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Intersections maximales de quadriques réelles
Arnaud Tomasini
To cite this version:
Arnaud Tomasini. Intersections maximales de quadriques réelles. Géométrie algébrique [math.AG].
Université de Strasbourg, 2014. Français. �NNT : 2014STRAD035�. �tel-01076980v2�
UNIVERSITÉ DE STRASBOURG
ÉCOLE DOCTORALE ED269
Institut de Recherche Mathématique Avancée
THÈSE
présentée par :
Arnaud TOMASINI
soutenue le :
10 novembre 2014
pour obtenir le grade de :
Docteur de l’université de Strasbourg
Discipline/ Spécialité
: MATHÉMATIQUES
Intersections maximales de
quadriques réelles
THÈSE dirigée par :
KHARLAMOV Viatcheslav
Professeur, Université de Strasbourg
RAPPORTEURS :
FINASHIN Sergey
Professeur, Middle East Technical University
MANGOLTE Frédéric
Professeur, LAREMA
AUTRES MEMBRES DU JURY :
ITENBERG Ilia
Professeur, Université Pierre et Marie Curie
(VC, τ ) VC τ 16 VC τ (VC, τ ) bi(VR) i VR Hi(VR) Z2 bi(VC) VC Hi(VC) bi(VK) Hi(VK)
X i≥0 bi(VR) ≤ X i≥0 bi(VC) XC (r + 1) PN C XC hj,i(XC) = hi,j(XC) i ≤ j δi,j+ r+1 X k=1 r + 1 k "kX−1 ℓ=0 k − 1 ℓ k − 1 j − i + ℓ j + ℓ k − 1 # i + j = N − r − 1 δi,j n m = 0 n < m hj,i(X C) = hi,j(XC) = δi,j i + j 6= N − r − 1 |i − j| > r + 1 r + 1 (r + 1) (Q0, . . . , Qr) Rr+1 −→ S2(RN+1)∗ (ω0, . . . , ωr) 7−→ Pri=0ωiqi S2(RN+1)∗ RN+1 Pr i=0ωiqi = ωq ωq Sr ind imin imax Sr
Ωi = {λ ∈ Sr ind(λ) + corang(ωq) ≤ i} ∅ = Ω−1⊂ Ω0⊂ · · · ⊂ ΩN+1= Sr RN+1 Λi(N ) = {q ∈ P(S2(RN+1)∗) λi(Q) 6= λi+1(Q)} {λi(Q)}0≤i≤N Q q ξi Λi(N ) q ∈ Λi(N ) RN+1 λ0(Q), . . . , λi(Q) Ri+1 // E i+1 ξi Λi(N ) γi π : ω 7→ ωq ω ∈ Sr XR (r+1) PN R Hi+j(PN R \ XR) E2i,j= Hi(Ω N−j) di,N2 −j : Hi(Ω j) δ(γj+1)∪ // Hi+2(Ω j+1, π−1(Λj+1)) // Hi+2(Ωj+1) δ H1(π−1(Λj+1)) → H2(Ω j+1, π−1(Λj+1)) (Ωj+1, π−1(Λj+1)) Z2 Z2 0 0 0 0 0 0 Z2 Z2 0 0 0 j = imin j = imax i = 0 i = r i j ∗ Ωi6= Sr
(q1, q2) PN C q1 0 ∈ CN+1 CN+1 (q 1, q2)
(λq1+ µq2)(x) = (λa1+ µb1)x21+ (λa2+ µb2)x22+ · · · + (λam+ µbm)x2m+
+ m+MX i=m+1 (λαi+ µβi)(xi+ ıxi+M)2+ (λ ¯αi+ µ ¯βi)(xi− ıxi+M)2 ai bi αiβ−1i 6∈ R m m = m1+ · · · + m2k+1 m mi Pm m PN R Pm m 1 ≤ m ≤ N + 1 m = N + 1 2 m 1 ≤ m ≤ N + 1 m = N +1 2 XR PRN N ≥ 3 XR imin= N 2 m = N + 1 m C ⊂ P1 R N + 1 1 N 1 2 N N N
2⌊N4⌋−1+ 1 N + 1 X m|N +1 2 m ϕ(m)2N +12m −1 ϕ(m) N N + 1 1 2 N = 7 8 = 2+2+2+1+1 8 = 2 + 1 + 2 + 1 + 2 16 d0,N −imin 2 d2 XR PRN N = 2k + 1 XR imin= k
XR PRN N = 2k XR d2 imin= k − 1 d2 imin= k (M − 1) d2
imin= k Himin−1(XR) → Himin−1(P
N R)
(M − 2) d2
imin = k Himin−1(XR) → Himin−1(P
N R) (C, θ) C θ 3 4 P3 R P3 R 8 7 P3 R 7 F1 F2 f : PR3× [0, 1] → PR3
f (F1, 0) = F1 f (F1, 1) = F2 t ∈ [0, 1] f (F1, t) 7 P3 R P3 R P4 R P4 R
P3 R
P4 R
G p 1 Z/pZ Zp K G G n n K G (g0, . . . , gn) G (v0, . . . , vn) K g ∈ G i g(vi) = gi(vi) ZpG = X gi∈G nigi, ni∈ Zp KG= { } K KG g G σ = 1 + g + g2+ · · · + gp−1 τ = 1 − g
ZpG στ = 0 = τ σ G p gp = 1 σ = τp−1 τ = σ p = 2 ρ = τk 1 ≤ k ≤ p − 1 ρ = τ¯ p−k L ⊂ K G LG = L ∩ KG ρC(K, L) ⊂ C(K, L) 0 −→ ¯ρC(K, L) ⊕ C(KG, LG)−→ C(K, L)i −→ ρC(K, L) −→ 0ρ i ker ρ = i C(K, L) ρ i C(K, L) Zp n K L s n K L s ∈ KG τ (s) = 0 s i ρ s 6∈ KG P gi∈Gnigi(s) Λ = ZpG 0 −→ ¯ρΛ−→ Λi −→ ρΛ −→ 0ρ
dim ¯ρΛ + dim ρΛ = dim Λ = p
ker(τ : Λ → Λ) = hσi dim τ Λ = dim Λ − 1 ker τ = Zpσ ⊂ τ Λ σ = τp−1 = τkτp−k−1
dim τk+1Λ = dim τ (τkΛ) = dim τkΛ − 1
dim τkΛ = p − k H∗ρ(K, L) = H∗(ρC(K, L; Zp)) Hn(K, L) ρ∗ xx Hρ n(K, L) // Hnρ¯(K, L) ⊕ Hn(KG, LG) i∗ ii
K G L ⊂ K n ≥ 0 ρ = τk (Hρ n(K, L)) + X k≥n (Hk(KG, LG)) ≤ X k≥n (Hk(K, L)) k (Hkρ(K, L)) + (Hk(KG, LG)) ≤ (Hk+1ρ¯ (K, L)) + (Hk(K, L)) (Hk+1ρ¯ (K, L)) + (Hk+1(KG, LG)) ≤ (Hk+2ρ (K, L)) + (Hk+1(K, L)) ρ ρ¯ k ≥ n K VC c : VC→ VC (VC, c) VC c VR G = { , c} G VC L = ∅ n = 0 dim(H0c(VC; Z2)) + X i≥0 dim(Hi(VCG; Z2)) ≤ X i≥0 dim(Hi(VC; Z2)) VG C VR VC X i≥0 dim(Hi(VR; Z2)) ≤ X i≥0 dim(Hi(VC; Z2)). WC ⊂ VC VC c WC X i≥0 dim(Hi(VR, WR; Z2)) ≤ X i≥0 dim(Hi(VC, WC; Z2)) VR X i≥0 dim(Hi(VR; Z2)) = X i≥0 dim(Hi(VC; Z2)) VR (M − k) X i≥0 dim(Hi(VR; Z2)) = X i≥0 dim(Hi(VC; Z2)) − 2k
PN C (PN C , conj) PN R PRN (r + 1) XK PKN K= R C (r + 1) q0, . . . , qr XK= {x ∈ PKN q0(x) = · · · = qr(x) = 0}. (r + 1) (r + 1) 2 (q0, . . . , qr) Kr+1 −→ S2(KN+1)∗ (ω0, . . . , ωr) 7−→ Pr i=0ωiqi , S2(KN+1)∗ KN+1 Pr i=0ωiqi = ωq ˆ q : Pr K −→ P(S2(KN+1)∗) (ω0, . . . , ωr) 7−→ ωq . (q0, . . . , qr) ω ∈ Pr K x ∈ KN+1 x ∈ ker(ωq) i qi(x) = 0 ω ∈ Pr K ω ˆ q (r + 1) PN K N − r − 1 r < N
q = (q0, . . . , qr) RN+1 q ω0∈ Pr R ρ > 0 q B(ω0, ρ) ω0 ∈ PRr ker(ω0q) = {0} ω0q ρ > 0 ω ∈ B(ω0, ρ) ker(ωq) = {0} q B(ω0, ρ) ω0 ∈ Sr ker(ω0q) 6= {0} q ω0 x ∈ ker(ω0q) i 0 r qi(x) 6= 0 k dim ker(ωq) ≤ k U ω0 ω ∈ U dim ker(ωq) ≤ dim ker(ω0q)
U ε > 0 x′ ∈ ker(ωq) ω ∈ U x ∈ ker(ω 0q) ||x′ − x|| < ε x′ ∈ ker(ωq) ω ∈ U x ∈ ker(ω 0q) ||x′−x|| < ε q i qi(x) 6= 0 qi C qi(x′) = qi(x) + ||x′− x||C + o(||x′− x||2). ε U ω0 qi(x′) 6= 0 ρ > 0 q B(ω0, ρ) ⊂ U q = (q0, . . . , qr) RN+1 q ω 0∈ PRr q+ ρ > 0 ε > 0 τ 0 < τ < ε qτ = (q0− τ q+, q1, . . . , qr) B(ω0, ρ) q+ > 0 RN+1 XR⊂ RN+1 q SN ρ > 0 1 ρ = supx∈SN |q+(x)|. q ω0 x ∈ SN \ XR ε > 0 ωx ∈ B(ω0, ρ) |ωxq(x)| ≥ ε > 0 SN ε x ∈ SN \ X R q′ B(ω0, ρ) (ω, x) ∈ B(ω0, ρ) × SN x ∈ ker(ωqτ) q+ x 6∈ XR ωx∈ B(ω0, ρ) |ωxq(x)| ≥ ε τ < ε |ωxqτ(x)| ≥ |ωxq(x)| − τ ||ωx||.|q+(x)| > 0
qτ B(ω0, ρ) QN C = {x20+ x21+ · · · + x2N = 0} ⊂ PCN PN C bk= dim Hk(QNC; Z2) QNC QN C PCN i ≤ 2N bi(QNC) = 2 i = N − 1 bi(QNC) = 1 i 6= N − 1 bi(QNC) = 0 i k 6= N − 1 Hk(QN C; Z2) ≃ Hk(PCN; Z2). bi(QNC) i 6= N − 1 bN−1(QNC) QNC PCN−1 x2 0= −(x21+ · · · + x2N) Q N−1 C χ(QN C) = 2χ(P N−1 C ) − χ(Q N−1 C ) = 2N − χ(QNC−1). QN C χ(QNC) = N X i=0 (−1)kbk. bN−1= dim HN−1(QNC; Z2) = 2 N − 1 0
QR⊂ PRN (N++1, N−+1) N++ 1 N−+ 1 QR N+ ≥ N− N = N++ N−+ 1 QR QR ⊂ PRN (N+ + 1, N− + 1) N+≥ N−+ 1 bi(QR) = 1 0 ≤ i ≤ N− N+≤ i ≤ N − 1 0 N+= N− bi(QR) = 2 i = N+ 1 0 ≤ i ≤ N − 1 i 6= N+ N−= 0 QR SN−1 N− > 0 QR q(x) = x20+ x21+ · · · + x2N+− x 2 N++1− · · · − x 2 N q(x) = 0 x2 0+ · · · + x2N = 1 x2 0+ · · · + x2N+ = 1 2 x 2 N++1+ · · · + x 2 N = 1 2 q(x) = 0 SN+× SN− x 7→ −x SN+× SN− SN+ SN− SN+× SN−→ SN− π : QR→ P N− R SN+ Hi(Q R) E2p,q = Hp(P N− R ; H q(SN+))
p q N+ 0 N− 0 Z2 0 0 Z2 Z2 0 0 Z2 . . . . . . . . . Z2 0 0 Z2 Z2 0 0 Z2 N−≤ N+ Hi(Q R; Z2) = M p+q=i Hp(PN− R ; Hq(SN+)) N+ = N− N+ = N− + 1 XR r r (M − r + 1) X i≥0 bi(QR) ≥ X i≥0 bi(QC) − 2(r − 1). PN C VN−r−1 (r +1) di PN C VN−r−1 P (y; VN−r−1) = NX−r−1 i=0 dim Hi(V N−r−1; C)yi
