AG for NT 10
Goal:
To understand the theory behind blowdown for arithmetic surfaces, learn minimal model and maybe canonical
model.
Give intuition and where hypothesis (normality, regularity)
Everything here will be Noetherian and of nite type
1
Prerequisites
1.1 Sheaves of dierentials
f : A → B,
Start with rings:
and where
Example.
d
B = A[x1 , . . . , xn ]/(f1 , . . . fm ).
Leibniz rule). This is a B -module.
write
is evaluated using
A = k[y] = k[xn ] ,→ B = k[x],
so
B = A[t]/(tn − y).
Ω1B/A
Let
Ω1B/A =
P
Bdxi / hdfj , dai
(for all
a∈A
Then
= Bdt/d(tn − y)
= Bdt/(ntn−1 dt − dy)
= Bdt/ntn−1 dt
= B/ntn−1
B -module correspond to a sheaf on Spec B
0
−1
Taking B = k[x, x
] then Ω1B 0 /A = 0
This
The idea is that
Let
f :X→Y
Ω1B/A
supported at
0 only.
This is exactly where the map
x 7→ xn
ramies.
detects smoothness and ramication.
be a morphism of schemes. Then this construction sheaes and gives
Ω1X/Y ,
a sheaf on
X.
Properties
• f : X → Y equidimensional ber of dimension n
Ω1X/Y is locally free of dimension n around x.
So
of
f is smooth if and
f are all smooth)
• i : Z ,→ X
Ω1X/Y
only if
and
x∈X
is locally free of rank
a closed immersion with dening sheaf of ideal
0→
i∗ (I/I 2 )
| {z }
n.
a point. Then
(Note that
I ⊂ OX .
f
f
is smooth over
Y,
then
Z
if and only if
In general, there is a sequence
→ Ω1X/Y ⊗Ox OZ → Ω1Z/Y → 0
is smooth if and only if this sequence is left exact.
1
x
is smooth if and only if the bers
=:CX/Y the conormal sheaf
X
is smooth at
1.2 Local complete intersections
f : X → Y
f
morphism of schemes.
neighborhood
x∈U
local complete intersection
is a
such that there exists
Z
if it for every
reg immesion
A
smooth
Z
Where a regular immersion is dened as follows: On rings this corresponds to
A/(x1 , . . . , xi−1 )
for all
Geometrically, this means that
Y = Spec k ,
Intuition:
then
Z
U
there is a an open
/Y
?
f
U o
BO
a zero divisor in
x ∈ X
and
i ≤ d.
B = A/(x1 , . . . , xd )
where
xi
is not
Successive quotient by non-zero divisors.
is dened by a number of equations equal to its codimension in
is a smooth variety
K
(such as
Ank )
and
X
Z.
is locally dened by the appropriate
number of equations
Complete intersection :
Example.
Same denition as above except with
U = X.
This is much more restrictive
of a local complete intersection that is not a complete intersection is the twisted cubic, i.e, Proj(k[x, y, z, w]/
While this is a curve, it needs
3
equations to dene it and not
2.
Example.
1. Curves over a eld are local complete intersections except if they have embedded point. An example of non
local complete intersection:
2. Let
3.
R
be a Dedekind ring,
f :X→Y
Example.
X
F ∈ R[x, y].
Then
R[x, y]/ hF i
is a local complete intersections.
morphism of regular schemes is an local complete intersection.
Denition 1.1.
i∗ det Ω1Z/Y .
k[x, y]/(x2 , xy).
The
canonical sheaf
of a local complete intersection
This is locally free of rank
is a curve smooth over
X → Y
is
∨
ωX/Y = det CX/Z
⊗O X
1.
Y = Spec k ,
then
ωX/Y = Ω1X/k
Properties:
• ωX/Y
is stable under at base change
p
X0
Y0
(either
α
or
Additionally
•
β
is at), then
f :X→Y
For composition:
α
/X
/Y
β
∗
ωX 0 /Y = p ωX/Y
is a local complete intersection if and only if it is so berwise
f : X → Y , g : Y → Z . ωX/Z = ωX/Y ⊗OX f ∗ (ωY /Z ).
formula in disguise)
The sheaf
ωX/Y
gives Serre duality:
H 0 (X, ωX/Y ) = H d (X, OX )∨ .
2
(This is the Riemann-Hurwitz
xz − y 2 , y
2
Arithmetic surfaces
Denition 2.1.
A
bered surface
dimension Dedekin scheme (like
is an integral surface
X
with a projective at map
π:X→S
where
S
is a one
Spec Z).
