Remarks for a classification of ideals with fixed codimension in the local rings for irreducible curve singularities

ۂઢͷಛҟ఺ʹ෇ਵ͢Δہॴ؀ͷ༨࣍ݩʹΑΔ ΠσΞϧͷ෼ྨʹ͍ͭͯͷ஫ҙ

౉རਖ਼߂

Remarks for a classification of ideals with fixed codimension in the local rings for irreducible curve singularities

Masahiro Watari

We discuss a classification problem of ideals by codimension in case the ring is the local ring for irreducible curve singularity. Our aim is to intoroduce an effective method for computing order sets for fixed

codemension. The order sets admit us to decomposse the set of all ideals with given codimension into some families. We determine the normal forms of such families for theA2d singularity.

Key words: local ring, ideal, codimension, irreducible curve singularity

1. ಋ ೖ

جૅମkΛඪ਺0ͷ୅਺ดମͱ͢Δɽط໿ಛҟۂઢ ͷಛҟ఺ʹ෇ਵ͢Δہॴ؀OΛߟ͑ΔɽຊߘͰ͸ɼO

͸׬උͰ͋Δͱ͠ɼ༨࣍ݩʹΑΔہॴ؀OͷΠσΞ ϧͷ෼ྨ໰୊Λߟ͑Δɽ͜ͷΑ͏ͳΠσΞϧͷ෼ྨ͸ɼ r఺ͷώϧϕϧτεΩʔϜͷݚڀʹͱͬͯͱͯ΋ॏཁ Ͱ͋Δɽ(r఺ͷώϧϕϧτεΩʔϜʹ͍ͭͯ͸ɼࢀߟ จݙ1)ͱ4)Λࢀর.) ಛʹO͕࣮ࡍʹ༩͑ΒΕͨ࣌ɼ ༩͑ΒΕͨ༨࣍ݩΛ࣋ͭΠσΞϧΛͲͷΑ͏ʹͯ͠ٻ Ί͍͔ͯ͘ͱ͍͏͜ͱ͕໰୊ͱͳΔɽ͜ͷΑ͏ͳܭࢉ

͸ɺࢀߟจݙ2)΍3)ʹ͓͍࣮ͯߦ͞Ε͍ͯΔ͕ͦͷ

΍Γํʹ͍ͭͯ͸ৄ͍͠આ໌͕ͳ͞Εͳ͔ͬͨɽຊߘ Ͱ͸ͦͷํ๏Λ঺հ͢Δɽ

·ͣୈ2અͰ͸ه߸΍ิ୊ͷ४උΛߦͳ͏ɽ·ͨΠ σΞϧͷҐ਺ू߹Λఆٛ͠ɼ༩͑ΒΕͨ༨࣍ݩʹର͠

ͯͦͷ༨࣍ݩΛ࣮ݱ͢ΔΠσΞϧͷҐ਺ू߹͸༗ݶݸ

͔͠ͳ͍͜ͱΛূ໌͢Δɽୈ3અͰ͸Ґ਺ू߹ͷΓ-Ճ

܈ͱͯ͠ͷੜ੒ݩͷܭࢉΞϧΰϦζϜͱ༨࣍ݩ͕iͷ ΠσΞϧશମͷҐ਺ू߹͔Β༨࣍ݩ͕i+ 1ͷΠσΞϧ શମͷҐ਺ू߹ΛٻΊΔΞϧΰϦζϜΛ঺հ͢Δɽ࠷

ޙʹୈ4અͰ۩ମྫͱͯ͠A6ܕͷಛҟ఺Λߟ͑ɼલ અ·Ͱͷ݁ՌΛ࢖ͬͯہॴ؀OͷΠσΞϧͷ༨࣍ݩʹ ΑΔ෼ྨΛߟ͑Δɽ

2. ४ උ

·ͣ࠷ॳʹه߸ͷ४උΛ͢Δɽہॴ؀Oͷਖ਼نԽΛ Oͱද͢ɽ͢ͳΘͪO=k[[t]]Ͱ͋Δɽہॴ؀Oͷۃ

ݪߘड෇ ฏ੒23೥8݄31೔

∗ઐ໳ֶՊڞ௨Պ໨

େΠσΞϧΛmOͰද͢ɽہॴ؀OͷݩͷҐ਺ͷू߹

Γ :={ord(f)|f ∈ O}

Λɼہॴ؀Oʹ෇ਵ͢Δ൒܈ͱݺͿɽ൒܈Γͷ෦෼

ू߹S͕Γ-Ճ܈Ͱ͋Δͱ͍͏͜ͱΛɼ࣍ͷ2ͭͷ৚

݅Λຬͨ͢͜ͱͰఆٛ͢Δɽ

(i) s1+s2∈S for∀s1, s2∈S, (ii) s+γ∈S for∀s∈S,∀γ∈Γ.

