Birationally rigid varieties with a pencil of Fano double covers. I

V

π:V→P1

V/P1

V

Q

χ:V−−→V′,

V′

V′

χ∗∈BirV

c(Σ)≤c(Σ′),

Σ=(χ◦χ∗)∗Σ′

χ◦χ∗

:V−−χ∗

→V−−χ

→V′,

|·|

c(·)

c(Λ)=sup{ε∈Q+D∈Λ

|D+εK∈A1+(·)},

ΛK

A1+(·)⊂A1(·)⊗R

χ∗

V

=idQ

π:V→P1

V

V

π:V→P1

V

V

PicV=ZKV⊕π∗PicP1

Σ′

Σ′

V

V

V

Ft=π−1(t)

l∈Z+

t∈P1F∈PicV

D∼−nKV+lF

KV∈IntA1+V.

V

V

V

π

χ

τ:W→T

χ:V−−→W

Vπ↓P

−−→W

↓τ−−→

χ

1

α

T

τ:W→T

τ∗PicP1χ:V−−→W

α:P1→T

PicW=ZKW⊕

χ

V

V

V/P

1

V/P1

C

P

PM+1

G=P(H0(P,OP(m)))

2l

P

m3≤m≤M−1Wm+l=M+1

F={F|σ:F→G}

2:1

W∩GW∈W

F∈Fsm

regFsm⊂Fsm

Fsm⊂F

(G,W∩G)

G∈G

PicF=ZKF

F∈Fsm

FregF∈Fsm

reg

sm

codimFreg

sm(Fsm\\Fsm

)≥2.Fsing=F\\FsmcodimFFsing=1

Fregsing

codimF(Fsing\\Freg

sing)≥2.

F

V/P1

Freg=Fsmreg

∪Fregsing.

codimF(F\\Freg)≥2P1→F

F

Ftreg

1

∈Ft∈P1

V/PV/P1

KV2∈IntA2+V,

A2+V⊂A2

V⊗R

V/P1Ai♯♯

V/P1

BirV=AutV=Z/2Z={id,τ},

τ∈AutV

Fsing

Ft=π−1(t)t∈P1

V/P1

V/P1

V/P1

K2

i

V/Q

V

V/P1

m=1

1

P

m=2

V/P1

P2

V4⊂P4

V/P1

K2

AutP1

P1

K2

K2

K2

P

1

K2

π:V→P1

Fπ(Y)=P1

degV=degF

PicV=ZKV⊕ZF,

degY=(Y·F·(−KV)dimY−1),

−1Y⊂π(t)

degY=(Y·(−KV)dimY).

V

V

Y⊂V

V/P1

Y⊂π−1(t)=Ft

Y

o∈Ft

multo

degV

;

V

o∈Ft

Y

multo

󰀎tF

Y⊂Ft

Ft󰀎tx∈F󰀎⊂F󰀎toϕ(x)=oY

󰀎t→Ftϕ:F

degV

,

󰀎multxY

degV

;

o∈Y

Y

multo

degV

.

dimV≥4

V

2

A2V=ZKV⊕ZHF,

HF=(−KV·F)

A2F=Z(HF·HF)F

2

A2+V⊂ARV

F=Ft⊂V

2∼2A2RV=AV⊗R=R

{λ∆|λ∈R+,∆

}.

V/P1

K2

2

KV∈IntA2+V.

a≥1

b≥1

K2

2∆(a,b)=aKV−bHF

A2+V

N≥1

HF∈A2+V

2

KV∈IntA2+V

∆(N,1)∈

K2

V/P1

V/P1

F∈F

reg

V

E

Q⊂X

aQ∈Z+

aWπ−Q1(∈t)

Z+

π

P

1

V/P1

V/P1

|−KFσF:F→G⊂P

πM+2P1

∗O(−KV|)P

Q

P1

P(π∗O(−KV))

ME=

󰀋+1i=0

OP1(ai)

a0=0≤a1≤...≤ai≤ai+1≤...≤aM+1.

