න㗄ᦛ✢ߩ․⇣ὐߦኻߔࠆ
r
ὐߩࡅ࡞ࡌ࡞࠻ࠬࠠࡓߦߟߡᷰᱜᒄ
∗On the Hilbert schemes of r points for monomial curve singularities
Masahiro Watari
Pfister and Steenbrink studies Hilbert schemes ofrpoints for monomial curve singularities. Our aim in this present paper is to show that if the Hilbert scheme ofrpoints for given monomial curve singularity is irreducible, then it is a rational projective variety.
Key wordsHilbert schemes ofrpoints, monomial curve singularities
1. ዉ
ၮ␆kࠍᮡᢙ0ߩઍᢙ㐽ߣߔࠆ㧚ᣢ⚂․⇣ᦛ✢
ߩ․⇣ὐߦኻߔࠆዪᚲⅣO߇Ⅳ
k[[ta1, . . . , tam]] (ai∈N)
ߣ ห ဳ ߢ ࠆ ߽ ߩ ࠍ 㧘න 㗄 ᦛ ✢ ⧘ߣ ߱ 㧚ߎ ߎ ߢ gcd(a1, . . . , am) = 1ߣቯߒߡ߽৻⥸ᕈࠍᄬࠊߥ㧚 ዪᚲⅣOߩరߩᢙߩ㓸ว
Γ :={ord(f)|f ∈ O}
ࠍ㧘ዪᚲⅣOߦઃ㓐ߔࠆඨ⟲ߣ߱㧚એਅ㧘ᧄⓂߢ↪
ࠆਥߥ⸥ภࠍᰴߩ᭽ߦḰߔࠆ㧚 O:=k[[t]]
I(n) :=
f ∈ O |ordf ≥n (n∈N) I(n) :=I(n)∩ O
δ:= dimkO/O=�(N\Γ) c:= min
n|I(n)⊂ O Gr
δ, O/I(2δ):=
O/I(2δ)ߩδᰴరߩㇱಽⓨ㑆 ᢙሼߩ0ߩᢙࠍ∞ߣቯ⟵ߔࠆߎߣߦࠃࠅ㧘ᢙߩ 㓸วI(n)ߣI(n)ߪߘࠇߙࠇ㧘OߣOߩࠗ࠺ࠕ࡞ߦߥ ࠆ㧚⸥ߩᱜᢛᢙδߣcࠍ㧘ߘࠇߙࠇዪᚲⅣOߩδ- ਇᄌ㊂㧘ዉߣ߱㧚ߎࠇࠄߦߟߡ㧘ᰴߩ㗴߇ᚑ
┙ߔࠆ㧚
㗴 1(cf. [3], p. 80, Prposition 7). ዪᚲⅣOߩδ-ਇ ᄌ㊂δߣዉcߦኻߒߡ㧘㑐ଥᑼ
δ+ 1≤c≤2δ
߇ᚑ┙ߔࠆ㧚․ߦc= 2δߣߥࠆߩߪO߇ࠧࠗࡦࠪࡘ