Γ-Ճ܈S͕ɼΓ-Ճ܈ͱͯ͠ू߹A={α1, . . . , αm}ʹ Αͬͯੜ੒͞ΕΔͱ͖(i.e. S = {γ+_m

i=1αi|γ ∈ Γ}), S = A = α₁, . . . , αmͱද͢ɽΓͷಋମΛɼ c:= min

n|I(n)⊂ O

ʹΑΓఆٛ͢ΔɽIΛ0Ͱͳ

͍OͷΠσΞϧͱ͢Δɽ͜ͷ࣌ɼτ(I) := dim_kO/I ΛɼIͷ༨࣍ݩͱ͍͏ɽ༨࣍ݩ͕rͰ͋ΔOͷΠσΞ ϧͷू߹Λ

Ir:={I|I͸τ(I) =rΛຬͨ͢OͷΠσΞϧ} ͱ͓͘ɽ·ͨΓ(I) :={ord(f)|f ∈I}ΛΠσΞϧIͷ Ґ਺ू߹ͱݺͼɼG(I) := Γ\Γ(I)ͱ͓͘ɽҐ਺ू߹

Γ(I)͸ɼΓ-Ճ܈ͷߏ଄Λ࣋ͭ͜ͱʹ஫ҙ͢Δɽ

ิ୊ 1. ΠσΞϧ I ͷ༨࣍ݩ͕ r Ͱ͋Δ͜ͱͱɼ G(I) =rͰ͋Δ͜ͱ͸ಉ஋Ͱ͋Δɽ

Proof. ΠσΞϧIͷ༨࣍ݩ͕rͰ͋Δ͜ͱ͸ɼ࣍ͷ৚

͕݅੒Γཱͭ͜ͱͱಉ஋ɽ (1) O/I=

_r−1

i=0

ait^di|ai ∈k, d0= 0, di< di+1

৚݅(1)͸ɼG(I) =rΛҙຯ͢Δɽ

ิ୊ 2. IrͷݩͷҐ਺ू߹ͱ੒ΓಘΔΓ-Ճ܈͸༗ݶ ݸͰ͋Δɽ

Proof. ೚ ҙ ͷ Ir ͷ ݩ I ʹ ର ͠ ͯ ɼG(I) = {0, d1, . . . , dr−1} ͸ ৚ ݅ (1) Λ ຬ ͨ ͢ɽ໌ Β ͔ ʹ

͜ ͷ Α ͏ ͳ ਺ d1, . . . , dr−1 ͸ ༗ ݶ ݸ Ͱ ͋ Δ ͔ Β ɼ Γ(I) = Γ\G(I)΋༗ݶݸͰ͋Δɽ

Γ-Ճ܈SΛҐ਺ू߹ͱͯ࣋ͭ͠Α͏ͳΠσΞϧશମ ΛI(S)Ͱද͢ɽ͜ͷ࣌ɼI(S)ʹଐ͢Δ೚ҙͷΠσΞ ϧ͸શͯಉ͡༨࣍ݩΛ࣋ͭࣄʹ஫ҙ͢ΔɽIrͷݩͷҐ

਺ू߹ͱ੒ΓಘΔΓ-Ճ܈ΛS1, . . . , Slͱ͢Δͱ,Ir͸

࣍ͷ༷ʹ෼ղ͞ΕΔɽ

(2) I_r=

l i=1

I(Si).

