MMP for surfaces
X smooth projective surface
Kymis not
Kk
X Xo X Xm70
Ehmnot net
Xi is smooth m is a FaroXm is fibationXi Blp it a minimal HLNSurface
Kxm net Kym is semi ample
4 0 1dk.nl Xm ZELP
Z Pro R k dim 2 KCA
KI is birational Kz ample
2 the canonical modelXm the
miming
Z has mild singularitiesk Of X Z smooth curveM
with genus 1 fibers ellipticfibutionsK of X pt X is K trivialM M
i e.dk NOXm
k D e g Kxm
is not nef
Xm 7C a ruledsurfacesFaro fibation overPFE a curve
22 Xm P Faro variety i e
Faroa fibrationover
hCwvesonSwfa VI Hartshorne
X smooth pvj surfacea curve in X D is a Cartier diviso
D C deg ICD lcEZProf This product extends to a uniquepairingDiv x x Divx Z
D CDC1 3 D C
if D C intersect transverselyeffectiveher D c Dnc
2 The product depends only on Nclasses PidNxpic 2
Adjunction DEX is a curvethen KatD ID KD
Wx D ID WD
Riemann Roch offsetting on Xd
2 0 43 3 D D Kitt Xcxl
x QCD b'K.QCDDtbcx 9co
Nun equivalence
D D D c D c foraycurves C
Pic IR Q N x
N'Cx N'Cx IR nondegenerate
Paladin N'CX s N Picard Park
HodgeindextheoremThe intersection paining on N x
has signature Cbp Df T
one of Curves
NEW Eai Teil is guidedED numerical
eq class
TECH closure of NE E N'G
A divisor L is not if L C70
for any curve c effective
Netcong is the dual coreunder intersection to NTICH
Thin Kleiman's criterion
L ample L 70an NICK TH
Thin Nakai MoisheZon
L is ample L.co foreffectiveo
crues
Factsuppose L is semi ample
DL is base point free
4 0114 X ZERN dL fH
CE p L C O
R If is birational EEO Kp
EZ 40 exceptional curves
40 x z curveDE 0 CP DIO
132 0 D MFF 4 p
2 Blowups of smooth surfaces
U smoothX BlpX X op
pmjotmpdd70
Facts I 3 H
E µ p
D EEP z E I
NEE gpamixXlp
4 Pick z Pic ZEp p ti
5 ka n't k t E Yestiftians
DI F EX is a GD curveif F Elp t E2 I
E is a CD Curve
E LO
Kx E do
Thm_ Castelnuovo's Contraction Theorem
X smooth proj surfaceUl
F is a GD curve
The F u X x proper birationalset µ E P X is smooth
it is the blowup of X at p
proof want ar L sit4,4 is a morphism 4µ n
L E o Y C is not zerofor any otherA very anpuwA E k O E I
At KE satisfies L E OLalso L C 0 bc H C o
E c o
Ln A th E n o K
Steph H X O Ln 0for all OE ne k
Ln A
Wloy H X 0 97 0
by Serre vanishing
I induct
At E Q Ame OECAHEHO
o QC E Oe 0700 44
H 4 d H AHEDO 0
F P de Athe pCKon
Spep2At ne is bpf for allOE ne k
Bs Atn E E E
IAthEEIAthElwait to show no basepoints
in E o
Ho Aine HoldefAxnet itO 4 SpPEE
z s sit de AthElSep toQpick h
A HE is bpf 70
Stapf 4 4 meX X EPN
is on isomorphism away from Eb c it containsOf E P 1Altne w is
very amplepi E away from E
ta t
Step w X X
Xo f is an isomorphismaway from Eg t fl E PX X
f p EProperf is birational between normal varieties
f O Oxosteps Xo is smooth at PTh i III H IH
Io p 0 5 Housden
En f Ge osPqnpn
V IE Konig
Tin Hok ovcmpnqpztinkik.fr fEn
tick HYE de kE is C D curve Ie I
NEG Opi lt I Iz Cfp l
t E is Cartier I locally generated
IYIhtiafn E Cfp n by one element
n T HEIMIIIa a E E Opo I ntl deny DE
o s HolITInt H Oeg H'loan 70
G y k 1 k 70Lx y
by induction
Holdp ulk 9 kE o
gUH Gee
Il
formnot degree
p finn k1,4 k 413Xo is smooth at P
Step6 BlpXoti
X E Xo
Check that µ is an iso
µ Ox OBlpXo
Def X is a minimal surfaceif it contains ops no C 1 curves
GI for any smooth projectiveSurface z finite
blowupan sequence
X Xo X XmSot g Xi BlpXit
2 Xi is smooth3 Xm is minimal
1ft if Xi has a C D curveCastelnuovo Xi Xin
Phi f xi t1 70
classify Xm based on whether
Kym is ref or not