࠲ࠗࡦⅣߩᤨ㧘߆ߟߘߩᤨߩߺߦ㒢ࠆ㧚
ේⓂฃઃ ᐔᚑ22ᐕ831ᣣ
∗ኾ㐷ቇ⑼ㅢ⑼⋡
ߎߎߢⅣO/I(2δ)ࠍ⠨߃ࠆ㧚ߎߩⅣߩࡌࠢ࠻࡞
ⓨ㑆ߣߒߡߩᰴర߇㧘2δߢࠆߎߣߦᵈᗧߔࠆ㧚ᰴర ߇δߩㇱಽࡌࠢ࠻࡞ⓨ㑆ోߩ㓸วࠍGr
δ, O/I(2δ) ߣߔ㧚W ∈Gr
δ, O/I(2δ)߇ࠃ⁁ᘒ (good)ߢ
ࠆߣ߁ߎߣࠍ㧘ዪᚲⅣOߩరߣߩⓍ
O ×W �(f, v)�→f v∈W
߇ቯ⟵ߐࠇ㧘ߎߩⓍߦኻߒߡW ߇O-ㇱಽട⟲ ߢ
ࠆߣ߈ߣቯࠆ㧚ߎߎߢGr
δ, O/I(2δ)
ߩࠃ⁁ᘒ ߢࠆㇱಽࡌࠢ࠻࡞ⓨ㑆ߩ㓸วࠍ
M:=
W ⊂Gr
δ, O/I(2δ) W ߪࠃ⁁ᘒ ߢߔ㧚߹ߚ⸥ภMrߢᰴర߇rߢࠆOߩࠗ࠺
ࠕ࡞ߩ㓸วࠍߔ߽ߩߣߔࠆ㧚ߔߥࠊߜ Mr:=
I ⊂
idealO dimkO/I =r
ߣߔࠆ㧚⺰ᢥ[1]ߦ߅ߡ㧘౮
φr:Mr−→ M
∈ ∈
I�−→t−rII(2δ)
߇ ቯ ⟵ ߢ ߈ ࠆ ߎ ߣ ߇ ␜ ߐ ࠇ ߚ 㧚߹ ߚ ⸥ ภ ψ ߢ Gr
δ, O/I(2δ)߆ࠄᓇⓨ㑆PN (N =2δ δ
−1)߳ ߩPl¨uckerၒㄟߺࠍߔ߽ߩߣߒ㧘Gr
δ, O/I(2δ) ߩరWߩψߦࠃࠆࠍ
ψ(W) = (π1,2,...,δ,· · · , πi1,...,iδ,· · · , πδ+1,...,2δ) ߣߒ㧘ߎࠇࠍPl¨uckerᐳᮡߣ߱㧚◲නߩߚᷝ߃ ሼ㓸ว{i1, . . . , iδ}ࠍ㧘⸥ภΛߥߤߢߔ㧚
Vr:= (ψ◦φr)(Mr)
ߣ߅ߊ㧚ߎߩVrࠍ㧘ਈ߃ࠄࠇߚ․⇣ᦛ✢⧘ߦኻߔࠆr ὐߩࡅ࡞ࡌ࡞࠻ࠬࠠࡓߣ߱㧚PfisterߣSteenbrink ߪ⺰ᢥ[1]ߦ߅ߡ㧘ߎߩ౮φr߇ᰴߩᕈ⾰ࠍᜬߟߎ ߣࠍ␜ߒߚ㧚
― 62 ―
津 山 高 専 紀 要 第 5 2 号 ( 2 0 1 0 )
― 63 ―
ቯℂ 2 (Pfister, Steenbrink [1]). ౮φrߪනߢ
ࠆ㧚․ߦr≥2δߩᤨ㧘౮φrߪోනߢࠆ㧚ᦝߦ છᗧߩ⥄ὼᢙrߦኻߒߡ㧘Vrߪࠩࠬࠠ㐽㓸วߢࠆ㧚
ߎߩቯℂࠃࠅ⋥ߜߦޔᰴߩ㗴߇ዉ߆ࠇࠆ㧚
♽ 3. ߽ߒr≥2δߥࠄ߫㧘Vr=V2δ߇ᚑ┙ߔࠆ㧚 Gr�
δ,O/I(2δ)�
M
Mr φr
PN ψ
ψ◦φr