͜͜Ͱi=jͳΒ͹ɼI(Si)∩ I(Sj) =∅. ෼ղ(2)ʹ ରͯ͠ɼIrͷҐ਺ू߹શମΛ

(3) S_r:={S₁, . . . , Sl}

ͱද͢ɽ·֤ͨSiͷੜ੒ू߹ΛAiͱͨ͠ͱ͖ɼͦͷ શମΛAr:={A₁. . . , Al}ͱද͢ɽ

IΛɼI= (1)͔ͭI= (0)Λຬͨ͢OͷΠσΞϧͱ

͢ΔɽG1(I) :=G(I)\ {0}ͱ͓͘ɽb1:= min{G1(I)} ʹରͯ͠ɼू߹B1(I) :=G1(I)∩(b1+ Γ)Λߟ͑Δɽ

΋͠G1(I) = B1(I)Ͱ͋Ε͹ɼb2 := min{G1(I)\ B1(I)},B2:={G₁(I)\(b1+ Γ)}∩(b2+ Γ)ͱ͓͘ɽҎ Լɼ͜ͷաఔΛؼೲతʹ܁Γฦ͢ɽ͢ͳΘͪG1(I)=

∪^j−1_i=1Bi(I)Ͱ͋Ε͹ɼbj := min{G1(I)\ ∪^j−1_i=1Bi(I)}

͔ͭBj(I) :={G₁(I)\ ∪^j−1_i=1Bi(I)} ∩(bj+ Γ)ͱ͓͘ɽ G1(I) < ∞Ͱ͋Δ͔Βɼ͋Δਖ਼੔਺n͕ଘࡏͯ͠ɼ G1(I) = ∪ⁿ_i=1Bi(I)ͱͳΔɽ͜ͷͱ͖֤iʹରͯ͠ɼ di:= max{Bi(I)}ͱ͢Δɽ

3. Ґ਺ू߹ͷܭࢉ๏

Γ-Ճ܈ͷੜ੒ݩͷܭࢉ๏(ҎԼɼܭࢉ๏ͱͯ͠ࢀর)

༩͑ΒΕͨΓ-Ճ܈Sͷ(Γ-Ճ܈ͱͯ͠ͷ)ੜ੒ݩΛٻ ΊΔɽ·ͣα1 := min{S}ʹରͯ͠ɼू߹α1+ Γ = {α1+γ|γ∈Γ}Λߟ͑Δɽ΋͠S=α1+ ΓͰ͋Ε͹ɼ α2 := min{S\(α1+ Γ)}ͱ͓͖ɼू߹∪²_i=1(αi+ Γ) Λߟ͑Δɽ΋͠S = ∪²_i=1(αi+ Γ)Ͱ͋Ε͹ɼα3 :=

min{S\∪²_i=1(αi+ Γ)}Λߟ͑Δɽ͜ͷૢ࡞Λؼೲతʹ

܁Γฦ͍ͯ͘͠ɽͭ·ΓS=∪^j−1_i=1(αi+ Γ)Ͱ͋Ε͹ɼ αj := min

S\∪^j−1_i=1(αi+Γ)

ͱ͓͖ɼू߹∪^j_i=1(αi+Γ) Λߟ͑Δɽ͜ͷ݁Ռɼ࣍ͷΑ͏ͳ߱ԼྻΛಘΔɽ

SS\(α1+ Γ)· · ·S\ ∪^j_i=1(αi+ Γ)· · · ࠓɼແݶू߹{a∈N|a≥α1+c}͕Sͱα1+ Γͷ྆

ํʹؚ·ΕΔ͜ͱ͔Βɼ{S\(α1+ Γ)}<∞Ͱ͋Δ͜

ͱ͕ै͏ɽΑͬͯS=∪^m_i=1(αi+ Γ)Λຬͨ͢ਖ਼੔਺m

͕ଘࡏ͢Δɽ͜ͷૢ࡞ͰಘΒΕͨA = {α1, . . . , αm}

͕ɼS=AͱͳΔ࠷খͷੜ੒ू߹Ͱ͋Δ͜ͱ͸໌Β

͔Ͱ͋Δɽ

͜͜Ͱ࣍ͷิ୊Λূ໌͢Δɽ

ิ୊3. Ґ਺ू߹Γ(I) =α1, . . . , αmΛ࣋ͭIr−1ͷ ݩIʹରͯ͠,G(I) =G(I)∪ {αi}(i͸೚ҙ)ͱͳΔ ΠσΞϧI͕I_rͷݩͱͯ͠ଘࡏ͢Δ. ٯʹG1(J) =

∪ⁿ_i=1Bi(J)Λຬͨ͢ɼIrͷݩJΛߟ͑Δɽ֤iʹର͠

ͯɼG(J) = G(J)\ {di}Λຬͨ͢Α͏ͳJ͕I_r−1 ͷதʹଘࡏ͢Δɽ

Proof. IΛΓ(I) =α₁, . . . , αmΛ΋ͭI_r−1ͷݩͱ͢

Δ. ೚ҙͷiʹରͯ͠

S= [Γ(I)\ {αi}]∪ {αi+γ|γ∈Γ\ {0}}

͸ɼΓ-Ճ܈ͷߏ଄Λ΋ͭ. ͜͜Ͱ fγ =t^γ (γ∈S)