n ∈ N Vn (r + 1) di PCn+r+1 ∞ X n=0 P (y; Vn)zn+r+1= 1 (1 + zy)(1 − z) r+1Y i=1 (1 + zy)di− (1 − z)di (1 + zy)di+ y(1 − z)di (r + 1) di PCN hp,q(V N−r−1) = dim Hp,q(VN−r−1; C) hp,q0 (Vn) = hp,q(V n) = dim Hp,q(Vn, C) p 6= q, hp,p(V n) − 1 = dim Hp,p(Vn, C) − 1 . hp,q0 (Vn) d = (d0, . . . , dr) H(d; y, z) = X p,q≥0 hp,q0 (Vp+q)ypzq Vp+q r di PCp+q+r r H(d; y, z) = (1 + y) d−1− (1 + z)d−1 (1 + z)dy − (1 + y)dz H(d; y, z) = X P⊂[0,r] " (1 + y)(1 + z)|P |−1Y i∈P H(di; y, z) # 0 r Vn d PCn+1 X n≥0 P (u; Vn)vn= 1 (1 + uv)(1 − v)v× (1 + uv)d− (1 − v)d (1 + uv)d+ u(1 − v)d
y = uv z = −v X n≥0 P (u; Vn)vn = 1 (1 + y)(1 + z)× (1 + y)d− (1 + z)d −z(1 + y)d+ y(1 + z)d X n≥0 P (u; Vn)vn= X n≥0 n X p,q=0 (−1)qhp,q(Vn)upvn =X n≥0 n X p,q=0 (−1)qhp,q0 (Vn)upvn+ X n≥0 n X p=0 (−1)pupvn = n X p,q=0 (−1)qhp,q0 (Vp+q)upvp+q+ X n≥0 n X p=0 (−1)pupvn = n X p,q=0 hp,q0 (Vp+q)(uv)p(−v)q+ X p≥0 X n≥p (−1)n(uv)p(−v)n−p y = uv z = −v X n≥0 P (u; Vn)vn= n X p,q=0 hp,q0 (Vp+q)ypzq+ X p≥0 X n−p≥0 (−1)pyp(−1)n−pzn−p = n X p,q=0 hp,q0 (Vp+q)ypzq+ 1 (1 + y)(1 + z) X n≥0 P (u; Vn)vn= H(d; y, z) + 1 (1 + y)(1 + z) H(d; y, z) = 1 (1 + y)(1 + z) (1 + y)d− (1 + z)d y(1 + z)d− z(1 + y)d − 1 (1 + y)(1 + z) = (1 + y) d−1− (1 + z)d−1 y(1 + z)d− z(1 + y)d H(d; y, z) = 1 (1 + y)(1 + z) " r Y i=0 (1 + y)d− (1 + z)d y(1 + z)d− z(1 + y)d − 1 # r Y i=0 ai− 1 = X P⊂[0,r] Y i∈P (ai− 1)
H(d; y, z) = 1 (1 + y)(1 + z) X P⊂[0,r] Y i∈P (1 + y)di− (1 + z)di y(1 + z)di− z(1 + y)di − 1 = X P⊂[0,r] (1 + y)(1 + z)|P |−1Y i∈P H(di) (r + 1) PN C (r + 1) 2 XC(r) (r + 1) PN C XC(r) p < q p + q = N − r − 1 hp,q0 (XC(r)) = r+1 X k=1 r + 1 k "kX−1 ℓ=0 k − 1 ℓ k − 1 q − p + ℓ q + ℓ k − 1 # , n m = 0 n < m XC(r) p + q 6= N − r − 1 XC(r) p + q 6= N − r − 1 2 H(2) = 1 1 − yz = X k≥0 (yz)k (r + 1) PN C XC(r) H(2) = r+1 X k=1 r + 1 k (1 + y)(1 + z)k−1 H(2)k H(2)k = 1 1 − yz k =X ℓ≥0 ℓ + k − 1 k − 1 (yz)ℓ (1 + y)(1 + z)k−1= k−1 X ℓ=0 ℓ X m=0 m X p=0 k − 1 ℓ ℓ m m p yℓ−pzm
p < q p + q = N −r −1 H(2) hp,q0 (XC(r)) r+1 X k=1 q X ℓ=q−k+1 k−1 X m=q−ℓ r + 1 k ℓ + k − 1 k − 1 k − 1 m m q − ℓ q − ℓ ℓ + m − p hp,q0 (XN−r−1) = r+1 X k=1 r + 1 k "k−1X ℓ=0 k − 1 ℓ k − 1 q − p + ℓ q + ℓ k − 1 # 6 P12 C PC13 • • • • • • • p hp,6−p0 (XC) 0 1 6 r = 5 N = 12 • • • • • • • • p hp,7−p0 (XC) 0 1 7 r = 5 N = 13 (r + 1) PN C (r + 1) p = q = N−r−1 2 (r + 1) XC PN C X i≥0 bi(XC) = 2N N 2(N − 1) N
hp,q(X C) p + q = N − 2 = dim(XC) p = q ≤ N − 2 hp,p(X C) = 1 2p = N − 2 N = 2k X i≥0 bi(XC) = hk0−1,k−1(XC) + 2k − 1 = 2k + 1 + 2k − 1 = 4k = 2N. N = 2k + 1 X i≥0 bi(XC) = 2hk0−2,k−1(XC) + 2k = 2k + 2k = 4k = 2(N − 1). XC PN C X i≥0 bi(XC) = N2− 4 N N2− 1 N hp,q(X C) p + q = N − 3 = dim(XC) p = q ≤ N − 3 hp,p(X C) = 1 2p = N − 3 N = 2k X i≥0 bi(XC) = 2hk−2,k−1(XC) + 2k − 2 = 4k2− 2k − 2 + 2k − 2 = 4k2− 4 = N2− 4. N = 2k + 1 X i≥0 bi(XC) = hk−1,k−1(XC) + 2hk−2,k(XC) + 2k − 2 = 3k2+ 3k + 2 + k(k − 1) + 2k − 2 = 4k2+ 4k = N2− 1. q = (q0, q1, . . . , qr) (r + 1)
ˆ q : Pr C −→ P(S 2(CN+1)∗) (ω0, . . . , ωr) 7−→ ωq . CC CC= {ω ∈ PCr det(ωq) = 0}. CC PCr CC= PCr XC CC PCr N + 1 q0, q1, . . . , qr R ˆ q : Pr R −→ P(S2(RN+1)∗) (ω0, . . . , ωr) 7−→ ωq , CR CC R ω CC PN C {ωq = 0} (q0, q1) q1 C ω [1 : 0] PC1 (q0+ λq1)λ∈C {q0= 0} x0∈ CN+1 ker(q0) q1(x0) 6= 0 x⊥0 x0 q1 CN+1 λ ∈ C CN+1 = Cx0⊕ x⊥0
det(q0+ λq1) = det((q0+ λq1)|Cx0) det((q0+ λq1)|x⊥0)
[1 : 0] ∈ P1 C λ = 0 det(q0+λq1) det((q0)|x⊥ 0) = 0 ker(q0) x0 x06∈ x⊥0
q0 x0 ∈ ker(q0) q1 q1(x0) = 0 x0 λ = 0 det(q0+ λq1) ω q1 q1 x⊥0 x1∈ CN+1 q1(x0, x1) = 1 x1 x1 x0 q1(x1) = 0 x0 q1 x⊥1 V = x⊥0 ∩ x⊥1 2 x0 x1 (x2, . . . , xN) V v ∈ V q1(x1, v) = q1(x0, v) = 0 (x0, . . . , xN) CN+1 q 0+ λq1 0 λ 0 . . . 0 λ c1 c2 . . . cN 0 0 c2 cN M ci = q0(x1, xi) M (q0+ λq1)|V det(q0+ λq1) CC (r + 1) XC XC ω ∈ CC ωq XC ω ∈ CC CC CC ω ∈ CC ω {ωq = 0} XC C CC r ≥ 3 Rr+1 → S2(RN+1)∗ r S2(RN+1)∗ S2(RN+1)∗ D(N ) CR ˆ q 1 ≤ k ≤ N + 1 Dk(N ) = {q ∈ S2(RN+1)∗ corang(q) = k}
q0∈ S2(RN+1)∗ V0= ker(q0) U q0 S2(RN+1)∗ Φ : U → S2(V0)∗ q ∈ U Φ(q0) = 0; corang(q) = corang(Φ(q)); dq0Φ(q) = q|V; i −(q) = i−(q 0) + i−(Φ(q)). i−(q) γ q0 γ C γ q0 U q0 S2(RN+1)∗ q ∈ U q C\ γ q ∈ U πq Vγ(q) γ πq0|V0= idV0 Φ(q) = q ◦ πq|V0. Dk(N ) k(k + 1)/2 S2(RN+1)∗ 1 ≤ k ≤ N + 1 Dk(N ) =Sj≥kDj(N ) Dk+1(N ) D(N ) 3 (r + 1) r ≥ 3 C R x2 0+ x21− 2x22+ x23= 0 −x2 0− x21+ x22+ x23= 0 2x0x1− 2x22+ x23= 0 −2x0x1+ x22+ x23= 0 R C
C ˜ C ⊂ Sr C R Pr R Sr ˜ C = {ω ∈ Sr det(ωq) = 0} indq : Sr→ Z ωq ind indq XR ind ind Sr\ C ind ±1 C
ind(−ω) = N + 1 − [ind(ω) + corang(ωq)]
imin= minω∈S2ind(ω) imax= maxω∈S2ind(ω) = N + 1 − imin
ind XR PN R \XR L+R = {(ω, x) ∈ S r× PN R (ω.q)(x) > 0} ⊂ Sr× PRN. Sr× PN R → PRN L+R → PRN \ XR p Sr× PN R → PRN x ∈ XR L+R L+R PRN \ XR L+R → PRN\XR x ∈ PRN\XR p−1(x) = {ω ∈ Sr (ω.q)(x) > 0}
Sr ∅ = Ω−1 ⊂ Ω0⊂ · · · ⊂ ΩN+1 = Sr Ωi = {ω ∈ Sr ind(ω) + corang(ωq) ≤ i} U × [0, 1] → S2(RN+1)∗ (ω, t) 7→ ωqt U ⊂ Sr q t t ∈ [0, 1] Lt= {(ω, x) ∈ U × PRN qˆt(ω)(x) > 0} L0 L1 L0 L1 Ft Zt U × PRN F0 = id Ft(L0) ⊂ Lt Gt G0= id Gt(L1) ⊂ L1−t Zt U × PRN Zt= Xt+ Yt ω, x) ∈ U × PRN Xt(ω, x) PN R Yt(ω, x) U Zt (ω, x, t) ∈ U ×PN R ×[0, 1] ωqt(x) = 0 ωqt(x) > 0 Zt(ω, x) = 0 x 6∈ ker(ωqt) Yt(ω, x) = 0 Xt(ω, x) PN R x PN R ωqt(x + Xt(ω, x)) > 0 x ∈ ker(ωqt) Xt(ω, x) = 0 qt i qt,i(x) 6= 0 qt = (qt,0, . . . , qt,r) Yt(ω, x) qt,i (t, ω, x) ωqt(x) = 0 F0= id Ft(L0) ⊂ Lt Gt G0= id Gt(L1) ⊂ L1−t F1◦ G1 : L1 → L1 G1◦ F1 : L0 → L0 L0 L1 q U ⊂ Sr Lq(U ) = {(ω, x) ∈ U × PRN ωq(x) > 0}. Lq(U ) Lq′(U )
q q′ q q ω0 ∈ Sr ρ0 > 0 U1 ⊂ U2 diam(U2) < ρ0 Lq(U1) ⊂ Lq(U2) q ω0 ∈ Sr ρ0 > 0 q B(ω0, ρ0) ρ1 < ρ2 ≤ ρ0 B(ω0, ρ1) ⊂ U1⊂ U2⊂ B(ω0, ρ2) ρ1≤ ρ ≤ ρ2 q B(ω0, ρ) ρ B(ω0, ρ2) q = (q0, . . . , qr) q ω0∈ Sr U ω0 Lq(U ) N − ind(ω0) − corang(ω0q) ker(ω0q) = {0} ker(ωq) = {0} ρ > 0 ω ∈ B(ω0, ρ) ker(ωq) = {0} ρ q B(ω0, ρ) ρ Lq(B(ω0, ρ)) Lq({ω0}) Lq({ω0}) = {x ∈ PN R ω0q(x) > 0}
N − ind(ω0) = N − ind(ω0) − corang(ω0q)
ker(ω0q) 6= {0} q qi ω0 = (1, 0, . . . , 0) ∈ Sr q+ ρ > 0 ε > 0 0 < τ < ε qτ = (q0− τ q+, q1, . . . , qr) B(ω0, ρ) q+ τ > 0 q0− τ q+
ker(ω0qτ) = {0} indqτ(ω0) = indq(ω0) + corang(ω0q)
U ω0 Lqτ(U )
N − indq(ω0) − corang(ω0q)
Lq(U ) Lqτ(U )
RN+1