ψ: Pl¯uckerၒㄟߺ N=�2δ
δ
�−1
r≥2δߥࠆ⥄ὼᢙrߦኻߒߡ, V :=Vr
ߣ ߅ ߊ㧚ߎ ߩ ᓇ ઍ ᢙ ᄙ ᭽ V ࠍ Pfister- Steenbrink ᄙ ᭽ (PS ᄙ ᭽ ) ߣ ߱ 㧚[1] ߢ ߪ㧘᭽ޘߥන㗄ᦛ✢⧘ߦኻߒߡߩPSᄙ᭽V ߩ᭴ㅧ ߇ߒߊ⎇ⓥߐࠇߚ㧚߹ߚ[4]ߦ߅ߡ㧘ᰴߩᣢ⚂ߥ න⚐․⇣ὐ
A2d:x=t2, y=t2d+1 (d≥1), E6:x=t3, y=t4,
E8:x=t3, y=t5㧘
ߦኻߒߡ㧘ઍᢙᄙ᭽Vr (1≤r≤2δ)ߩቯ⟵ᣇ⒟ᑼ ߇⸘▚ߐࠇߚ㧚
ᧄⓂߩ⋡⊛ߪ㧘ᰴߩቯℂࠍ⸽ߔࠆߎߣߢࠆ㧚 ቯℂ4. છᗧߩන㗄ᦛ✢⧘ߦኻߔࠆrὐߩࡅ࡞ࡌ࡞࠻
ࠬࠠࡓVr(r≥1)ߪ㧘ᣢ⚂ߥࠄ߫ℂ⊛ᓇઍᢙᄙ
᭽ߢࠆޕ
2. ቯℂ 4 ߩ⸽
߹ߕቯℂ4ߩ⸽ߦᔅⷐߥᨩࠍḰߔࠆ㧚Pl¨ucker ၒㄟߺψࠍᰴߩࠃ߁ಽ⸃ߔࠆ㧚
ψ:M −→ Gr(δ,2δ) −→ Mδ,2δ(k)/∼ −→ PN
∈ ∈ ∈ ∈
W �−→ �a1,· · ·,aδ�k �−→ AW �−→ (πi1···iδ) MߩరW ࠍW =�f1,· · ·, fδ�kߣߔ㧚ߎߎߢfi=
�2δ−1
j=0 aijtj ∈ O/I(2δ)ߪW ߩk-ࡌࠢ࠻࡞ⓨ㑆ߣߒ ߡߩၮᐩߢࠆ㧚ߎߩၮᐩߩଥᢙ߆ࠄቯ߹ࠆk2δߩᮮ ᢙࡌࠢ࠻࡞ࠍ
ai= (ai0,· · ·, ai2δ−1)
ߣ߅߈㧘ߩวᚑ౮ߦࠇࠆᦨೋߩ౮ߪߎߩ⥄ὼ ߥኻᔕߦࠃࠆ߽ߩߢࠆ㧚ᰴߦδ×2δⴕ
AW =
⎛
⎜⎜
⎜⎜
⎜⎜
⎜⎜
⎝ a1
...
ai
...
aδ
⎞
⎟⎟
⎟⎟
⎟⎟
⎟⎟
⎠
=
⎛
⎜⎜
⎜⎜
⎜⎜
⎜⎜
⎝
a1,0 · · · a1,j · · · a1,2δ−1
... ... ...
ai,0 · · · ai,j · · · ai,2δ−1
... ... ...
aδ,0 · · · aδ,j · · · aδ,2δ−1
⎞
⎟⎟
⎟⎟
⎟⎟
⎟⎟
⎠
ࠍδᰴరk-ࡌࠢ࠻࡞ⓨ㑆W ߩⴕߣ߱㧚W ߩ
ⴕߪၮᐩࠍขࠅᣇߦଐሽߔࠆ߇㧘หߓW ࠍ
ߔߎߣߦᵈᗧߔࠆ(ߘࠇࠄߪߦ⋧ૃߢࠆ)㧚วᚑ ౮ߦࠇࠆ2⇟⋡ߩ౮ߪ㧘k-ࡌࠢ࠻࡞ⓨ㑆W ߦ ߘߩⴕࠍኻᔕߐߖࠆ౮ߢࠆ㧚ߎߎߢห୯㑐 ଥ∼ߪ⋧ૃࠍߔ㧚ᦨᓟߩ౮ߪ㧘AW ߦPl¨uckerᐳ