શମͰੜ੒͞ΕΔΠσΞϧΛIͱ͢Δͱɼ໌Β͔ʹ Γ(I) =SͰ͋Δɽͭ·ΓG(I) =G(I)\ {α_i}. I ∈ Ir−1Ͱ͋Δ͜ͱ͔Β,ิ୊1ʹΑΓG(I) =r−1Ͱ

͋Δ. ΑͬͯG(I) =r. ࠶ͼิ୊1ΑΓɼI ∈ IrΛ ಘΔ.

࣍ʹJ ΛIrͷݩͰ,G(J) =∪ⁿ_i=1Bi(J)Λຬͨ͢΋

ͷͱ͢Δɽ͜͜Ͱ֤iʹରͯ͠Γ(J)∪ {d_i}͸ɼΓ-Ճ

܈ͱͳΔ͜ͱʹ஫ҙ͢Δɽ͜ͷ࣌ɼ

fγ=t^γ (γ∈Γ(J)∪ {di})

Ͱੜ੒͞ΕΔΠσΞϧJΛߟ͑Ε͹ɼΓ(J) = Γ(J)∪ {di}͔ͭG(j) =r−1Λຬ͍ͨͯ͠Δɽิ୊1ʹΑ ΓJ∈ I_r−1ͱͳΔ.

ิ୊3Λ༻͍ͯɼIr−1ͷҐ਺ू߹શମSr−1͔ΒIr

ͷҐ਺ू߹શମSrΛٻΊΔํ๏Λߟ͑Δɽ Sr−1͔ΒͷSrߏ੒๏(ҎԼɼߏ੒๏ͱͯ͠ࢀর) Sr−1 = {S1, . . . , Sl}͕༩͑ΒΕͨͱ͖ɼ֤ݩSiͷ ੜ੒ू߹Λܭࢉ๏ʹΑΓٻΊ͓ͯ͘ɽͦΕΛ Ai = {αi,1,· · · , α_i,m(i)} (i = 1, . . . , l)ͱද͢ɽิ୊3ͷূ

໌ʹ͋ΔΑ͏ʹɼ֤i ∈ {1, . . . , l}, j ∈ {1, . . . , m(i)} ʹରͯ͠ɼ

(4) S:= [Si\ {αi,j}]∪ {αi,j+γ|γ∈Γ\ {0}}

Λߟ͑Ε͹ɼ͜Ε͸S_rͷݩͰ͋Δɽ࠶ͼิ୊3ͷূ໌

ΑΓɼIrͷݩIͷҐ਺ू߹Γ(I)ʹରͯ͠ɼΓ(I)∪{di}

͸I_r−1ʹର͢ΔҐ਺ू߹Ͱ͔͋ͬͨΒSr͸ɼ(4)ͷ ܗͷݩͰͭ͘͞ΕΔɽ

ܭࢉ๏ͱߏ੒๏Λ૊߹ΘͤΔ͜ͱʹΑΓɼ༨࣍ݩr ʹର͢ΔҐ਺ू߹શମSrΛ࣍ͷఆཧʹΑΓٻΊΔ͜

ͱ͕Ͱ͖Δɽ

ఆཧ4 (Sr ͷܭࢉ๏). ू߹Sr͸࣍ͷૢ࡞1ͱૢ࡞

i͔ΒɺؼೲతʹಘΒΕΔɽ

ૢ࡞1 S1 ={Γ(mO)}ͱ͓͖ɼܭࢉ๏ʹΑΓΓ(mO) ͷੜ੒ू߹ΛٻΊΔɽ

ૢ࡞i (i = 2, . . . , r): S_i−1͔Βɼߏ੒๏Λ༻͍ͯ

SiΛٻΊΔɽ࣍ʹAiΛܭࢉ๏ʹΑΓٻΊΔɽ Proof. ໌Β͔ʹ I₁ = {m_O} Ͱ͋ΔɽΑͬͯS₁ = {Γ(mO)}ɽͦͷଞͷ෦෼͸ɼܭࢉ๏ͱߏ੒๏͔Βै͏ɽ

