Thin Non Vanishing theoremX projective Danef Cartier

divisor

G a divisorsuch that

1 AD G K is big turf forCartier some a

0

2 CX G is Ktt

Then for my O

HIX MD tri 1 0

Thin basepoint free theorem

X projective CX D kelt pair

with A effective D pet

Cartier divisor such that

AD Kato is big net

for some cis0

Ther D is semiamplelb Dl is bpf for bro

Non Vanishing bpf rationalitycone

bpf cone contraction

Strategy Bpf

X smooth D O M is ample

F effectiveintegral

70 12 14 3k tFtM d LkFtM

o

F If KE

By Kodaira Vanishing Hx CK tH o

HOCK QC Ft I Hoff lketMHl

BD K Mtf4

if we can write

Need Kawamata viehweg vanishing holds

for K tM

Obs 1

f YX proper

birational

map

b f Dt E

E effective f exceptional

Hock O BD HEYdo bf'D

wantHOLY dy bf'D EH

bf D Ky N t Dyt F F Effective

bigInet felt boundothirreducible

as tDyFff SuppDyNytDFObs 2 a fat fort

satisfiescan assume f exc hypothesis

me notf Imp IL t I r E t Irie Fk exc

i mmmoving pot fixedbpf

rj he O

ke cb EEDt n

b conto t f't D KylZa E

t I rjEjbig t hot

I f Fk

Ky f K TajEjB c

mmmm

Car a Ej t 2 erk Fk

Dy t F E

pick c s t

Max 9 co aj erk

min C 1 I I

F L B o reducedintesse divisor

B c L B GD Dy Er eeffective

Need Dq to be a blt boundary

E effective exceptional

since L 1314 LBC EO

Dy pet boundary

E 2 Car a E is exceptional

Crj Aj Eo fSuppHY

So we've writtenintegral

effective

bf D Ky N AptF E exceptional

7 abig thef kiltboundary

issues f f is not irreducible

2 Nlp doesn't haveto be bigthef

solution we perturb by

p F to make

F irreducible GN ample

o Pj I

proor thutnonvorishing 7b.pe

D net CX D proskit

at Kab is bigthief for

some a O

Stept MD form

o

f Y 7 X logresolution

s.fiKy f Kitts t 2 aj F a I2 f k t jF is ample

for osp cal

al texts bigtnef effectiveAD D Ah INV

9ample

D Ky tIg

Aj P 7 Ieffective

claim fol is f exceptional

Uj P OF E foliff aj Pj

0

Fj exceptional b cD is effective

H Y Ou mf'D fed HoH 9GDI

itby nonvanishing b c

of Dt G Ky is bigthief

Stepf since Imp 0 for mO

Stable base locus Noetherianinduction

n BschnDl BD BsClmDDq

MEN set theoretic somemiso

fix such an m

suppose B IMDb

pick log resolutionsatisfying

1 1 2 from steph as

well as

3 f lmDl ILIt ENT r o

e Tbpf fixed

a F BsllmDl Bsf4mD as

sets

b BsumDl U f Fj 1 rj o

proof by contradictionfind an Fn ul Rmsit Fm BsumDl

steps apply the strategy

µ b c b f D Ky Elaor PIE

E b cm a FDrn

big ther

ichbp f

t f't AD D 2 Pj5

as long asample

es o b Cmt a

Ncb 47 af D Ky E Taj of PIF7Pick op c sit FAI F

min a or PI t achieved

Er

a uniquej k

2 og crj Pj Fj A FnF Fk E DYeffExe T put boundary

step lifting sections

K.ytfnlb.cl I b Dt TATF

Oy bf'DFAT F dy bF'Dtm

ly OfHDtattle

o

Ky tramplerH OylKytrauplet 00 K

HOLY dy bf DtfAHoff dpkf.DE

D

bf DtfAXfKFCbf DtfAI kx F f

NCas4IEis ample

6 fAfp 3 nonvanilhing on F

4494121o SE Ho Oy

bf DtfAI

ISlp

0effective

F Fu FAI exceptional

Ffa B Ibm forall large

b DO

B