{λi(Q)}0≤i≤N Q q ξi Λi(N ) q ∈ Λi(N ) RN+1 λ0(Q), . . . , λi(Q) Ei Ri+1 // Ei ξi Λi(N ) γi ξi XR (r + 1) PRN L + R Hi+j(L+ R) E2i,j= Hp(Ω N−j) di,N2 −j: Hi(Ωj) δ(γj+1)∪ −−−−→ Hi+2(Ω j+1, ˆq−1(Λj+1)) → Hi+2(Ωj+1) δ H1(ˆq−1(Λ j+1)) → H2(Ωj+1, ˆq−1(Λj+1)) (Ωj+1, ˆq−1(Λj+1)) βg : L+R → Sr L+R E2i,j= Hi(Sr, Fj) Fj U 7→ Hj(β−1 g (U )) U ⊂ Sr ω ∈ Sr Fj ω lim −−−→ ω∈U Fj(U ) β−1 g (U ) = Lq(U ) lim −−−→ ω∈U Fj(U ) = HjPN−ind(ω)−corang(ωq) R . ω ∈ Sr Z 2 j ≤ N − ind(ω) − corang(ωq) ω ∈ ΩN−j ω 6∈ ΩN−j E2i,j= Hi(ΩN−j)
Ωi XR (r + 1) PN R XR χ(XR) = 1 2 1 + (−1) N+1+ N X j=0 (−1)j+1χ(Ωj). (PN R, XR) χ(XR) = χ(PRN) − (−1)Nχ(PRN\ XR) PN R \ XR E2i,j χ(PN R \ XR) = N X j=0 (−1)j+Nχ(Ωj). Ωi {ω ∈ Sr ind(ω) ≤ i} E20,j = E r,j 2 = Z2 j < imin E2i,j= 0 j < imin 1 ≤ i ≤ r − 1
E2i,j= 0 imin≤ j < imax i ≥ r
E2i,j= 0 j ≥ imax
E2i,j = Hi(ΩN−j)
j < imin N − j > N − imin = imax− 1
Ωimax= S
r⊂ Ω
Z2 Z2 0 0 0 0 0 0 Z2 Z2 0 0 0 q = imin q = imax p = 0 p = r p q ∗ Ωi6= Sr (r + 1) PN R (r + 1) PN R imin= imax r ≤ N2+1 Hi(L+ R) = Hi(PR\XR) Hi(L+ R) = 0 i > N L+R r + imin− 1 = r + N + 1 2 . PN R \ XR
ℓ
(X
R)
ℓ(XR) Hp(XR) → Hp(PRN) Hp(XR) → Hp(PRN) p ≤ ℓ(XR) q µ = min{N−+ 1, N++ 1} > 0 N−+ 1, N++ 1 Hi(Q) → Hi(PRN)Q ⊂ PN R q(x) = 0 0 ≤ i ≤ µ − 1 q −q Q µ = N−+ 1 Hi(Pµ−1 R ) → Hi(Q) 0 ≤ i ≤ µ − 1 π : Q → PRN− SN+ π∗: Hi(Pµ−1 R ) → Hi(Q) 0 ≤ i ≤ µ − 1 H Q ⊂ PN C x H1(Pµ−1 R ) π∗(t) = [HR]∗ [HR]∗ ∈ H1(Q R) [HR] ∈ HN−2(QR) ([HR]∗)i= π∗(ti) 0 ≤ i ≤ µ − 1 i ≥ µ XR (r + 1) PN R imin ℓ(XR) imin6= imax imin− r ≤ ℓ(XR) < imin q0, . . . , qr XR ω ∈ Sr ind(ω.q) = imin Q ω.q = 0 QR⊂ PRN Hi(Q) → Hi(PRN) 0 ≤ i ≤ imin− 1 Hi(XR) → Hi(PRN) i ≥ imin ℓ(XR) < imin (XR, PRN) → HN−i−1(L+ R) → Hi(XR) → Hi(P N R ) → HN−i(L + R) → HN−i(L+ R) = 0
Z2 Z2 0 0 0 0 0 0 Z2 Z2 0 0 0 q = imin q = imax p = 0 p = r p q ∗
Ωi6= Sr HN−i(L+R) = 0 N −i ≥ imax+r−1
i ≤ N + 1 − r − imax= imin− r XR (r + 1) PN R XR imin 6= imax ℓ(XR) ≥ dim(X R) 2 = N − r − 1 2 ⌊x⌋ x X (r + 1) Hi(XC) → Hi(PCN) i < dim(XC) = N − r − 1 i = dim(XC) → Hi(PCN) ϕi → Hi(PCN, XC) → Hi−1(XC) → Hi−1(PCN) ϕi−1 → 0 → ker ϕi→ Hi(PCN) ϕi
−→ Hi(PCN, XC) → Hi−1(XC) → ker ϕi−1 → 0.
dim H∗(PCN, XC) = dim H∗(PCN) + dim H∗(XC) − 2 dim ker ϕ∗
≤ dim H∗(PCN) + dim H∗(XC) − 2 N − r − 1 2 + 1 ℓ(XR)
PN R XR ℓ(XR) ≥ N − r − 1 2 XR (r + 1) (r + 1) (M − r + 1) X i≥0 dim(Hi(QR; Z2)) ≥ X i≥0 dim(Hi(QC; Z2)) − 2(r − 1) XR (r + 1) imin > j dim(XR) 2 k = N−r−1 2 (r + 1) XR (r + 1) PN R r ≤ N+12 p q imin 0 r 0 Z2 Z2 0 0 0 0 . . . . . . . . . 0 0 0 Z2 Z2 0 ξ : L+R → Sr PN R \ XR
ℓ(XR) (r + 1) r ξ wr(ξ) ∈ Hr(Sr, Z2) wr(ξ) = 0 ℓ(XR) = N2−1 ℓ(XR) =N2−1− r (XR, PRN) Hi(XR) → Hi(PRN) → Hi(PR, XR) ℓ(XR)
HN−i(PRN) → HN−i(PRN\ XR) ∼= HN−i(L+R)
t H1(PN R) tN−i HN−i(PN R ) tN−i p∗(tN−i) p : L+ R → PRN ℓ(XR) = imin− 1 p∗(timin) = 0 p∗(timin) = w r(ξ) (r + 1) XR (r + 1) PRN r = N2+1 wr(ξ) 6= 0 XR r < N2+1 1 wr(ξ) = 0 bi(XR) = 1 0 ≤ i ≤ N − 3 2 − r bi(XR) = 2 N − 1 2 − r ≤ i ≤ N − 1 2 bi(XR) = 1 N + 1 2 ≤ i ≤ N − r − 1 2 wr(ξ) 6= 0 bi(XR) = 1 0 ≤ i ≤ N − 1 2 − r bi(XR) = 0 N + 1 2 − r ≤ i ≤ N − 3 2 bi(XR) = 1 N − 1 2 ≤ i ≤ N − r − 1
L+R ℓ(XR) p q imin 0 r 0 Z2 Z2 0 0 0 0 . . . . . . . . . 0 0 0 Z2 Z2 0 L+R dim Hi(L+ R) = 1 0 ≤ i ≤ r − 1 dim Hi(L+R) = 2 r ≤ i ≤ N − 1 2 dim Hi(L+ R) = 1 N + 1 2 ≤ i ≤ N − 1 2 + r (XR, PRN) ℓ(XR) i < ℓ(XR) 0 → HN−i−1(L+ R) → Hi(XR) → Hi(P N R) → 0 i = ℓ(XR) 0 → Hℓ(XR)+1(P N R ) → HN−ℓ(X R)−1 (L+R) → Hℓ(XR)(XR) → Hℓ(XR)(P N R ) → 0 i > ℓ(XR) 0 → Hi+1(PRN) → HN−i−1(L+R) → Hi(XR) → 0 XR bi(XR) = dim HN−i−1(L+R) + 1 0 ≤ i < ℓ(XR) bi(XR) = dim HN−i−1(L+R) i = ℓ(XR) bi(XR) = dim HN−i−1(L+R) − 1 ℓ(XR) < i ≤ N − r − 1 r = (N + 1)/2 HN(L+ R) 1
H0(XR) → H0(PRN) ℓ(XR) < 0 wr(ξ) 6= 0 (r+1) PN R N = 3 r = 1 x2 0− x21+ x22− x23= 0 2x0x1+ 2x2x3= 0 . P3 R S1
q0, q1 C CC⊂ PC1 XR Ωi Ω˜i = {ω ∈ Sr ind(ω) ≤ i} ˜ Ωi Hp(Ω i) 1 N ≥ 2 0 ≤ p ≤ 1 Z2 Z2 Z2 Z2 q = imin q = imax p = 0 p = 1 p q
L+R PRN\ XR q0, q1 XR PN R XR imin< (N − 1)/2 bi(XR) = 1 i ≤ imin− 2 bimin−1(XR) = dim H 0(Ω imin)
bi(XR) = dim H0(Ωi+1) − 1 imin≤ i ≤ imax− 3
bimax−2(XR) = dim H 0(Ω imax−1) bi(XR) = 1 imax− 1 ≤ i ≤ N − 2 imin= (N − 1)/2 = imax− 2 N XR bi(XR) = 1 i ≤ imin− 2 bimin−1(XR) = dim H 0(Ω imin) bimin(XR) = dim H 0(Ω imin+1) bi(XR) = 1 imin+ 1 ≤ i ≤ N − 2 imin= N/2 = imax− 1 N XR bi(XR) = 1 i ≤ imin− 2 bimin−1(XR) = dim H 0(Ω imin) + 1 bi(XR) = 1 imin≤ i ≤ N − 2 Hi(L+ R) r = 1 ℓ(XR) = imin− 1 ℓ(XR) Hi(XR) → Hi(PRN) (XR, PRN) → HN−i−1(L+ R) → Hi(XR) → Hi(PRN) → HN−i(L+R) → HN−i−1(L+ R) HN−i(L + R) Hi(XR) Hi(PRN)
Z2 Z2 Z2 Z2 q = imin q = imax p = 0 p = 1 p q N − i − 1 ≥ imax
i ≤ N −1−imax= imin−2
Hi(XR) = Z2 i ≤ imin− 2 Himin−1(XR) 0 → Himin(P N R) → HN−imin(L + R) → Himin−1(XR) → → Himin−1(P N R) → 0 Himin(XR) → Himin(P N R) imin> ℓ(XR) Himin−1(XR) → Himin−1(P N R) bimin−1(XR) = dim H N−imin(L+ R) = dim H0(Ωimin) i > ℓ(XR) 0 → Hi+1(PRN) → HN−i−1(L + R) → Hi(XR) → 0 i > imin− 1 bi(XR) = dim HN−i−1(L+R) − 1 L+R dim HN−i−1(L+ R) = 2 dim HN−i−1(L + R) Ωj 6= S1 imin < (N − 1)/2 bi(XR) = 1 i ≤ imin− 2 bi(XR) = 1 imax − 1 ≤ i ≤ N − 2 bimin−1(XR) = dim H 0(Ω imin)
bi(XR) imin− 1 ≤ i ≤ imax− 2 i 6= imax− 2
bi(XR) = dim HN−i−1(L+R) − 1 = dim H 0(Ω
i = imax− 2
bimax−2(XR) = dim H
imin(L+
R) − 1
= dim H0(Ωimax−1) + dim H
1(Ω imax) − 1 = dim H0(Ωimax−1) PN R q1(x) = a0x02+ · · · + aNxN2= 0 q2(x) = b0x02+ · · · + bNxN2= 0 XR Ai = abii ∈ R2 S = {Ai, i = 0, . . . , N } R2 (H1) : R2 [Ai, Aj] S XR N −2 R2 (H1) (H1) R2 (H1) XR (H1) m m1, . . . , mk m = m1+ · · · + mk k 0 0 = 0
N + 1 (H1) N + 1 (H1) R2 (H1) S A, B ∈ R2 S S A ∼S B A ∼ B C ∈ S A B C S = {A, C1, C2, C3} O A C1 C2 C3 B B′ A ∼ B A 6∼ B′ O ∈ [AB′C 2] S k k S S k (q1, q2) RN+1 (q1, q2) S = {B0, . . . BN} (q1, q2) (A1, A2, . . . , Ak) R2 k k = 2ℓ + 1 k > 1 Ai Ai+1 j = i + ℓ + 1 k j O Ai Ai+1 Aj R2 k > 1 A1, Ak Aℓ+1
O [A1, Ak, Aℓ+1] i = 1, . . . , ℓ
As(i) O [Ai, Ai+1, As(i)]
s(i) i s(i) [ℓ + 2, k] ℓ ≤ k − ℓ − 1 i = ℓ + 1, . . . , k − 1 ℓ ≥ k − ℓ − 1 k = 2ℓ + 1 s(i) s(i) = i + ℓ + 1 N + 1 N + 1 = k X i=1 Ni Ni = ♯ {C ∈ S C ∼ Ai} A1 Ak S (2k+1) Ai (2k + 1) (q1, q2) RN+1 q1(x) = m X i=1 aixi2+ m+MX i=m+1 2xixi+M ; q2(x) = m X i=1 bixi2+ m+MX i=m+1 (βi(xi2− xi+M2) + 2αixixi+M) . M = N−m+1 2 ai, bi αi, βi (q1, q2) q1 Q1, Q2 q1, q2 RN+1 (q1, q2) Q1−1Q2 C Q1−1Q2 CN+1 q1 q2 q1(x) = NX+1 i=1 aizi2 , q2(x) = NX+1 i=1 bizi2.