ఆཧ4ʹΑΓٻΊΒΕͨS_rʹରͯ͠ɼS_rͷ֤ݩΛ Ґ਺ू߹ʹ࣋ͭΑ͏ͳΠσΞϧΛܾఆ͢Δඞཁ͕͋Δ

͕Ұൠʹ͜Ε͸؆୯Ͱͳ͍ɽ͕ͩA2dܕͷಛҟ఺ʹ෇

ਵ͢Δہॴ؀k[[t², t^2d+1]]ͷΠσΞϧʹؔͯ͠ɼ࣍ͷ

ิ୊͕੒Γཱͭɽ

ิ୊ 5. ہॴ؀k[[t², t^2d+1]]ͷΠσΞϧIʹରͯ͠ɼ Γ(I) =α1, α2(α1< α2)Ͱ͋Ε͹ɼI͸

(5) f1=t^α1+

j∈G(I),j>α1

ajt^j, f2=t^α2.

ʹΑΓੜ੒͞ΕΔɽଞํɼ΋͠Γ(I) =α1Ͱ͋Ε͹ɼ I͸্هͷf1ͷΈͰੜ੒͞ΕΔ.

4. ྫ

લষ·Ͱͷ݁ՌΛ࢖͍ɼA6ܕͷಛҟ఺ʹ෇ਵ͢Δ ہॴ؀k[[t², t⁷]]ͷΠσΞϧΛɼ1≤τ≤6ͷ༨࣍ݩʹ

ؔͯ͠෼ྨ͢Δ. ͜ͷઅͰ͸Γ-Ճ܈ͷू߹(3)Λ S_r={S_r,1,· · · , Sr,l}

ͱද͠ɼ͜ͷදهʹରͯ͠Irͷ෼ղ(2)ͷཁૉΛ Ir,i=I(Sr,i) (i= 1,· · · , l)

ͱ͓͘͜ͱʹ͢ΔɽࠓɼΓ ={0,2,4,6,7,8, . . .}Ͱ͋Δ.

ఆཧ4Λ༻͍Δ͜ͱʹΑΓɼ࣍ͷදΛಘΔɽ τ order sets

1 S1,1=2,7

2 S2,1=4,7,S2,2=2 3 S3,1=6,7,S3,2=4,9

4 S4,1=7,8,S4,2=6,9,S4,3=4 5 S5,1=8,9,S5,2=7,10,S5,3=6,11 6 S6,1=9,10,S6,2=8,11,S6,3=7,12

S6,4=6

্هͷදͷҐ਺ू߹ʹରͯ͠ɼิ୊5Λ༻͍Δ͜ͱ ʹΑΓ࣍ͷ෼ྨ݁ՌΛಘΔɽ

τ ideals 1 I1,1= (t², t⁷)

2 I2,1= (t⁴, t⁷),I2,2= (t²+at⁷) 3 I3,1= (t⁶, t⁷),I3,2= (t⁴+at⁷, t⁹) 4 I4,1= (t⁷, t⁸),I4,2= (t⁶+at⁷, t⁹)

I_4,3= (t⁴+at⁷+bt⁹)

5 I_5,1= (t⁸, t⁹),I_5,2= (t⁷+at⁸, t¹⁰) I5,3= (t⁶+at⁷+bt⁹, t¹¹)

6 I6,1= (t⁹, t¹⁰),I6,2= (t⁸+at⁹, t¹¹), I_6,3= (t⁷+at⁸+bt¹⁰, t¹²)

I_6,4= (t⁶+at⁷+bt⁹+ct¹¹)

ࢀߟจݙ

1) G. Pfister, J.H.M. Steenbrink: Reduced Hilbert schemes for irreducible curve singularities, J. Pure and Applied Algebra. (1992)77103-116

2)૬അ๕ل: ฏ໘ۂઢͷ୯७ಛҟ఺ʹର͢Δr఺ͷώ ϧϕϧτεΩʔϜ, ࡛ۄେֶཧ޻ֶݚڀՊ਺ཧిࢠ৘

ใઐ߈ म࢜࿦จ(2010)

3)૬അ๕ل,౉རਖ਼߂: Punctual Hilbert schemes for irreducible curve singularities of E6 and E8 types, arXiv:math.AG/1109.4991

4)౉རਖ਼߂: ୯߲ۂઢͷಛҟ఺ʹର͢Δr఺ͷώϧϕ ϧτεΩʔϜʹ͍ͭͯ,௡ࢁߴઐلཁ(2010) 63-65