See diagram on slides for intuition
Divisors on
X
comes in two avors:
•
Vertical one (components of special bers
•
Horizontal ones (closures of points in
XS )
Xη ).
Properties:
• Xη
•
If
is geometrically integral (like a smooth curve), hence
Xη
∼
OS → π∗ OX
is smooth, then there are only nitely many non-smooth bers (proof: Smooth locus is open, use
and non-empty. Its complement is closed hence so is its image under
• ωX/S |XS = ωXS |k(S) which
π
(proper), therefore this image is nite)
follows from base change
• ps (XS ) = pa (Xη ), where pa of a curve is 1 − dim H 0 (C, OC ) + dim H 1 (C, OC )
0
smooth, we have = dim H (C, ωC ) = g(C) the usual genus of C )
Denition 2.2.
π:X→S
Ω1X/S ,
is called
normal
if
X
is, and
regular
(or an
(and in the case the curve is
arithmetic surface ) if X
is regular.
2.1 Desingularisation
Process of nding
If
Xη
Y 99K X
birational (isomorphism
Yη → Xη )
such that
Y
is regular.
is smooth, one can do:
XO
XO 1
normalisation of X
X10
O
blowup of remaining sing in X1
XO 2
X20
Fact. This stops and gives a regular surface at some points. After further blowup, all special bers can be taken to
have normal crossings.
2.2 Contraction
X→S
arithmetic surface,
E
component of special bers
X
f
XS .
/Y
S
3
We want typically to construct a morphism:
contracting
E
means we want
f (E)
f is an isomorphism outside E .
L an invertible sheaf on X : H 0 (X, L) = Rs0 ⊕ · · · ⊕ P
Rsn . This gives a
x 7→ (s0 (x) : · · · : sn (x)). This is well dened as long as Lx = i si OX,x , i.e., L
to be a point and
This is done by using invertible sheaves.
morphism
fL : X → Pn
dened by
should be generated by its global sections.
f = fL , suppose Z ⊂ XS is a projective component of bers, then f (Z) is a point if and only if L|Z = OZ .
⇒: suppose the point is (1 : 0 : · · · : 0), then by construction s0 generates on all points above y .
⇐: Restricting to Z the space H 0 (Z, L) is nite (Z is projective), so we get f |Z : Z → H 0 (Z, L) → PnA .
Fact. Birational maps
X → Y between normal bered surfaces are sequences X 99K X1 99K X2 99K · · · 99K Y
(where the map is either a blowup of a point or a contraction of a curve to a point).
Contraction Criteria
Let
E
be a set of vertical divisors: Contraction of
E
exists if and only if there exists a carties divisor
D
on
X
such
that
• deg(D|Xη ) > 0
• OX (D)
generated by global sections
• OX (D)|E ∼
= OE
Over ane
if
n 0.
D+E
S,
for
E
vertical if and only if
E∈E
for any eective horizontal Cartier divisors
D,
may not be generated by global sections even if
D
the sheaf
and
E
OX (nD)
is generated by its global sections
are.
2.3 Intersection Theory
Let
X
D, E divisors on X . Suppose that D and E have no common component. D, E then interP x ∈ X is a point of intersection, we set ix (D, E) = lengthOX,x (OX,x /(OX,x (−D)+
i(D, E) =
x∈X ix (D, E). (With the convention that if x is not an intersection point the
be a bered surface,
sect in nitely many points. Suppose
OX,x (−E)) and
ix (D, E) = 0)
let
D|E = j ∗ (D)
Alternatively:
j : E ,→ X , then ix (C, D) =multiplicity of x on D|E .
s ∈ S , is : Div(X) × Divs (X) → Z. If E is a component of XS
where
On a bered surface we get for
then
degk(S) OX (D)|E .
Properties
• Xs
ber of
• X→Y
•
then
a contraction,
is
is negative denite and
Γi → y .
Look at
P
a point of
Xη .
TakeK such that
Consider
x·x=0
ni nj Γi Γj ≤ 0,
Hodge index theorem for ordinary surfaces Pic(X)
• P
•
X→S
x ∈ ZXs .
with equality if and only if
× Pic(X) → Z,
{P } · Xs = [K(P ) : K(S)],
ωX|S = OX (K),
implies
hence it is
has signature
1
if
P
then
2pa (Xη ) − 2
=
deg ωXη |k(η)
= −2χk(η) (OXη )
=
2Xk(S) (OXS )
=
deg(ωXs |k(s) )
=
deg(OX (K)|XS )
= K · XS
X
=
di (KX|S · Γi )
where
Γi
are components of
XS
and
di
are the length of
4
OX,Γi .
ni = 0.
(1, −1, . . . , −1).
is a rational point
is (D, E) =