q1 q2 Q1−1Q2 e e¯ (z1, z2) λq1+ µq2(z1e + z2e) = (λa + µb)z¯ 12+ (λ¯a + µ¯b)z22 a b z1e + z2e¯ z2= ¯z1 z1= u + iv, a = a1+ ia2, b = b1+ ib2.
λq1+ µq2(z1e + ¯z1e) = 2(λa + µb)(u¯ 2− v2) + 4(λa2+ µb2)uv
b/a = α + iβ a = −i/2 λq1+ µq2(z1e + ¯z1e) = µβ(u¯ 2− v2) + 2(λ + µα)uv PN R m m N + 1 m ≡ N + 1( 2) q1 q2 RN+1 (q1, q2)
λq1+ µq2(x) =(λa1+ µb1)x21+ (λa2+ µb2)x22+ · · · + (λam+ µbm)x2m+
+ m+MX i=m+1 (λαi+ µβi)(xi+ ıxi+M)2+ (λ ¯αi+ µ ¯βi)(xi− ıxi+M)2 M = N−m+1 2 ai, bi∈ R βj−1αj6∈ R
(λa1+ µb1)x21+ (λa2+ µb2)x22+ · · · + (λam+ µbm)x2m.
m m
q1
q2
m ≤ N + 1
m m CR⊂ PR1 0 S1 S1 m = m1+ · · · + mk mi • O • • • • • • imin imin+ 1 imin+ 2 imin+ 3 imin+ 2 imin+ 3 imin+ 4 = imax 6 6 = 3 + 2 + 1 q1 q2 RN+1 (q1, q2)
λq1+ µq2(x) =(λa1+ µb1)x21+ (λa2+ µb2)x22+ · · · + (λam+ µbm)x2m+
+ m+MX i=m+1 (λαi+ µβi)(xi+ ıxi+M)2+ (λ ¯αi+ µ ¯βi)(xi− ıxi+M)2
(λ, µ) ∈ S1 M = N−m+1 2 ai, bi∈ R β−1j αj 6∈ R c > 0 det(λq1+ µq2) = 2c m Y i=1 (λai+ µbi). m+MY i=m+1 |λαi+ µβi|2 = 2c m Y i=1 (λai+ µbi).A A C f1: R2 −→ S1 (a, b) 7−→ √ b a2+b2, −a √ a2+b2 f2: R2 −→ S1 (a, b) 7−→ √−b a2+b2, a √ a2+b2 Ai = (ai, bi) ∈ R2 C˜ ω ∈ S1 ind(ω.q) = i min S1 ω 1 f1 1 f2 Ai, Aj Ai∼ Aj Ai Aj Aℓ f1(Aℓ) f1(Ai) f1(Aj) AiAj Aℓ •f1(Ai) • f1(Aj) i i + 1 i + 2 Aℓ f1(Aℓ) f1(Ai) f1(Aj) Aℓ f2(Aℓ) f1(Ai) f1(Aj) R2 [AiAjAℓ]
XR PRN i =N 2 N b0(Ωimin) = m N b0(Ωimin) + b0(Ωimin+1) = m m C ⊂ P1 R N i = N 2 i + i = N + 1 ˜ C 2m dim H0(Ωimin) = m N i =N 2 i + i = N + 1 ˜ C i +1 i +2 i i + 1 Ωimin+1 Ωimin S 1 2m dim H0(Ω imin) + dim H 0(Ω imin+1) = m XR PRN N ≥ 3 XR imin= N 2 m = N + 1 m C ⊂ P1 R N = 2k XR 1 bk−1(XR) = 2k + 2 N = 2k + 1 XR 1 bk−1(XR) = bk(XR) = k + 1 N N 1 N NX−2 i=0 bi(XR) = N − 1 + b0(Ωimin)
dim H0(Ω imin) = m NX−2 i=1 bi(XC) = 2N N − 1 + m = 2n m = N + 1 dim H0(Ω imin) = m = N + 1 2 N N−2X i=0 bi(XR) = N − 3 + b0(Ωimin) + b0(Ωimin+1) b0(Ωimin) + b0(Ωimin+1) = m NX−2 i=1 bi(XC) = 2(N − 1) N −3+m = 2N −2 m = N +1 b0(Ωimin) + b0(Ωimin+1) = m = N + 1 b0(Ωimin+1) = b0(Ωimin) XR PN R XC bi(XR) = hi,N−2−i(XC). XR PRN N N ≥ 3 2⌊N4⌋−1+ 1 N + 1 X m|N +1 2 m ϕ(m)2N +12m −1
ϕ(m) XR⊂ PRN imin= N 2 m = N + 1 m C ⊂ P1 R N 1 N imax= imin+1 1 m 1 m = N + 1 N + 1 = 1 + · · · + 1 | {z } N+1 2 N imax= imin+ 2 1 2 m = N + 1 1 2 N + 1 1 2 N+1 2 imin = N 2 1 N +1 8 = 2+2+2+1+1 P7 R 8 = 2 + 1 + 2 + 1 + 2 • O • • • • • • • • imin imin+ 1 imin+ 2 imin+ 1 imin+ 2 imin+ 1 imin imin+ 1 imin+ 2 = imax P7 R
N + 1 2 N+1 2 #k k N = 2k + 1 #1= 1, #2= 2, #3= 2, #4= 4, #5= 5, #6= 9, #7= 12, #8= 23, #9= 34, #10= 63, #11= 102, #12= 190. #k≥ 2⌊ k 2⌋−1+ 2 k−1 k + 1.
q0, q1, q2 ω ∈ P2 R ωq = ω0q0+ ω1q1+ ω2q2 C det(ωq) = 0 C d d CR CC x ∈ PN R C x C 16 (P2 R, CR) C d
P2 R P2 R PR2 P2 R σ σ′ σ ≺ σ′ σ σ′ σ0 m σ0≺ σ1≺ · · · ≺ σm−1≺ σm 0 p C n CR C d ℓ
CR 1 − (−1)d 2 ≤ ℓ ≤ (d − 1)(d − 2) 2 + 1 d ℓ d ℓ d ≥ 3 d ≤ 2 C d ≥ 3 CR ℓ = (d−1)(d−2)2 + 2 ℓ − 1 C1, . . . , Cℓ−1 Cℓ i 1 ℓ − 1 pi∈ Ci q1, . . . , qd−3∈ Cℓ d − 2 d(d−1)2 > ℓ + d − 4 D d − 2 pi qi CR DR c c ≥ 2(ℓ − 1) + d − 3 = (d − 1)(d − 2) + 2 + d − 3 > d(d − 2) ℓ ≤ (d−1)(d−2)2 + 1 1−(−1)d 2 ≤ ℓ g = (d−1)(d−2)2 g + 1 d = 2k p − n ≡ k2 ( 8) CR CC CC ⊂ PC2 : P2 C → PC2 CR CC (CC, ) CR CC
CC\ CR C CR CR C CR C C D = CC\ CR 2g + 2 CC χ(D) = 2 − 2g ˜ D 1 χ( ˜D) = 2 − 2g + 2(g + 1) = 4 2 CR CC CR CR Π+ Π−
Λ+ Λ− C d ℓ 2(Π+− Π−) = ℓ −d2 4 C d ℓ Λ+− Λ−+ 2(Π+− Π−) = ℓ −d2− 1 4 m M+ M− C Π+− Π−= n − 2(m + M+− M−) Ci C Π+Ci = Ci Π−Ci = Ci mCi = 1 Ci 0 MC+i = Ci MC−i = Ci
2k 2k + 1 Π+Ci− Π−Ci = 1 − 2(mCi+ MC−i− M + Ci) Ci −2(MC−i− MC+i) Ci 2(mCi+ MC−i− M + Ci) m + M+− M−≡ 0 ( 2) q0, q1, q2 PRN C C CR6= ∅ imin 6= 0 Ωi Ωfi f Ωi Hp(Ω i) L+R L+R = (ω, x) ∈ S2× PN R (ωq)(x) > 0 E2p,q= Hp(ΩN−q) Ωi = {ω ∈ S2 (ωq) ≤ i} ωq
Z2 Z2 Z2 Z2 p q imin imax ∗ Hi(Ω j) dp,N2 −q: Hp(Ωq) δ(γq+1)∪ −→ Hp+2(Ωq+1, π−1(Λq+1)) −→ Hp+2(Ωq+1) δ H1(π−1(Λ q+1)) → H2(Ωq+1, π−1(Λq+1)) (Ωq+1, π−1(Λq+1)) d0,N −q2 : H0(Ωq) → H2(Ωq+1) q ≥ imax−1 q > imax−1 H2(Ωq+1, π−1(Λq+1)) π−1(Λq+1) = Ωq+1= S2 π−1(Λq+1) Ωq+1 d0,imin 2 : H0(Ωimax−1) → H 2(S2) d2
ℓ(XR) d2 imin< imax−1 d2 ℓ(XR) = imin− 1 ℓ(XR) = imin− 2 XR imin < imax− 2 bimin−1(XR) = bimax−3(XR) ε1 d2 ε1= 0 d2 ε1= 1 ε2= 0 ℓ(XR) = imin− 1 ε2= 1 imin− 2 ≤ ℓ(XR) < imin ε1 = ε2 (PN R, XR) 0 → Himin(P N
R) → Himax−1(PRN\ XR) → Himin−1(XR) → Himin−1(P
N R ) ℓ(XR) < imin Himin(XR) → Himin(P N R ) bimin−1(XR) = dim H imax−1(PN R \ XR) − 1 + (1 − ε2)
= dim H0(Ωimin) + dim H
1(Ω imin+1) − ε2 (PN R , XR) 0 → Himax−2(P N R) → Hi min+1(PN R \ XR) → Himax−3(XR) → 0 imax− 3 ≥ imin > ℓ(XR) bimax−3(XR) = dim H imin+1(PN R \ XR) − 1
= dim H0(Ωimax−2) + dim H
1(Ω
imax−1) + 1 − ε1− 1
= dim H1(Ωimin+1) + 1 + dim H
0(Ω
imin) − 1 − ε1
= dim H1(Ωimin+1) + dim H
0(Ω imin) − ε1 ε1 = ε2 imin = imax− 2 N > 3 bimin−2(XR) = bimax−2(XR) N > 3 imin= 1 bimin−2(XR) = dim H imax(PN R \ XR) + 1 − ε2 = dim H1(Ωimin) + 1 − ε2
bimax−2(XR) = dim H imin(PN R \ XR) − 1 = dim H0(Ω imax−1) + 1 − ε1− 1 = dim H1(Ωimin) + 1 − ε1 ε1= ε2 imin= imax− 2 N = 3 ℓ(XR) = imin− 1 = 0 H3(LR) b1(XR) = 0 XR 0 imin= imax−1 d2 ℓ(XR) = imin− 2 imin= imax− 1 d2 ℓ(XR) = imin− 2 (PN R, XR) Hi(XR) → Hi(PRN) Hi(PRN) → Hi(PRN, XR) HN−i(PN R ) → HN−i(PRN\ XR) ∼= HN−i(L+R) x ∈ H1(PN R) xN−i p∗(xN−i) p : L+ R → PRN Hi(XR) → Hi(PRN) p∗(xN−i) = 0
p∗(xN−(imin−1)) = p∗(ximax) 6= 0