πX=P(E)

Proj

X:X→P1LX

XsQ∈H0X,LX⊗π∗

XOP1(aQ),

πQ:Q→P1

󰀁⊗m󰀂mπX|QP

W⊂X

Q/P1

sW∈H

0

X,L⊗X

⊗

π∗

XOP1(2aW)

,WQ=W∩Q

󰀁

2l

󰀂

Q

t∈P1

Gt

G

σ:V→Q

π−1WQ

(t)

Ft

F

P1

PicV=ZKV⊕ZF

OP1(k)k∈Z

E

π∗O(−KV)

LX|Q

LX∈PicXQ

LXLQ=

PicQ=ZLQ⊕ZG.

LV=σ∗LQ

KV=−LV+(a1+...+aM−2+aQ+aM)F.

HF=(−KV·F)

2

A2V=ZKV⊕ZHF,

HF

A1F

V

2

(KV·LM−1)=2m(4−a1−...−aM+1−aQ−aW)+2aQ.

(HF·LM−1)=2m

2

LM−1)≤0KV∈IntA2+V

|LV|

2

A2+V⊂AV⊗R

2

(KV·

σ:F→G⊂P

F∈F

F

F

m·2l

P(1,1,...,1,l),

󰀇󰀉

M+2

˜(x0,...,xM+1)=0,f

u2=g˜(x0,...,xM+1),

x∗

o∈F

W∩P

u

G⊂P=P(1,...,1)g˜

˜lf

o=(0,0,...,0,1).󰀇󰀉

M+2

o

AM+2

zi=xi/x0,i=1,...,M+1,

y=u/xl0

z∗

(0,...,0)F

(z∗,y)

f=q1+...+qM=0,

y2=g=w0+...+w2l,

qiwjp=σ(o)∈Gp∈W

p

z∗

W

i

j

w0=1

w0=0

F

z∗

a≥1

√

p=σ(o)

󰀍

f=qa+...+qm=0,

y2=g=1+w1+...+w2l,

2ii!

=(−1)i−1

(2i−3)!

g]i=1+

󰀊iΦi(w∗),

g(i)=g−[

√

j=1

(z∗,y)

q1,...,qm,gl+1,...,g2l−1

Op,P

a=1

p∈G

2l≥m+1

M−1

q2,...,qm,gl+1,...,g2l−1,

2l≤m

q2,...,qm−1,gl+1,...,g2l

PM=P(TpP)

M

q2=...=qm−1=gl+1=...=g2l=0

Z∗

PM

deg(P∩Z∗)<λm!(2l−1)!

m,l=

(l+1)!

(l−2)

m=3Z∗

λm,l

w0=y(0)=1

y−[

√(l−1)!

,

o∈F

multoC∗≥m!

(2l)!

P⊂PM

o∈F

F

Z∗

L∋oσ:L→σ(L)

C∗

σ(L)⊂P

Z∗=P(ToC∗).

P⊂P

{q2=...=qm=gl+1=...=g2l=0}∩σ−1(P)

λm,lC∗

−1

λm,lσ−1(P)

−1

F

󰀍

f=q1+...+qm=0,y2=g=w1+...+w2l.

o∈F

q1,...,qm

Op,P

q2

{q1=w1=0}

o∈Fq1

w1

w1=0}

W∩Go∈F

σF:F→G

W∩G

p=σp

(o)q1=0

WG

w1=λq1,

λ∈C

q1=zM+1

w¯2=w2|{zM+1=0}

(z∗,y)

{q1=σF

WG=

G

󰀎→GϕG:G

p

EG∼=PM−1

z1,...,zM

EG

WE={w¯2=0}.

qi

EG

qM+1=0

q¯i

EG∼=PM−1

(z1:...:zM)

p=σ(o)∈WG

q¯2=...=q¯m=0

Z2·...·m

(m−1)

q¯2=0

WE

F∈F

F∈Freg

Fsm

o∈F

F∈Fsing

p=σ(o)∈WG

F∈F

codimF\\Freg

sing(Fsingsing)≥1

codimF(Fsing\\Fregsing)≥2.

