ m.

τ(R, (a+m)t−ε) ⊆ m A #4.3. C ⇒ B.

4.12. FPR 4.10 {Pn}B

Pn = (xn, yn) x xn 1 3

1 =∞∑

i=1

2

3i

n+ 1 y

yn = fpt(R, axn ; b) := inf{y ∈ R!0 | τ(R, axnby) ̸= R}

76

9

C q

4.13. t ∈ R!0 q ∈ Z!2, n ∈ Z t q(1) ⟨t⟩n,q := ⌈tqn − 1⌉/qn ∈ Q!0.(2) t(n,q) = ⌈tqn − 1⌉ − q⌈tqn−1 − 1⌉.

q ⟨t⟩n, t(n)

t(n) t q n ⟨t⟩n, q n+ 1

Lemma 4.14. t, q, n

(1) 0 " t(n) < q.(2) n

10

Lemma 4.19. p t ∈ Q>0(1) a, b pa(pb − 1)t ∈ Z(2) e > 0 q = pe q(q − 1)t ∈ Z.(3) (2) q t q n t(n,q) n ! 2

Proof. (1) s, a > 0 p r > 0 t = s/(par)(Z/rZ)× b pb ≡ 1 (mod .r) (2)e := ab (3) s

t = s/(q(q− 1)) s q− 1 a, l s = a(q− 1) + l1 " l " q − 1 t = (a/q) + (l/q(q − 1)) n ! 2 t(n) = l

#4.20. t ∈ Q>0 q = pe q(q − 1)t ∈ Z l := t(2)

s := ql/(q − 1) ∈ Q>0 a, b ⊆ R n0 (as, q, n0) Dq, n0 N n ! n0 + 2

fpt(R, a⟨t⟩n+1,q ; b) ! fpt(R, a⟨t⟩n,q ; b)−N/qn.Proof. N := qn0+1 n , yn := fpt(R, a⟨t⟩n,q ; b) ! 0, y′n :=⌈qn−n0−1yn⌉/qn−n0−1 yn " y′n < yn+(1/qn−n0−1) τ(R, a⟨t⟩n,qby

′n) ⊆ m

sα := qn−n0⟨t⟩n,q − ⟨s⟩n0,q

α = qn−n0⟨t⟩n+1,q − ⟨s⟩n0+1,q

1.15 (1), (2), D

m ⊇ τ(R, a⟨t⟩n+1,qby′n+1)

= Tre(n−n0)(F e(n−n0)∗ τ(R, aqn−n0 ⟨t⟩n+1,qbq

n−n0y′n+1))

= Tre(n−n0)(F e(n−n0)∗ aαbq

n−n0y′n+1τ(R, a⟨s⟩n0+1,q ))

= Tre(n−n0)(F e(n−n0)∗ aαbq

n−n0y′n+1τ(R, a⟨s⟩n0,q ))

= Tre(n−n0)(F e(n−n0)∗ τ(R, aqn−n0 ⟨t⟩n,qbq

n−n0y′n+1))

= τ(R, a⟨t⟩n,qby′n+1).

yn " y′n+1 " yn+1 +N/qn. #4.21. b C

b m D ⇒ C[Sat17] C D

4.5. a∞ ⊆ R# D ⇒ {am}m D. D Catapower

4.22. s > 0 n0 > 0 m am ⊆ Ra∞ := [am]m ⊆ R# (as∞, q, n0) D

a > 0 Ta ⊆ N m ∈ Taτ(R, a⟨s⟩n0,q ) ⊆ τ(R, a⟨s⟩n0+1,q ) +ma.

Proof. 1.15 (2) [Sat17, Proposition 2.10 (5)] n

τ(R#, a⟨s⟩n,q∞ ) = Tr

en(F en∗ a⌈sqn−1⌉∞ )

= [Tren(F en∗ a⌈sqn−1⌉m )]m

= [τ(R, a⟨s⟩n,qm )]m ⊆ R#

78

11

3.9 #4.23. D a∞ D ⇒

{am}m D implication D

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