L+R|Ωimin → Ωimin. Himax(L+ R) = H 1(Ω imin) = H imax(L+ R|Ωimin) ξimax−1 S2 p∗(ximax) ξimax−1 p∗(x imax) = w 1(ξimax−1) d2 p∗(ximax) = w 1(ξimax−1) 6= 0 ℓ(XR) = imin− 2 PN R
q0, q1, q2 XR PN R ε d2 ε = 0 d2 i < i − 2 bi(XR) = 1, i < i − 2 i > i − 2; bi −2(XR) = b1(Ωi ) + 1 − ε; bi −1(XR) = b0(Ωi ) + b1(Ωi +1) − ε; bi(XR) = b0(Ωi+1) + b1(Ωi+2) − 1, i ≤ i ≤ i − 4; bi −3(XR) = b0(Ωi −2) + b1(Ωi −1) − ε; bi −2(XR) = b0(Ωi −1) − ε. i = i − 2 bi(XR) = 1, i < i − 2 i > i ; bi −2(XR) = b1(Ωi ) + 1 − ε; bi −1(XR) = b0(Ωi ) + b1(Ωi +1) + 1 − 2ε; bi (XR) = b0(Ωi −1) − ε. i = i − 1 ℓ(XR) = imin− 1 bi(XR) = 1, i < i − 2 i > i − 1; bi −2(XR) = b1(Ωi ) + 2; bi −1(XR) = b0(Ωi ) + 1. i = i − 1 ℓ(XR) = imin− 2 bi(XR) = 1, i < i − 2 i > i − 1; bi −2(XR) = b1(Ωi ) + 1 − ε; bi −1(XR) = b0(Ωi ) − ε. imin6= imax (PRN, XR) → HN−i−1(PN R \ XR) → Hi(XR) → Hi(PRN) → HN−i(PRN \ XR) → i N − i > i + 1 i < i − 2 HN−i−1(PN R \ XR) HN−i(PRN\ XR) bi(XR) = 1 i < i − 2 i ≥ i Hi(XR) → Hi(PRN) ℓ(XR) (PN R , XR) 0 → Hi+1(PRN) → HN−i−1(PRN \ XR) → Hi(XR) → 0
bi(XR) = dim HN−i−1(PRN\ XR) − 1
i ≥ imax− 1 dim HN−i−1(PRN\ XR) = 2 bi(XR) = 1
imin< imax− 2 bi(XR) = b0(Ωimax−1) − ε i = imax− 2
b0(Ωimax−2) + b1(Ωimax−1) − ε i = imax− 3
b0(Ωi+1) + b1(Ωi+2) − 1 imin≤ i ≤ imax− 4
imin= imax− 2
bimax−2(XR) = b0(Ωimax−1) − ε
bimin−1(XR) bimin−2(XR) imin < imax− 1
ℓ(XR) = imin− 1 ε = 0 (PN R, XR) 0 → Himin(P N R ) → Hi max−1(PN R \ XR) → Himin−1(XR) → Himin−1(P N R ) bimin−1(XR) = bimax−1(P N R \ XR) − 1 + (1 − ε) =
b0(Ωimin) + b1(Ωimin+1) − ε imin< imax− 2
b0(Ωimin) + b1(Ωimin+1) + 1 − 2ε imin= imax− 2
Himin−1(XR) → Himin−1(P N R) → Himax(PRN\ XR) → Himin−2(XR) → → Himin−2(P N R) → 0 bimin−2(XR) = bimax(P N R \ XR) + 1 − ε = b1(Ωimin) + 1 − ε
imin= imax−1 bimin−2(XR)
bimin−1(XR) ℓ(XR) = imin− 1 d2 bimin−2(XR) = b0(Ωimin) + 1 bimin−1(XR) = b1(Ωimin) + 2 ℓ(XR) = imin− 2 d2 bimin−2(XR) = b0(Ωimin) − ε bimin−1(XR) = b1(Ωimin) + 1 − ε
N > 3 N = 3 imin= imax− 2 = 1 XR 0 b0(XR) bimin−1(XR) CR ℓ i ≤ i − 2 X i≥0 bi(XR) = i −1 X i=i b0(Ωi) + b1(Ωi) ! + 4(i − ε) − N − 1 i −1 X i=i b0(Ωi) + b1(Ωi) = 2ℓ 2ℓ + 1 i = i − 1 = k N = 2k X i≥0 bi(XR) = 2ℓ + 2k − 2(1 + ε)ν ν = 0 ℓ(XR) = imin− 1 ν = 1 PN R N = 2k + 1 XR 2k − 2 XR PRN XR i = k N − 3 2 ≤ ℓ(XR) < i
⌊x⌋ i ≤N+1 2 N = 2k + 1 k − 1 < i ≤ k + 1 i = k + 1 i = i XR PRN XR i = k XR i = k i = i − 2 X i≥0 bi(XR) = 2ℓ + 2k − 4ε − 2 ε d2 ℓ X i≥0 bi(XC) = hk−1,k−1(XC) + 2hk−2,k(XC) + 2k − 2 = 3k2+ 3k + 2 + k(k − 1) + 2k − 2 = 4k2+ 4k XR 2ℓ + 2k − 4ε − 2 = 4k2+ 4k ℓ = 2k2+ k + 1 + 2ε = 2k(2k + 1) 2 + 1 + 2ε = g + 1 + 2ε ℓ ≤ g + 1 ℓ = g + 1 ε = 0 PN R N = 2k XR 2k − 3 XR PRN XR i = k − 1 i = k
N − 3 2 ≤ ℓ(XR) < i ⌊x⌋ i ≤N+1 2 N = 2k k − 2 < i ≤ k i = k − 1 i = k XR PRN XR i = k − 1 d2 i = k d2 (M −1) i = k d2 ℓ(XR) = imin− 2 (M −2) i = k ℓ(XR) = imin− 1 XR i = k − 1 i = k i = k − 1 i < i − 2 X i≥0 bi(XR) = 2ℓ + 2k − 4ε − 4 ε d2 ℓ X i≥0 bi(XC) = 2hk−2,k−1(XC) + 2k − 2 = 4k2− 2k − 2 + 2k − 2 = 4k2− 4 XR 2ℓ + 2k − 4ε − 4 = 4k2− 4 ℓ = 2k2− k + 2ε = 2k(2k − 1) 2 + 2ε = g + 2ε
ℓ + 1 ≤ g + 1 ℓ = g ε = 0 i = k i = i − 1 X i≥0 bi(XR) = 2ℓ + 2k − 2(1 + ε)ν ε d2 ℓ ν = 0 ℓ(XR) = imin− 1 X i≥0 bi(XC) = 4k2− 4 XR 2ℓ + 2k − 2(1 + ε)ν = 4k2− 4 ℓ = 2k2− k − 2 + (1 + ε)ν = 2k(2k − 1) 2 − 2 + (1 + ε)ν = g − 2 + (1 + ε)ν (ε, ν) (1, 1) (0, 1) (0, 0) ε = 1 ν = 1 ℓ = g ε = 0 ν = 1 ℓ = g − 1 ν = 0 ℓ = g − 2 C g θ CC θ θ⊗2= K C KC Σ(C) Σ C h0(C, θ)
dim H0(C, θ) > 0 (C, θ) C d θ CC d d (C,12(d − 3)H) H d ≡ ±1 8 d ≡ ±3 8 C g H0(C, K C) 1 CC ι : H1(C, Z) −→ H0(C, KC)∨ γ 7−→ ι(γ)(ω) =Rγω Λ 2g H0(C, K C)∨ Jac(C) = H0(C, KC)∨/Λ g CC P ic0(C) 2 2 Jac2(C) Jac2(C) ∼= H1(C, Z2) H1(C, Z) × H1(C, Z) −→ H2(C, Z) ∼= Z H1(C, Z) ∼= Z2g α1, . . . , αg, β1, . . . , βg ω1, . . . , ωg 1 CC Z αi ωj= δi,j τi,j= Z βi ωj τ = (τi,j) τ = Re(τ ) + ıIm(τ ) tτ = τ, Im(τ ) > 0
Cg C C (ε, δ) ∈ Zg2 θ ε δ (z, τ ) = X r∈Zg exp ıπ t (r +1 2ε)τ (r + 1 2ε) + 2 t (z +1 2δ)(r + 1 2ε) θ ε δ (−z, τ ) = exp(ıπtεδ) θ ε δ (z, τ ) CC −→ C P 7−→ θ ε δ (u(P ), τ ) u(P ) = RPP 0ω1, . . . , RP P0ωg ∈ Cg P 0 θ ε δ (0, τ ) = 0 d ≡ ±1 8 τ (C, θ) C d θ CC (C, θ) θ d (C, θ) θ CC L 2 θ ⊗ L qθ: Jac2(C) −→ Z2 L 7−→ h0(C, θ ⊗ L) − h0(C, θ) qθ
θ 7→ qθ C Jac2(C) Jac2(C) ∼= H1(C, Z2) qθ H1(C, Z2) u1, . . . , ug, v1, . . . , vg H1(C, Z2) qθ(ui) = αi, qθ(vj) = βj (α, β) ∈ Zg2 Arf (qθ) =tαβ 2 Z2 q q′ H1(C, Z2) h., .i v ∈ H1(C, Z2) u ∈ H1(C, Z2) q′(u) − q(u) = hv, ui q′ = q + v H1(C, Z2) q′ = q + v q′ Arf (q′) = Arf (q) + q(v) q0 q = q0+ v q0(v) = 0 C g 22g C C 2g−1(2g+1) 2g−1(2g−1) C g θ CC s θ ˆ s = s ⊗ s θ ⊗ θ = KC 1 CC K∗ C T C τ T C τ (ˆs) 1 CC ˆ s τ Ω ⊂ CC τ γ c ∈ H1(C; Z2) γ : S1→ CC\ Ω p ∈ S1 T γ(p)C \ {0} γ(p)
0 S1= {(x, y) ∈ R2 x2+y2= 1} (1, 0) ∈ S1 γ′(p) τ (γ(p)) S1 f : S1→ S1 n = deg(f ) (τ, γ) = n 2 H1(C; Z2) → Z2 c 7→ τ γ γ c ∈ H1(C; Z2) (τ, γ) γ τ s θ H1(C; Z2) → Z2 c 7→ θ c C g θ CC c ∈ H1(C; Z2) qθ(c) = θ c + 1 2 (C, ) : CC → CC c : Σ → Σ ΣR θ τ : θ → θ θ τ // θ CC // CC θR→ CR C0, . . . , Cr CR ε = (ε0, . . . , εr) ∈ Zr+12 εi= 0 θR Ci εi ε0+ ε1+ · · · + εr≡ g + 1 ( 2) ε0 (ε1, . . . , εr) Ci CR pi∈ Ci γi p0 pi γi⊂ CC C0 Ci Gi = γi∪ (γi) δ = (δ1, . . . , δr) ∈ Zr2 δi = qθ([Gi]) [Gi] Gi H1(C, Z2) θ CR