F∈F

F

o∈F

F

V/P1

multo

degV

,

o∈F=Ft

Y⊂F

T=σ−1(TpG∩G)

p=σ(o)

T⊂F

degT=degV

multo

degV

Y=T

YT

Y∩T

FY

T

Z

multo

degV

.

Y⊂V

o∈F

multo

degV

.

π(Y)=P1

Y=FY◦F)

F

Z=(o∈F

◦F)

F

multZ=(Yo

8

deg

Z>

deg

σ(Z)>

4

multoT=2

Z=(Y◦T)

o∈F

p∈G

p∈G

q2,...,qm

degG≤dimG+1

󰀎→Fϕ=ϕF,o:F

󰀎→GϕG,p:G󰀎EG⊂G

G

oϕG=

󰀎p=σ(o)E=EF⊂F

ϕ∗(|kHF−(k+1)E|)

F

(ϕG)∗(|kHG−(k+1)EG|)

G

Λk=ΛFk

Λk

k

ΛGk

|kHF|

G

󰀎k⊂|kHF−(k+1)E|,Λ

σ∗ΛGk⊂Λk,

p∈WG

σ:F→G

p∈WG

󰀎→G󰀎F

p∈WG

ΛEk

󰀎ΛEk=Λk|E

󰀎GΛEk=Λk|EG,

󰀊(BsΛk◦E)=BsΛEk

z1,...,zM+1

p

p∈G

G

f=qa+qa+1+...+qm=0,

a=1

ΛG

k⊃󰀈

󰀈󰀈󰀊

ksk−ifi󰀈󰀈󰀈q󰀈

󰀈

i=a

f󰀈,i=a+...+q󰀈i,

k≥a

sj

z∗

fi|G=(−qi+1−...−qm)|G.

p∈WG

Λk⊃σ∗ΛGk

Λk

σ

p∈WG

Wt=W∩Pt⊂P

g(z∗)=1+w1+...+w2l=0,

wi(z∗)

i

√

g]j=1+

󰀊jΦi(w∗(z∗)).

i=1

Λk⊃󰀈

󰀈󰀈kmin{󰀈󰀈󰀈󰀊sk−ifi+󰀊k,2l−1}

s∗k−i

(y−[√i=ai=l

j

Λk

s∗k−i

z∗

k−i

k≤l−1

z∗

√

(y−[

2

wi+Ai(w1,...,wi−1).p=σ(o)∈WG

p

Wt

g(z∗)=w1+...+w2l=0,

q1w1

F

{w1|G=0}

p=σ(o)

󰀈󰀈

k󰀈󰀈󰀊

󰀈󰀈

sk−ifi+sk−1w1󰀈.Λk⊃󰀈

󰀈󰀈

i=1

WG

ΛGk

Λk⊃σ∗ΛGk

o∈F

[a,b]⊂R

ba=multpG∈{1,2}p=σ(o)

M=[a,m−1]∩Z+={a,...,m−1},

L=[l,2l+a−3]∩Z+={l,...,2l+a−3}.

o

ML

oML

e=max{m−1,2l−1}Λ∞

Λe

p=σ(o)∈WG

G

codimoBsΛk≥codimEBsΛEk≥♯[1,k]∩M+♯[1,k]∩L,

p=σ(o)∈WG

dimoBsΛ∞≤1,

G

codimBsΛk≥codimEBsΛEk≥♯[2,k]∩M+♯[2,k]∩L,

σ−1(P∩G)

P⊂PP∋p

BsΛ∞⊂C∗

Λk

PF

FΛPk=Λk|PF

k≤max{m,2l}−2

PF=

codimPFBsΛPk≥♯[2,k]∩M+♯[2,k]∩L,

dimBsΛP∞≤1,

BsΛP∞

p=σ(o)∈WG

λm,l

WG

codimoBsΛk≥codimEBsΛEk≥♯[1,k]∩M+1,

p=σ(o)∈WG

WG

codimoBsΛk≥codimEBsΛEk≥♯[1,k]∩M.