ᮡࠍኻᔕߐߖࠆ౮ߢࠆ㧚ߎߎߢπi1···iδߪAW ߩ i1, . . . , iδ ࠍㆬࠎߢߢ߈ࠆዊⴕߩ୯ࠍߔ㧚
GࠍടᴺࠍṶ▚ߣߔࠆඨ⟲ߣߒ㧘ടᴺߦ㑐ߔࠆන
ర0ࠍᜬߟ߽ߩߣߔࠆ㧚ടᴺࠍṶ▚ߣߔࠆඨ⟲M ߇ G-ඨ⟲ߢࠆߣߪ㧘GߩరߣM ߩరߣߩടᴺ
+ :G×M −→M
∈ ∈
(a, x)�−→a+x ߇ቯ⟵ߐࠇߡߡ㧘Gߩනర0ߦኻߒߡ
0 +x=x
߇છᗧߩM ߩరxߦኻߒߡᚑࠅ┙ߟߎߣߢࠆ㧚߹
ߚSࠍM ߩㇱಽ㓸วߣߔࠆߣ߈㧘SࠍMߩోߡ ߩG-ㇱಽඨ⟲ߩᣖ{Nλ}λ∈Λߦኻߒߡ
[S]G:= �
λ∈Λ
Nλ
ࠍSߢ↢ᚑߐࠇߚG-ㇱಽඨ⟲ߣ߁㧚․ߦG-ㇱಽඨ
⟲M ߦኻߒߡ㧘M = [S]Gߣߥࠆࠃ߁ߥㇱಽ㓸วS ࠍM ߩGߩ↢ᚑ♽ߣ߁㧚
ዪᚲⅣOߩࠗ࠺ࠕ࡞Iߦኻߒߡ㧘㓸ว Γ(I) :={ord(f)|f ∈I}
ࠍ㧘Iߩඨ⟲ߣ߱㧚ߎߩΓ(I)ߪΓ-ඨ⟲ߢࠆ㧚Γ(I) ߩΓߩ↢ᚑ♽߇{p1,· · · , ps}ߢࠆߣ߈㧘ࠗ࠺ࠕ࡞I ࠍ(p1,· · · , ps)ဳߢࠆߣ߁㧚(p1,· · · , ps)ဳߢࠆ
ࠗ࠺ࠕ࡞ోߩ㓸วࠍ⸥ภJ(p1,· · · , ps)ߣߔ㧚2ߟ ߩࠗ࠺ࠕ࡞I1ߣI2߇หߓဳࠍᜬߟߩߪ㧘Γ(I1) = Γ(I2) ߇ᚑ┙ߔࠆᤨ㧘߆ߟߘߩᤨߩߺߦ㒢ࠆߎߣߦᵈᗧߔࠆ㧚
㗴 5. ዪᚲⅣOߩࠗ࠺ࠕ࡞I߇Mrߦዻߔࠆߩߪ㧘 㑐ଥᑼ�{Γ\Γ(I)} =r߇ᚑ┙ߔࠆᤨ㧘߆ߟߘߩᤨߩ ߺߦ㒢ࠆ.
― 64 ―
津 山 高 専 紀 要 第 5 2 号 ( 2 0 1 0 )
― 65 ―
単項曲線の特異点に対するr点のヒルベルトスキームについて 渡利
㗴5ߩ⸽ߪ◲නߥߩߢ⋭⇛ߔࠆ㧚
ᵈᗧ6. 㗴5ߪ㧘ࠗ࠺ࠕ࡞ߩᰴరߪဳߦࠃߞߡ
ࠄࠇࠆࠍ␜ໂߒߡࠆ㧚
㗴 7. ߽ߒ J(p1,· · · , ps) ⊂ Mr ߢࠇ߫, (ψ ◦ φr)(J(p1,· · · , ps))ߪࠕࡈࠖࡦⓨ㑆ߢࠆ㧚
(⸽)I⊂ Oࠍ(p1,· · ·, ps)ဳߢᰴర߇rߩࠗ࠺ࠕ
࡞ߣߔࠆ㧚ߎߎߢδᰴరk-ࡌࠢ࠻࡞ⓨ㑆φr(I)ߩၮᐩ ࠍ�f1,· · · , fδ�kߣ߅ߊ㧚ߚߛߒ
fi=xdi+
2δ�−1 j=di+1
ai,jxj, (1≤i≤2δ).