ω θ Ω = ω ⊗ ω C Ω CR θ qθ qθ([Ci]) ≡ εi+ 1 ( 2) [Ci] Ci H1(C, Z2) Gi qθ([Gi]) = 1 Gi C0, Ci qθ(c) = (θ, c) + 1 c ∈ H1(C, Z2) (θ, [Ci]) = εi Gi C0, Ci (θ, [Gi]) = 0 C δi = 0 Ci θ conj∗: H1(C, Z2) → H1(C, Z2) C ([Ci], [Gi]) H1(C, Z2) Ik 0 0 0 0 0 0 0 Is 0 0 0 0 Is 0 0 0 0 0 0 0 Ik 0 0 0 0 0 0 0 Is 0 0 0 0 Is 0 C Ik 0 0 0 0 0 0 0 0 0 Is 0 0 0 0 0 0 Is 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 Ik 0 0 0 0 0 0 0 0 0 Is 0 0 0 0 0 0 Is 0 0 0 0 0 0 0 0 0 1 , Ik 0 0 0 0 0 0 0 0 0 0 0 Is 0 0 0 0 0 0 0 0 Is 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 Ik 0 0 0 0 0 0 0 0 0 0 0 Is 0 0 0 0 0 0 0 0 Is 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1
([Ci], [Gi]) C1 C2 C3 G1 G2 G3 C0 H1(C, Z2) qθ H1(C, Z2) conj∗ c ∈ H1(C, Z2) qθ(c) = qθ(conj∗(c)) CR= ∅ C r ≥ 1 (ε, δ) ∈ (Zr2−1)2 θ qθ([Ci]) ≡ εi+ 1 ( 2) qθ([Gi]) = δi C C qθ (qθ) = r X i=1 δi(1 + εi) C r g k = (g + 1 − r)/2 H1(C, Z2) ([C1], . . . , [Cr], X1, . . . , Xk, X1c, . . . , Xkc, [G1], . . . , [Gr], Y1, . . . , Yk, Y1c, . . . , Ykc) Xc i = conj∗(Xi) Yic = conj∗(Yi) C
Z2 [C1], . . . , [Cr] [G1], . . . , [Gr]) qθ qθ([Ci]) ≡ εi+ 1 ( 2) qθ([Gi]) = δi θ Xi Xic Yi Yic qθ(Xi) = qθ(Xic) qθ(Yi) = qθ(Yic) qθ(Xi) qθ(Yi) Arf (qθ) = r−1 X i=1 qθ([Ci])qθ([Gi]) + k X i=1 [qθ(Xi)qθ(Yi) + qθ(Xic)qθ(Yic)] = r−1 X i=1 (1 + εi)δi+ 2 k X i=1 qθ(Xi)qθ(Yi) = r−1 X i=1 (1 + εi)δi. g r g r k = (g − r)/2 H1(C, Z2) ([Ci], Xj, Xjc, X, [Gi], Yj, Yjc, Y )1≤i≤r−1 1≤j≤k Xc
i = conj∗(Xi) Yic = conj∗(Yi) conj∗(X) = X conj∗(Y ) = X + Y
([C1], . . . , [Cr], [G1], . . . , [Gr]) qθ qθ(Xi) = qθ(Xic) qθ(Yi) = qθ(Yic) qθ(conj∗(Y )) = qθ(Y + X) = qθ(Y ) qθ(conj∗(X)) = qθ(X) qθ(Y + X) = qθ(Y ) qθ(X) = hX, Y i = 1
qθ(Xi) qθ(Yi) qθ(Y ) Arf (qθ) = r−1 X i=1 qθ([Ci])qθ([Gi]) + k X i=1 [qθ(Xi)qθ(Yi) + qθ(Xic)qθ(Yic)] + qθ(X)qθ(Y ) = r−1 X i=1 (1 + εi)δi+ 2 k X i=1 qθ(Xi)qθ(Yi) + qθ(Y ) = r−1 X i=1 (1 + εi)δi+ qθ(Y ). qθ(Y ) (ε, δ) ∈ (Z2r−1)2 q0, q1, q2 C d CC K CC ω ∈ CC ker(ωq) 1 L = K(d − 1) L2 = O C(d − 1) H0(C, L(−1)) = 0 θ = L(−1) C A(x) m x ∈ P2 C
(m − 1) A(x) F (x)ℓ det A(x)
F (x)ℓ+1 p F (x)p det(A(x)) adj(A(x)) F (x)ℓ det(adj(A(x))) F (x)ℓm det(adj(A(x))) = [det(A(x))]m−1 F (x) [det(A(x))]m−1 p(m − 1) ℓm p(m − 1) ≥ ℓm ⇒ p ≥ m m − 1ℓ p p ≥ ℓ + 1 det(A(x)) F (x)ℓ+1
θ dim H0(C, θ) = 0 C d (q0, q1, q2) C θ φ11, . . . , φ1d H0(C, θ(1)) = H0(C, L) v 11 = φ211, . . . , v1d = φ11φ1d ∈ H0(C, L2) L2 = O C(d − 1) H0(P2 C, OP2 C(d − 1)) → H 0(C, O C(d − 1)) v1i d − 1 U (x0, x1, x2) = 0 C {v1i= 0} i ≥ 2 C {v11 = 0} vij w11ij d − 1 d − 2 v1iv1j= v11vij− U w11ij vij = vji V (x) = (vij) d x ∈ CR d − 1 dim H0(C, θ) = 0 r V (x) U (x)r−1 V (x) x ∈ CR
det V (x) Ud−1 deg(det V (x)) = d(d − 1) = deg Ud−1
c ∈ R det V (x) = cU (x)d−1 d − 1 U (x)d−2 M (x) = 1 U (x)d−2adjV (x) M (x) det M (x) = cd−1U (x) adjM (x) = cd−2V (x) d2
d2 q0, q1, q2 PRN CR P2 R\CR r N D1, . . . , Dr i Di ≺ Ci Di S2 → PR2 D′ i, D′′i Ci Γ′ i, Γ′′i Γ′i ⊂ ∂D′i Γ′′ i ⊂ ∂D′′i Ei′ ξm−1 D′i m D′ i Ei′′ ξN−m D′′ i Mi j ∈ {1, 2, . . . , r} j 6= i Cj ⊂ ∂Di j ∈ Mi Mi,j k ∈ {1, . . . , r} Cj ≺ Ck i ∈ {1, . . . , r} j ∈ Mi ω1(E′i)[Γ′j] = ε(Cj) + X k∈Mi,j ε(Ck) ω1(Ei′′)[Γ′′j] = ε(Cj) + X k∈Mi,j ε(Ck) ε(Ck) K ε(Ck) = 0 K Ck i ∈ {1, . . . , r} j ∈ Mi p(Cj) = p Cj ≺ Ck1≺ Ck2 ≺ · · · ≺ Ckp p p(Cj) = 0 Γ′ j D′ i Γ′ i
Mi,j = ∅ Ej′ D′ j Ei′|Γ′ j = E ′ j Γ′ j ⊕ K|Γ′ j, E ′ j Γ′ j = Ei′|Γ′ j⊕ K|Γ′j E′ j ω1(Ei′)[Γ′j] = ε(Cj) p(Cj) = 0 j ∈ Mi p(Cj) ≤ ρ j ∈ Mi p(Cj) = ρ + 1 Mj= {k1, . . . , ka} Γ′ j Γ′ k1 Γ′ ka Γ′ i D′ j p(Ck1) ≤ ρ . . . p(Cka) ≤ ρ ω1(Ej′)[Γ′k1] = ε(Ck1) + X k∈Mi,k1 ε(Ck) ω1(Ej′)[Γ′ka] = ε(Cka) + X k∈Mi,ka ε(Ck) E′ j D′j ω1(Ej′)[Γ′j] = a X m=1 ω1(Ej′)[Γ′km] = X k∈Mi,j ε(Ck) Ei′|Γ′ j = E ′ j Γ′ j ⊕ K|Γ′ j, E ′ j Γ′ j = Ei′|Γ′ j⊕ K|Γ′j ω1(Ei′)[Γ′j] = ω1(E′j)[Γ′j] + ε(Ck) q0, q1, q2 CR d2 i ∈ {1, . . . , r} D′ i Di′′ i j ∈ Mi ε(Cj) + X k∈Mi,j ε(Ck) = 0
d2 ξimax−1 d2: H0(Ωimax−1) δ(γimax)∪ // H2(Ω imax, π−1(Λimax)) // H 2(Ω imax) δ : H1(π−1(Λ imax)) → H 2(Ω i , π−1(Λimax) (Ωi , π−1(Λimax)) d2= 0 ⇔ δ(γi ) = 0 ⇔ γi = 0 H1(π−1(Λ imax)) = L H1(D′ i) ⊕ LH1(D′′ i) D′ i D′′i i γi = 0 ξimax−1 D′i Di′′ H1(D′ i) = L j∈MiH 1(Γ′ i) ξimax−1 D′i j ∈ Mi ω1(Ei′)[Γ′j] = 0 d2 K θ K K θ K θ ε = (ε0, . . . , εr) Ci CR ε(Ci) = εi Ci ε(Ci) = εi+ 1 Ci C0 qθ εi K Ci CR CR S2 CR
= i i − 1 i − 1 = i i + 1 i − 1 q0, q1, q2 CR q0, q1, q2 XR PN R N = 2k + 1 k ≥ 2 imin > 1 χ(XR) = (1 + (−1)k) + (−1)k−1(2p − 2n) 2 < imin< imax− 2 χ(XR) = N X j=0 (−1)j+1χ(Ωj). j < imin Ωj = ∅ j ≥ imax Ωj = S2
imin+ imax= N + 1 χ(XR) = (1 + (−1)imin−1) + imaxX−1 j=imin (−1)j+1χ(Ω j), χ(XR) = (1 + (−1)imin−1) + iXmax i=imin (−1)i(b 0(Ωi−1) + b1(Ωi)). i − k iXmax i=imin (−1)i(b 0(Ωi−1) + b1(Ωi)) = (−1)k−1(2p − 2n − 1 − (−1)k−imin). (−1) p − n VR PRN s N − s m1, . . . , ms VC σ(VC) σN s(m1, . . . , ms) σsN(m1, . . . , ms) = 1, s = 0; m1.m2. . . ms, 0 < s = N ; σsN(m1, . . . , ms) = msσsN−1−1(m1, . . . , ms−1) − ms X µ=1 σNs(m1, . . . , ms−1, µ, µ − 1), 0 < s < N . σ(VC) = σNs (m1, . . . , ms) σ32k+1(2, 2, 2) = (1 + (−1)k) + (−1)k−12(k + 1)2. VR 2k VR χ(VR) ≡ σ(VC) (16). VR χ(VR) ≡ σ(VC) ± 2 (16).
VR m1. . . ms ≡ 0 (8) Hk(VR, Z2) → Hk(PRN, Z2) χ(VR) ≡ ±σ(VC) (16). 2k p − n ≡ k2 (8). 2k p − n ≡ k2± 1 (8). C 2k imin= imax− 2 XR d2 d2 Hk(XR, Z2) → Hk(PRN, Z2) ℓ(XR) χ(XR) ≡ ±σ(XC) (16). p − n ≡ k2 (8). C imin= imax− 2 d2 XR χ(XR) ≡ σ(XC) ± 2 (16), p − n ≡ k2± 1 (8). VR 2k |χ(VR) − 1| ≤ hk,k(VC) − 1.