Λk

Y⊂F=Ft

m

x∈E

󰀎multxY

,

󰀎E⊂F

󰀎→Fo∈Fϕ:F

o∈Fp=σ(o)∈G

󰀎→GϕG:G

p

p∈W

σ

G=Gt

σ−1(p)={o,o+}

E

󰀎\\{o+}→G,󰀎σ˜:F

󰀎E⊂F

σ˜

ϕG

E

p∈P

◦F

M−1

oEP⊂P

󰀎F

Y

Y󰀎󰀎P=(󰀎◦P󰀎)P

HT=P(TpP)∼=P

M

ϕP:󰀎P

→PE⊂F

E⊂G

󰀎󰀎YP⊂PP∋pσ(Y)⊂P

P=(Y◦FPF)PF=σ−1(PG)PG=P∩G

multo

2m

.

YP

F

V

PFo∈PF⊂F

ϕP:P󰀎→PFP

F

E

Y

󰀎󰀎󰀎EPE

E֒→T

YP

P

Y

󰀎P∼aHP−bEP,󰀎b>

3

Z∼αHPP

P

β>

3

Z

{Cδ,δ∈∆}

TP

β>

3

−βE󰀎2

α.

T

P󰀎o∈F

TP=T∩PF

multo

3

.

{C

P

TP󰀎δ,δ∈∆}(Z·C

󰀎󰀎󰀎P⊂󰀎δ)=αdegCδ−βmultoCδ<0,bm=4

e≤2l≥3

ce=0

[a,b]⊂R

e≥max{m,2l}−1

ce=m+l−4m≥4

♯[4,e]∩M=0,

l=2

m=3

u∈U

ce+1≥ce.

χ:{1,...,m+l−4}→Z+

χ([ce−1+1,ce]∩Z+)=e.

ce−1=ce

[ce−1+1,ce]

ce+1−ce∈{0,1,2}

χ

ΛPi

Λi

PF

m+l−4ΛP=

󰀌ΛPχ(i).

i=1

Λe

D={Di∈ΛPχ(i),i=1,...,m+l−4}∈Λ

P

i

(Γu,u∈U)

Z

Γu

i

i=1,...,m+l−4

Z

Ri(D)=

ij󰀐

iD=1

∩TPD∈ΛP

i

TP

i=1,...,m+l−5

(Ri(D),D∈ΛP)

i

T

R0(D)=T

χ(j+1)=e

i=1,...,m+l−4

i≤j≤m+l−5j=0

Rj+1(D)=Rj(D)∩Dj+1,

Dj+1Pχ

∈Λe

Rj(D)

Rj+1(D)

ce=ce−1+2

j+1∈[ce−1+1,ce].

codimPFBsΛPe≥ce+1,

codimTPPBsΛe|TP≥ce,

codimTPRj(D)=j≤ce−1

PΛe

Rj(D)⊂Dj+1

j+1

Tj≤m+l−6

P

e≤max{m,2l}−2,

codimPFBsΛPe≥ce+2,

e=max{m,2l}−1

j+2∈[ce−1+1,ce],

codimTPRj(D)=j=ce−2.

Z

Rj(D)

j≤m+l−6

codimZBs(ΛPeΛP|Z)≥2,

e|Z

(R(D)=Rm+l−4(D),D∈ΛP).

R(D)

R(D)=(TP◦D1◦...◦Dm+l−4)=

δi∈∆

(Cδ,δ∈∆)

Φ

󰀊

Cδi+Φ,

R(D)D∈ΛP

Φ=BsΛP∞.

(Cδ,δ∈∆)

TF

m+l−41degR(D)=4m

󰀌χ(j)=4mj=1

󰀃m󰀌−j󰀆󰀃2󰀌

l−1j󰀆

=

j=4

j=l

=

2m!(2l−1)!