ߔࠆߣⴕAφr(I)ߪ㧘ᰴߩࠃ߁ߥ◲⚂㓏Ბⴕߦ ߣࠆߎߣ߇ߢ߈ࠆ㧚
⎛
⎜⎜
⎜⎜
⎜⎝
0· · · 0 1 · · · a1,d2 · · · a1,dδ · · · a1,2δ−1 1 · · · a2,dδ · · · a2,2δ−1
0
... ... ...1 · · · aδ,2δ−1
⎞
⎟⎟
⎟⎟
⎟⎠
ߎߩᤨ㧘ၮᐩf1߆ࠄfδ ߩߘࠇߙࠇߩవ㗡㗄ߦኻᔕ ߔࠆPl¨uckerᐳᮡߪ
πΞ= 1, (ߎߎߢΞ ={d1+ 1,· · ·, dδ+ 1}) ߣߥࠆ㧚
߹ߚછᗧߩᷝ߃ሼ㓸วΛߦኻߒߡ㧘 (1) πΛ∈k[ai,j|1≤i≤δ, d1≤j≤dδ]
ߣߥࠆߎߣߦᵈᗧߔࠆ㧚ᦝߦAφr(I)ߩછᗧߩᚑಽai,j
(1≤i≤δ, d1≤j≤dδ)ߦኻߒߡ (2) πΛ= (−1)lai,j
ߣߥࠆ⥄ὼᢙlߣᷝ߃ሼ㓸วΛ߇߆ߥࠄߕሽߔࠆ㧚
߃߫Λ = {d1 + 1, . . . , di−1 + 1, j + 1, di+1 + 1, . . . , dδ+ 1} ߣߔࠇ߫ࠃ㧚
ᐳᮡπΞߢหᰴᐳᮡࠍ㕖หᰴൻߒ㧘ߘߩ㕖หᰴᐳᮡࠍ
XΛ= πΛ
πΞ
ߣ߅ߊ㧚ߎߩᤨ㧘ઍᢙᄙ᭽(ψ◦φr)(J(p1,· · · , ps)) ߩࠕࡈࠖࡦᄙ᭽ߣߒߡߩቯ⟵ࠗ࠺ࠕ࡞ࠍJߣߔࠆߣ㧘 ߘߩᐳᮡⅣ
k[X1,2···,δ,· · · ,X∨Ξ,· · ·, Xδ+1,···,2δ]/J
ߪㆡᒰߥᄙ㗄ᑼⅣߣหဳߦߥࠆ㧚ࠃߞߡઍᢙᄙ᭽(ψ◦ φr)(J(p1,· · ·, ps))ߪ㧘ㆡᒰߥᰴరߩࠕࡈࠖࡦⓨ㑆ߣห
ဳߦߥࠆ㧚
ᰴߩ㗴ߪ㧘ࠃߊ⍮ࠄࠇߚታߢࠆ㧚
㗴 8 ([2], p.26, Corollary 4.5). છᗧߩᣢ⚂ߥ2ߟߩ ઍᢙᄙ᭽XߣY ߦኻߒߡ㧘ᰴߩ᧦ઙߪห୯ߢࠆ㧚 (1) XߣY ߪ㧘ℂห୯
(2) XߣY ߩߘࠇߙࠇߩ㐿㓸วU,V ߢ㧘U ∼=V ߣߥ ࠆ߽ߩ߇ሽߔࠆ
(ቯℂ4ߩ⸽)౮ψ◦φrߪනߥߩߢ㧘એਅߢߪ J(p1,· · ·, ps)ߣ(ψ◦φr)(J(p1,· · · , ps))ࠍห৻ⷞߔࠆ㧚 છᗧߩဳ(p1,· · · , ps)ߦኻߒߡJ(p1,· · · , ps)ߪ㧘㗴7 ࠃࠅㆡᒰߥࠕࡈࠖࡦⓨ㑆ߦหဳߢࠆ㧚ᦝߦ㗴8ࠃ ࠅJ(p1,· · · , ps)ߪㆡᒰߥᓇⓨ㑆ߣℂห୯ߣߥ
ࠆ.
ෳ⠨ᢥ₂
[1] G. Pfister, J.H.M. Steenbrink, “Reduced Hilbert schemes for irreducible curve singularities”, J.
Pure and Applied Algebra.77, 103-116, (1992).
[2] R. Hartshorne, “Algebraic Geometry”, Springer, (1977)
[3] J. P. Serre, “Groupes Alg´ebriques et Corps de Classes”, Hermann, Paris, 1959.
[4] ⋧㚍⧐♿, “ᐔ㕙ᦛ✢ߩන⚐․⇣ὐߦኻߔࠆrὐߩ ࡅ࡞ࡌ࡞࠻ࠬࠠࡓ”,ၯ₹ᄢቇℂᎿቇ⎇ⓥ⑼ᢙℂ 㔚ሶᖱႎኾ ୃ჻⺰ᢥ(2010)
― 64 ―
津 山 高 専 紀 要 第 5 2 号 ( 2 0 1 0 )
― 65 ―
単項曲線の特異点に対するr点のヒルベルトスキームについて 渡利