2k −3 2k(k − 1) ≤ p − n ≤ 3 2k(k − 1) + 1. C 2k XR 2(k − 2) |χ(XR) − 1| = (1 + (−1)k) + (−1)k−1(2p − 2n) − 1 = |1 − (2p − 2n)| ≤ hk−2,k−2(XC) − 1 ≤ 3k2− 3k + 2. PN R N = 2k + 1 XR 2k − 2 (C, θ) 2k + 2 2k Ci 2m + 1 εi+ X j∈ fMi εj = 0 f Mi= {j ∈ {1, . . . , g} Cj≺ Ci Cj 2m + 2}
(C, θ) imin= k S2\ eC 0 k + 1 0 k k + 2 1 k + 1 1 0 2k + 1 2k d2 i ∈ {1, . . . , g} Di imax j ∈ Mi εj+ X k∈Mi,j εk= 0 εi
C N + 1 N = 2k + 1 XR PN R C θ C C (C0, . . . , Cg) C δ = (δ1, . . . , δg) ε = (ε1, . . . , εg) Zg 2 δi = 0 Ci ε = 0 d2 a = δi = 1 2 (qθ) = g X i=1 δi(1 + εi) (qθ) = a (2) a (δ, ε) a Cj 2k δj= 0 2k δj= 1 Cj Cj (qθ) = a + 1 (2) a Cj 1 δj= 1 Ck 2 εj εk εj = εk = 1
1 δj= 1 εj= 0 εk= 0 δj= 1 εj= 1 εk= 1 εi d2 a Ck 2 3 1 δj= 0 δk= 0 δj= 0 δk= 1 a a a εk εj Cj Ck δj = 0 1 a m m a PN R N = 2k XR 2k − 3 imin = k − 1 εi imin = k
d2 imin= k i εi = 0 2k + 1 θ imin = k εi = 0 i C N + 1 N = 2k XR PN R C C i = k i εi = 0 εi = 1 i
P3 R PR4 Q1, Q2, Q3 PRN e C det(λ1Q1+ λ2Q2+ λ3Q3) = 0 Qi Qi
(Q1(t), Q2(t), Q3(t)) (Q1, Q2, Q3) 0 C (1 : 0 : 0) CN+1 Q1+ λQ2+ µQ3= λb00+ µc00 tBλ,µ Bλ,µ Cλ,µ Bλ,µ Q2 Q3 Cλ,µ λ µ Mλ,µ (1 : 0 : 0) b00 = c00 = 0 Q1+ λQ2+ µQ3= λbλb0001+ µc+ µc0001 λbλb0111+ µc+ µc0111 tB λ,µ Bλ,µ Cλ,µ Bλ,µ Q2 Q3 Cλ,µ λ = µ = 0 Mλ,µ det(Mλ,µ) = det Aλ,µ−tBλ,µCλ,µ−1Bλ,µ × det(Cλ,µ) Aλ,µ 2 (λbij + µcij)0≤i,j≤1 λ = µ =
0 2 det(Mλ,µ) = det(C0,0) × h λ2(b00b11− b201) + µ2(c00c11− c201) + λµ(b00c11+ + c00b11− 2b01c01) + i ∆ = (b00c11+ c00b11− 2b01c01)2− 4(b00b11− b201)(c00c11− c201) ∆ 6= 0 x y λ µ xe1+ ye2 e1, e2 t (xe1+ ye2)Mλ,µ(xe1+ ye2) = 0 λ (x2b00+ y2b11+ 2xyb01) | {z } P1(x,y) +µ (x2c00+ y2c11+ 2xyc01) | {z } P2(x,y) = 0 ∆ P1 P2 ∆ = 0 (Q1(t), Q2(t), Q3(t)) t
CC π : eCC → CC CC ωC CC U CC 1 τ π−1(U ) ⊂ eCC Pi,+ Pi,− Pi CC Pi,+τ + Pi,−τ = 0. CC CC F OC HomOC(F, ωC) Θ(CC) CC CC θ CC dim H0(C C, θ) ≡ 0 (2) θ dim H0(C C, θ) ≡ 1 (2) θ dim H0(C C, θ) = 0 q = (q0, q1, q2) PN R CC RN+1 M ∈ MN+1(OP2 R(1)) OCN+1(−1)−→ OM N+1 C EC 0 → EC→ ONC+1(d − 1)−→ OM NC+1(d) EC dim H0(E C(−1)) = 0 EC
EC ∼= Hom(EC, ωC(2)) EC(−1) CC θ dim H0(C C, θ) = 0 M ∈ MN+1(OP2 C(1)) CC θ M CC tAM A A ∈ ON+1(C) M θℓ = θ∗⊗ ω C dim H0(C C, θ(1)) = dim H0(CC, θℓ(1)) = N + 1. H0(C C, θ(1)) = he1, . . . , eN+1i H0(C C, θℓ(1)) = heℓ1, . . . , eℓN+1i θ(1) θℓ(1) (H0(θ(1)), H0(θℓ(1))) −→ H0(ωC(2)) ≈ H0(OC(N )). V (x) Vi,j (ei, eℓj) V (x) L L ⋔ C = {pt1, . . . , ptN+1} div(ei) ≥ X j6=i ptj6≥ pti, div(eℓi) ≥ X j6=i ptj 6≥ pti. k i 6= j Vi,j|ptk = 0 deg(Vi,j) ≥ N + 1
Vi,j PC2 N i 6= j Vi,j|L = 0 i Vi,i|pti 6= 0 Vi,i|L 6= 0 V (x) L M V (x) 2 CC adj(V (x)) fN−1 f (x) C M = adj(V (x))/fN−1 θ θ θℓ
M M CC (CC, conj) CC F CC Fconj ∼= F Fconj CC U ⊂ CC F conj(U ) ⊂ CC (CC, conj) θ M ∈ MN+1(OP2 R(1)) CC = {det M = 0} θ M f (x, y) H = {(x, y) ∈ C2 f (x, y) = 0} ⊂ C2 (0, 0) g(x, y; t) t ∈ Cµ µ g(x, y; 0) = f (x, y) g(0, 0; t) = 0 g ti t = (t1, . . . , tµ) C{x, y}/(∂ ∂xf, ∂ ∂yf )
Bη τ > 0 ∂Bη Ht= {(x, y) ∈ C2 g(x, y; t) = 0}, ||t|| < τ. Dτ Cµ 0 τ X = {(x, y; t) ∈ Bη× Dτ g(x, y; t) = 0} p : X → Dτ (x, y; t) 7→ t H = H0 ∆ = {t ∈ Dτ p−1(t) } p F = p−1(t 0) t0∈ Dτ\ ∆ H π1(Dτ\ ∆) H1(F ; Z2) = Zµ2 Γ Γ = Im π1(Dτ \ ∆) → Aut(H1(F ; Z2)) H Ti: H1(F ; Z2) → H1(F ; Z2) x 7→ x + hx, ξiiξi ξi H1(F ; Z2) h., .i (Q1, Q2, Q3) PRN XR C = {(x : y : z) ∈ PC2 det(xQ1+ yQ2+ zQ3) = 0} (0, 0, 1) C Ct= {(x : y : z) ∈ PC2 det(xQ1(t) + yQ2(t) + zQ3(t)) = 0} C0 = C θt (Q1(t), Q2(t), Q3(t)) t G(x, y, z; t) = det(xQ1(t) + yQ2(t) + zQ3(t)) F (x, y, z) = det(xQ1+ yQ2+ zQ3) (iii)
(Q1(t), Q2(t), Q3(t)) t = 0 |t| < η (ut, vt) Ct (0 : 0 : 1) ∈ P2 C f (ut, vt) = t2 f (u, v) = F (1, u, v) t (0 : 0 : 1) ∈ P2 C Ct det(Q1(t)+ λQ2(t) + µQ3(t)) = 0 t ∈] − η, η] η (Q1(t) + λQ2(t) + µQ3(t)) aa0001(t) + λb(t) + λb0001(t) + µc(t) + µc0001(t)(t) aa1101(t) + λb(t) + λb1101(t) + µc(t) + µc1101(t)(t) tB λ,µ(t) Bλ,µ(t) Cλ,µ(t) + A . Mλ,µ(t) Mλ,µ(0) = Mλ,µ t, λ µ det(Mλ,µ(t)) =t2(a00a11− a201) + λ2(b00b11− b201) + µ2(c00c11− c201) +λµ(b00c11+ c00b11− 2b01c01) + tλ(a00b11+ b00a11− 2a01b01) +tµ(a00c11+ c00a11− 2a01c01) + t ∆ ∆ = (b00c11− b11c00)2+ 4(b00c01− b01c00)(b11c01− b01c11) t Ct f (ut, vt) = t2 f t t G(x, y, z; t) = G(x, y, z; −t) |t| < η {f ((1 − λ)ut+ λu0, (1 − λ)vt+ λv0) = t2} ∆ t {f (ut, vt) = t2} {f (u0, v0) = t2}
{f (u0, v0) = t} C1 C2 C0 C → T ⊂ C Ct0 = C0 Ct1= C1 Ct2 = C2 γ T t1 t2 t1 t0 t2 γ ¯ γ Ct1 = C1 Ct2 = C2 C1 C2 γ C0 γ C0 Γ H1(C, Z2) CC C → T CC E(T ) = {(Cτ, F) τ ∈ T F ∈ Θ(Cτ)} (Cτ, F) ∈ E(T ) 7→ τ ∈ T Cτ 2 τ = 0
τ = 0 τ θ1 θ2 θs θ′ 1 θ′ s θ′2 p : C → [t1, t2] CC Ct0 t0∈]t1, t2[ θ1 C1 θ2 p θ1 c ∈ H1(C, Z2) qθ2(c) = qθ1(µ(c)) µ t0 C C0, . . . , Cℓ CR Gi C0 Ci θ θ′ qθ([Ci]) = 1 + εi qθ([Gi]) = δi qθ′([Ci]) = 1 + ε′i qθ′([Gi]) = δi′ (ε′ i, δ′i) (εi, δi)
[Ci] ∈ H1(C, Z2) δ′ k= δk+ δi,kεi ε′ k = εk [γi,j] ∈ H1(C, Z2) δ′ k = δk ε′ k= εk+ (δi,k+ δj,k)(1 + δi+ δj) δi,k c ∈ H1(C, Z2) qθ′(c) = qθ(µ(c)) µ 1 [Ci] ∈ H1(C, Z2) · Ci Ci Ci H1(C, Z2) (εi, δi) δ′k= qθ′([Gk]) = qθ(µ([Gk])) = qθ([Gk] − h[Ci], [Gk]i[Ci]) = qθ([Gk]) + h[Ci], [Gk]i2qθ([Ci]) + h[Ci], [Gk]i2 = δk+ δi,kεi δi,k [Ck] [Ck] [Ci] k δ′ k= δk+ δi,kεi ε′ k= εk 2 [γi,j] ∈ H1(C, Z2)
Ci Cj Ci Cj γij H1(C, Z2) γij = [Gi]−[Gj] (εi, δi) 1 + ε′k = qθ′([Ck]) = qθ(µ([Ck])) = qθ([Ck] − h[Ck], γijiγij) = qθ([Ck]) + h[Ck], γiji2qθ(γij) + h[Ck], γiji2 = 1 + εk+ (δi,k+ δj,k)(1 + qθ(γij)) = 1 + εk+ (δi,k+ δj,k)(1 + δi+ δj) δi,k δj,k [Gk] [Gk] γij k δ′ k = δk ε′ k = εk+ (δi,k+ δj,k)(1 + δi+ δj) (ei, fi) H1(C, Z2) ei Ci fi Gi i ≤ ℓ i > ℓ 3.3.4 q0 q0 g X i=1 αiei+ βifi ! = g X i=1 αiβi q H1(C, Z2) q = q0+ v v ∈ H1(C, Z2) q = q0+ v v ∈ H1(C, Z2)
q = q0+ v H1(C, Z2) u ∈ H1(C, Z2) q′ u q q′ = q0+ v + u hu, vi = q0(u) q0+ v h., .i q′ µ u x ∈ H1(C, Z2) q(µ(x)) = q′(x) µ(x) = x + hx, uiu q(µ(x)) = q(x + hx, uiu) = q(x) + hx, ui + hx, uiq(u)
= q0(x) + hv, xi + hx, ui + hx, ui(q0(u) + hv, ui)
= q0(x) + hv, xi + hx, ui(1 + q0(u) + hv, ui)
P3 R PR4
P
3 R P3 R 4 h4i 4 Ci Gi γij P3 R θ C θC0 C3 C2 C1 G1 G3 γ13 γ23 γ12 G2 γ13 G2 imin = 1 d2 ε1= ε2= ε3= 1 qθ([Ci]) = 1 + εi δi Ci εi= 1 δi= 0 i 1 ε3= 0 ε3 δ1+ δ3 δ1+ δ3 = 0 ε3 γ13 δ1+ δ3 = 1 δ1 = 0 δ3= 1 γ1 ε1= 1 C1 δ1= 1 δ1+δ3= 0 δ1 = 1 δ3 = 0 ε1 = 0 δ1 C1 0 = (qθ) = (1 + ε1)δ1+ (1 + ε2)δ2+ (1 + ε3)δ3 = 1 + (1 + ε2)δ2 ε2= 0 δ2= 1 δ1+ δ2= 0 γ12 ε1= ε2= 1 δ1 δ1+ δ3= 0 2 ε3= 1 ε2= 0 ε2