6(l−1)!

(Cδ,δ∈∆)

T

multo/deg

multoΦ=degΦ

multo

multoR(D)−degΦ

deg

Cδ=

l)!

4·

(2m!

deg

Cδ≥

(2ll−!1)!

−λm,l

3

3·

Cδ

D∈ΛP

l≥3

λm,l

m≥5

l=2

ce=♯[3,e]∩M+♯[3,e]∩L,

e∈Z+

ΛP2

ΛPlj≥j3

≥3

ce

≥3

m≥3

lD={Di∈ΛPl+i|i=1,...,l−1}.

ΛPl+1

m≥4

ΛPl

Rk(D)=

󰀃

j󰀐

kDj

=1

󰀆

∩T

k≤2l−2

Rk(D)

T

R2l−1(D)

Φ

degΦ<λ3,l=12(l−2)

(2l−1)!

χ

E=EF

σ˜󰀎→󰀎⊃EG⊂FG

ϕF↓↓ϕG

σ

F→G,

ϕF

ϕG

o∈F

p∈G

EF

EG

σ˜

WG

σ˜E=σ˜|E:E→EG∼=PM−1

WE=W

HE

E֒→P

M

󰀏G∩EG.PicE=ZHE

Wt=W∩Pt

h=w1+w2+...+w2l=0,

G

f=q1+q2+...+qm=0

=(z1,...,zM+1)

p

WG

p

z∗q1=zw2|{zM+1=0}

zM+1

1,...,zM

EG

σ˜E:E→EG∼=PM−1

WE={w¯2=0}

y∈EG\\WEC(y)⊂EG

yWE

πy:EG\\{y}→PM−2

y

πy|WE:WE→PM−2

Q(y)⊂PM−2

C(y)=

E

L⊂EG

WE

󰀏W

G⊂G󰀎w1=λqw¯1

2=

q¯i

qi

qM+1=0

q¯2=...=q¯m=0

EG

Z2·...·m

q¯2=0

(m−1)

WE

C(y)y∈EGy∈WE

\\WE

Z2·...·j={z∈PM−1|q¯2=...=q¯j=0}.

Z2·...·j

j

h0(OZ2(2))=...=h0(OZ2·...·j(2))=...=h0(OZ2·...·m(2)),

H0(OPM−1(2))→H0(OZ2·...·m(2))

Z2·...·m

C(y)y∈EG\\WE

H0(OPM−1(1))→H0(OZ2·...·m(1))

TZ2·...·m

yWEy∈WE

R

x∈E

R⊂F

σ˜E

σ˜(x)∈WE

x∈E

󰀎⊂F󰀎WE

µ=mult󰀎≤1xR

TZ2·...·m

yWE

Z2

µ>2k

∼kϕ∗HF−νE,RF

multoR=2ν

󰀎2k

ν>2kν≤multo

m

.

¯R

=σ(R)⊂Gσ:R→R

¯multo

m

.

p∈G

T1+=TpG∩G

multp

m

,

R¯=T1

+(R¯◦T1

+)pT1=2

mult+

multp(R¯◦T1

+)≥2multpR.¯deg(R¯◦T1+)¯Y2¯=degR

(R◦T1

+)multp

deg

R.

¯f=q1+q2+...+qm

z∗

f=q1+q2+...+qi,

ΛG

i=󰀈󰀈󰀈󰀈

󰀈

󰀊

ifjsi−jj=1

|G=0󰀈󰀈󰀈󰀈󰀈,Ti+={fi|G=0}∈ΛGi.

G⊂P

T1

+

p

G

Ti=σ∗Ti+,

Λi=σ∗ΛGi.

codimGBsΛG=i

iBsΛGi=T1+∩...∩Ti

+

mD=(D1,...,Dm−1)∈

󰀌−1ΛGi

j=1

YYii=1,...,m−1=R

¯1

YcodimGYi=i

Y2i+1Yi∩Di+1

⊂YiYi⊂Di+1Yi+1

multp

p

i+1

·

mult4deg

Y≥

m

󰀇

m−2·····

12󰀟m

󰀉multp

2m

.

multp/deg

deg

R,

¯TpG∩G

ν≤2k<µ

B=TxE∩E

󰀎≤multx(R󰀎◦E)≤deg(R󰀎◦E)=2ν.µ=multxR

E

PM

multBR

󰀎≥1B

T1∩...∩Ti

i=1,...,m−1

L=(L2,...,Lm−1)∈Λ2×...×Λm−1

o∈F

Lj

󰀎F

l≤m−2

Lj∈Λj

Y⊂F

󰀎j◦E).Lj=(L

󰀎jL

Y⊂E

Ll+1∈Λl+1

Ll+1∈Λl+1

󰀎j∩ELj=LY⊂Ll+1

Y⊂Ll+1

l≤m−2

codimFBsΛl+1=l+1,

codimEBsΛEl+1=l+1Lj∈ΛEj

Lj∈Λj

L

codimF(R∩L2∩...∩Lm−1)=m−1,codimE(RE∩L2∩...∩Lm−1)=m−1.

m−1

F

E

R+=(R◦L2◦...◦Lm−1)

L

+

RE=(RE◦L2◦...◦Lm−1),

+󰀎+◦E)RE=(R

R+

o

degR+=2km·(m−1)!=2km!,

+

multoR+=degRE=2ν·3·...·m=νm!.

Y

R+

Y⊂T1

Y=T1∩T2∩...∩Tm−1.

Li

f1si−1+f2si−2+...+fis0,

sj

Li

j

z∗

Y

s0=1

s0=0Y⊂T1

f1,f1s2,1f1s3,2

+f2,+f2s3,1

+...

f3,...

+fm−2sm−1,1+fm−1,

f1sm−1,m−2+

si,j

j

f1|Y≡f2|Y≡...≡fm−1|Y≡0,

Y⊂T1∩T2∩...∩Tm−1

Y

T=T1∩T2∩...∩Tm−1,

T=(T1◦...◦Tm−1)

T=T1∩T2∩...∩Tm−1.

T=(T1◦...◦Tm−1)E

degT=multoT=degT=2m!.

R+=aT+R♯,

♯+

RE=aT+RE,

a∈Z+

T

R♯

T

♯󰀎♯◦E)RE=(R

R♯

o

♯

RE

2multoR♯≤degR♯.

Y=T

Y

Y⊂T1

R♯

T1∩SuppR♯

m

R∗=(R♯◦T1)

2multoR♯≤multoR∗≤degR∗=degR♯,

B∩L2∩...∩Lm−1

+RE

m−1

E

B+

B

+RE

=

󰀊

i∈I

riYi,B=

+

i∈I,Yi⊂B

󰀊

riYi.

degB+≥(µ−ν)m!

RE=(µ−ν)B+∆,

∆

deg(B◦L2◦...◦Lm−1)=2·3·...·m=m!,

m−1

E

z∗sj

Y⊂L2∩...∩Lm−1

Y⊂T1Y=T

Li

q2si−1+...+qi+1,

Y

j

Y⊂T1

q2,q2s2,1

+q3,

......

+qm−1sm−1,1+qm,

q2sm−1,m−2+

degsi,j=j

q2|Y≡q3|Y≡...≡qm|Y≡0,

Y⊂T

Y=T

B∩L2∩...∩Lm−1

T⊂B

T1Y

Y⊂T1

Y=T

degR+=2km!=2am!+degR♯,multoR+=νm!=2am!+multoR♯.

B+

♯RE

♯

degRE≥degB+≥(µ−ν)m!.

♯

degRE=multoR♯

2(νm!−2am!)≤2km!−2am!.

νm!−2am!≥(µ−ν)m!.

k+a≥ν,2ν−2a≥µ,

µ>2k

Q