δ1+δ2 δ1+δ2= 0 ε2 γ12 δ1+ δ2= 1 δ1= 0 δ2= 1 γ1 ε1= 1 C1 δ1= 1 δ1+δ2= 0 δ1= 1 δ2= 0 0 = arf (qθ) = (1 + ε1)δ1+ (1 + ε2)δ2+ (1 + ε3)δ3 = 1 + ε1 ε1= 1 δ1 C1 δ1+ δ2= 0 3 ε2 = ε3 = 1 ε1 = 0 0 = arf (qθ) = (1 + ε1)δ1+ (1 + ε2)δ2+ (1 + ε3)δ3 = δ1 ε1 γ1 P3 R P3 R 8 P3 R 7 8 7 7 P3 R 7 7 8 7 P3 R 7 8 7 pi
pi 6 pi 3 7 pi X v X pi pi v pi X 7 X 4 7 6 7 7 F1 F2 7 f : P3 R× [0, 1] → PR3 f (F1, 0) = F1 f (F1, 1) = F2 t ∈ [0, 1] f (F1, t) 7 7 S = {q1, . . . , q7} 7 S (i, j, k) qi+ qj+ qk H1(C, Z2) S = {q1, . . . , q7} qS = q1+ q2+ · · · + q7
qθ 8 qS = qθ 4 7 P3 R 28 pi PR3 B 6 P3 R (pi, pj) pipj B P2 R p8 p2 p3 p4 p5 p6 p7 p1 • • • • • • • • • p6 • p7 • p5 • p8 • p1 • p2 • p3 • p4 8 280 p8
P3 R P3 R p8 p8 (q1, . . . , q7) pi qi q2 q5 q7 q4 q3 q6 q1 • • • • • • • • p8 p2 p5 p7 p4 p3 p6 p1 • • • • • • • • P3 R p8 PR3 qi, qj, qk i < j < k pi= qj∩ qk pj= qi∩ qk pk = qi∩ qj q2 q5 q7 q4 q3 q6 q1 • • • • • • • • • p5 • p7 • p2 • p8 • p4 • p3 • p6 • p1
7 P3 R (ei, fi) H1(C, Z2) ei Ci fi Gi q0 q0 3 X i=1 αiei+ βifi ! = 3 X i=1 αiβi qθ = q0+ e1+ e3+ f1+ f2+ f3 q1= q0+ e1+ e3+ f2+ f3 q2= q0+ e3+ f2+ f3 q3= q0+ e3+ f1+ f3 q4= q0+ e2+ e3+ f1+ f3 q5= q0+ e2+ e3+ f1+ f2 q6= q0+ e2+ f1+ f2 q7= q0+ e2+ f1+ f2+ f3 C0 C3 C2 C1 q1 q2 q3 q4 q5 q6 q7 qθ
S7 qi q qθ G1 f1 C0 C3 C2 C1 q1 1 q1 2 q1 3 q1 4 q15 q1 6 q1 7 q1= q0+ e1+ e3+ f2+ f3 q2= q0+ e3+ f2+ f3 q3= q0+ e3+ f1+ f3 q4= q0+ e2+ e3+ f1+ f3 q5= q0+ e2+ e3+ f1+ f2 q6= q0+ e2+ f1+ f2 q7= q0+ e2+ f1+ f2+ f3 ⇒ q1 1 = q0+ e1+ e3+ f2+ f3 q1 2 = q0+ e3+ f1+ f2+ f3 q1 3 = q0+ e3+ f3 q1 4 = q0+ e2+ e3+ f3 q1 5 = q0+ e2+ e3+ f2 q61= q0+ e2+ f2 q1 7 = q0+ e2+ f2+ f3 q1 1+ · · · + q17= qθ
qi C1 e1 C0 C3 C2 C1 q2 2 q2 3 q2 4 q2 5 q2 6 q2 7 q2 1 q1 1= q0+ e1+ e3+ f2+ f3 q1 2= q0+ e3+ f1+ f2+ f3 q1 3= q0+ e3+ f3 q1 4= q0+ e2+ e3+ f3 q1 5= q0+ e2+ e3+ f2 q1 6= q0+ e2+ f2 q1 7= q0+ e2+ f2+ f3 ⇒ q2 1 = q0+ e3+ f2+ f3 q2 2 = q0+ e3+ f1+ f2+ f3 q2 3 = q0+ e1+ e3+ f3 q2 4 = q0+ e1+ e2+ e3+ f3 q2 5 = q0+ e1+ e2+ e3+ f2 q2 6 = q0+ e1+ e2+ f2 q2 7 = q0+ e1+ e2+ f2+ f3 q2 1 + · · · + q72 = qθ γ12 f1+ f2 C0 C3 C2 C1 q3 1 q3 2 q3 3 q3 4 q3 5 q3 6 q73
q2 1= q0+ e3+ f2+ f3 q2 2= q0+ e3+ f1+ f2+ f3 q2 3= q0+ e1+ e3+ f3 q2 4= q0+ e1+ e2+ e3+ f3 q2 5= q0+ e1+ e2+ e3+ f2 q2 6= q0+ e1+ e2+ f2 q2 7= q0+ e1+ e2+ f2+ f3 ⇒ q3 1= q0+ e3+ f1+ f3 q3 2= q0+ e3+ f3 q3 3= q0+ e1+ e3+ f3 q3 4= q0+ e1+ e2+ e3+ f1+ f2+ f3 q3 5= q0+ e1+ e2+ e3+ f1 q3 6= q0+ e1+ e2+ f1 q3 7= q0+ e1+ e2+ f1+ f3 q3 1 + · · · + q73 = qθ e2 C2 q41= q0+ e2+ e3+ f1+ f3; q42= q0+ e2+ e3+ f3; q43= q0+ e1+ e2+ e3+ f3; q44; = q0+ e1+ e2+ e3+ f1+ f2+ f3; q4 5= q0+ e1+ e3+ f1; q46= q0+ e1+ f1; q47= q0+ e1+ f1+ f3. γ23 f2+ f3∈ H1(C, Z2) q15= q0+ e2+ e3+ f1+ f2; q25= q0+ e2+ e3+ f2; q5 3 = q0+ e1+ e2+ e3+ f2; q45= q0+ e1+ e2+ e3+ f1; q55= q0+ e1+ e3+ f1; q65= q0+ e1+ f1+ f2+ f3; q75= q0+ e1+ f1+ f2. C3 e3∈ H1(C, Z2) q6 1 = q0+ e2+ f1+ f2; q26= q0+ e2+ f2; q36= q0+ e1+ e2+ f2; q46= q0+ e1+ e2+ f1; q56= q0+ e1+ f1; q66= q0+ e1+ f1+ f2+ f3; q76= q0+ e1+ e3+ f1+ f2. G3 f3 q17= q0+ e2+ f1+ f2+ f3; q27= q0+ e2+ f2+ f3; q37= q0+ e1+ e2+ f2+ f3; q47= q0+ e1+ e2+ f1+ f3; q57= q0+ e1+ f1+ f3; q67= q0+ e1+ f1+ f2; q77= q0+ e1+ e3+ f1+ f2.
(q1, . . . , q7) S7 (q1, . . . , q7) S7 S7 (i, i + 1) qi qi+1 q4 q5 γ23 C0 C3 C2 C1 q1 q2 q3 q4 q5 q6 q7 γ23 q1 q2 C1 ei q1= q0+ e1+ e3+ f2+ f3 q2= q0+ e3+ f2+ f3 q3= q0+ e3+ f1+ f3 q4= q0+ e2+ e3+ f1+ f3 q5= q0+ e2+ e3+ f1+ f2 q6= q0+ e2+ f1+ f2 q7= q0+ e2+ f1+ f2+ f3 ⇒ q′ 1= q0+ e3+ f2+ f3= q2 q′ 2= q0+ e1+ e3+ f2+ f3= q1 q′ 3= q0+ e3+ f1+ f3= q3 q′ 4= q0+ e2+ e3+ f1+ f3= q4 q′ 5= q0+ e2+ e3+ f1+ f2= q5 q′ 6= q0+ e2+ f1+ f2= q6 q′ 7= q0+ e2+ f1+ f2+ f3= q7
q2 q3 f1+ f2 γ12 q′1= q0+ e1+ e3+ f2+ f3= q1 q′2= q0+ e3+ f1+ f3= q3 q′3= q0+ e3+ f2+ f3= q2 q′4= q0+ e2+ e3+ f1+ f3= q4 q′ 5= q0+ e2+ e3+ f1+ f2= q5 q′6= q0+ e2+ f1+ f2= q6 q′7= q0+ e2+ f1+ f2+ f3= q7 q3 q4 e2 C2 q′1= q0+ e1+ e3+ f2+ f3= q1 q′2= q0+ e3+ f2+ f3= q2 q′3= q0+ e2+ e3+ f1+ f3= q4 q′ 4= q0+ e3+ f1+ f3= q3 q′5= q0+ e2+ e3+ f1+ f2= q5 q′6= q0+ e2+ f1+ f2= q6 q′7= q0+ e2+ f1+ f2+ f3= q7 q4 q5 f2+ f3 γ23 q′ 1= q0+ e1+ e3+ f2+ f3= q1 q′2= q0+ e3+ f2+ f3= q2 q′3= q0+ e3+ f1+ f3= q3 q′4= q0+ e2+ e3+ f1+ f2= q5 q′5= q0+ e2+ e3+ f1+ f3= q4 q′6= q0+ e2+ f1+ f2= q6 q′7= q0+ e2+ f1+ f2+ f3= q7 q5 q6 e3 C3 q′1= q0+ e1+ e3+ f2+ f3= q1 q′2= q0+ e3+ f2+ f3= q2 q′3= q0+ e3+ f1+ f3= q3 q′4= q0+ e2+ e3+ f1+ f3= q4 q′ 5= q0+ e2+ f1+ f2= q6 q′6= q0+ e2+ e3+ f1+ f2= q5 q′7= q0+ e2+ f1+ f2+ f3= q7
q6 q7 f3 G3 q′1= q0+ e1+ e3+ f2+ f3= q1 q′2= q0+ e3+ f2+ f3= q3 q′3= q0+ e3+ f1+ f3= q2 q′4= q0+ e2+ e3+ f1+ f3= q4 q′5= q0+ e2+ e3+ f1+ f2= q5 q′6= q0+ e2+ f1+ f2= q6 q′ 7= q0+ e2+ f1+ f2+ f3= q7 q2 q3 f1+ f2 γ12 q′ 1= q0+ e1+ e3+ f2+ f3= q1 q′2= q0+ e3+ f2+ f3= q2 q′3= q0+ e3+ f1+ f3= q3 q′4= q0+ e2+ e3+ f1+ f3= q4 q′5= q0+ e2+ e3+ f1+ f2= q5 q′ 6= q0+ e2+ f1+ f2+ f3= q7 q′7= q0+ e2+ f1+ f2= q6 (i, i + 1) S7
P
4 R P3 R 5 5 C 5 θ dim H0(C, θ) 6= 0 Cθ 2 dim H0(C, θ) = 0 2 θ φ ψ (φ+tψ)t∈R K = OC(2) 2 P2 C Qt: (φ + tψ)2= 0 Qt C 2 (φ + tψ) θ t t′ C P3 R Ci γi C0 C1 C2 C3 C4 C5 C6 γ1 γ1,2 γ2,3 γ3 γ3,4 γ4,5 γ5,6 γ1,6 P4 R
θ C θ imin= 1 d2 imin = 2 d2 ε1 = ε2 = ε3 = 1 qθ([Ci]) = 1 + εi δi Ci εi = 1 δi= 0 i εi = 1 εi= 1 εj = 1 j 6= i 1 εi= 0 i εi= 1 i (δ1+ δ2) + (δ2+ δ3) + · · · + (δ6+ δ1) = 2(δ1+ · · · + δ6) i δi+ δi+1 = 0 γi,i+1 εi= 1 δi+ δi+1= 1 i δ1= 1 δ3= 1 γ1 γ3 ε1= 1 ε3= 1 i εi = 1 2 εi= 0 i 6= 5 εi= 1 δ1 δ3 δ3+ δ4 δ1+ δ6 2 i εi = 1 δ4 = δ6 = 0 C5 δ5 = 0 δ5+ δ4= δ5+ δ6= 0 γ4,5 γ5,6 ε4= ε5= ε6= 1 3 ε2= ε5= 1 δ3+ δ4 δ1+ δ6 εi= 1 δ1= δ3= 1 C2 δ2= 1 γ2,3 γ1,2 ε1= ε2= ε3= 1 4 ε2= ε3= ε5= 1 ε1= ε4= ε6= 0 δ4 δ4= 0 C3 δ3= 0 δ3+ δ4= 0 γ3,4 ε4 = 1 ε3 = 0 δ3 = 0
γ3 ε3= 1 δ4 = 1 C3 δ3 = 1 δ3+ δ4 = 0 γ3,4 ε4 = 1 ε3 = 0 arf (qθ) = 6 X i=1 (1 + εi)δi = δ1+ δ2+ δ3= δ1+ δ2+ 1 δ1+ δ2 = 1 δ1 = 0 γ1 ε1 = 1 δ1 = 1 C2 δ2= 1 δ1+ δ2= δ2+ δ3= 0 γ1,2 γ2,3 ε1= ε2= ε3 = 1 4 εi= 1 5 ε2= ε3= ε4= ε5= 1 arf (qθ) = 6 X i=1 (1 + εi)δi= (1 + ε1)δ1+ (1 + ε6)δ6 ε1= ε6= 0 δ1+ δ6= 0 γ1,6 ε1 = ε6 = 1 ε6 = 1 ε1 = 0 δ1 = 0 γ1 ε1= 1 εi= 1 i
m
Pn
