AG for NT Week 8
We will use the language of schemes to study varieties
1
Blowing Up Varieties
We will construct the blow up of a variety with respect to a non-singular closed subvariety. This tool/technique is
the main method to resolve singularities of algebraic variety.
Denition 1.1.
An at 0 is constructed as follows: Take the product An × Pn−1 . If {x1 , . . . , xn } are
{y1 , . . . , yn } are the homogeneous coordinates of Pn−1 , the blowup of An Bl0 (An ) is the
The blowup of
ane coordinated for
A
n
,
closed subset dened by
Bl0 (A
n
) := {xi yj = xj yi |1 ≤ i, j ≤ n, i 6= j}
We have the following commutative diagram
Bl0 (A
n
)

/ An × Pn−1
φ
projection
& n
A
For the next few pages, we let
φ
be the morphism dened as above.
Lemma 1.2.
p ∈ An , p 6= 0
A \ {0}
1. If
then
φ−1 (p)
consist of one point. In fact
phi
gives an isomorphism of Bl0 (A
n
) \ φ−1 (0) ∼
=
n
2.
φ−1 (0) ∼
= Pn−1
3. The points of
4. Bl0 (A
n
)
φ−1 (0)
are in
1-1
correspondence with the lines of
An
through the origin
is irreducible
Proof.
1. Let
p = (a1 , . . . , an ) ∈ An ,
assume
ai 6= 0.
So if
p × (y1 , . . . , yn ) ∈ φ−1 (p)
then for each
j , yj =
aj
ai
yi .
n−1
(y1 : · · · : yn ) is uniquely determined as a point in P
. By setting yi = ai , we have (y1 : · · · : yn ) =
(a1 : · · · : an ). Moreover setting ψ(p) = (a1 , . . . , an ) × (a1 : · · · : an ) denes an inverse morphism to φ.
An \ {0} → Bl0 An \ φ−1 (0)
So
2.
φ−1 (0)
consist of all points
3. Follows from
4. Bl0 (A
n
)=
0×Q
for
Q ∈ Pn−1
with no restrictions
2.
Bl0 (A
n
) \ φ−1 {0} ∪ φ−1 {0}.
The rst component, by part
φ−1 (0) is contained in the closure of some line L in Bl0 An \ φ−1 {0}.
n
and hence Bl0 (A ) is irreducible
1
1.
is irreducible, and each point in
Hence Bl0 A
n
\ φ−1 (0) is dense in Bl0 (An )
Denition 1.3.
If
Y ⊂ An \ 0
We see from Lemma 1.2
Fact 1.4.
φ
we dene Bl0 Y to be
Ye
is
φ−1 (Y \ 0)
induces a birational morphism of
Ye
to
Y.
Blowing up is independent of your choice of embedding.
Example 1.5.
(Node)
x, y be coordinates in A2 , and dene X : (y 2 = x2 (x + 1)). Let t, u be homogeneous coordinates for P1 .
2
2
2
1
2
2
Then Bl0 X = {y = x (x + 1), ty = ux} ⊂ A × P . On the ane piece t 6= 0, we have y = x (x + 1) and y = ux,
2 2
2
hence u x = x (x + 1). This factors, hence we get a variety {x = 0} = E (this is the preimage of 0 under φ and is
2
e (This is called the proper transform of X ).
called the Exceptional Divisors) and the variety {u = x + 1} = X
e
Note that X ∩ E consists of two points, u = ±1. Notice that this values for u are precisely the values of the
slopes of X through the origin. Blowups separates points and tangent vectors)
Let
Exercise 1.6.
Let
(Tacnode)
T : (y 2 = x4 (x + 1)).
Denition 1.7.
Blow this up at the origin and see what you get.
X ⊂ An
be an ane variety. Let Z ⊂ X be a
Z dened by the vanishing of the polynomials {f1 , . . . , fk } in An . Let (y1 : · · · : yk )
k−1
n
for P
. Dene BlZ (A ) = {yi fj = yj fi |1 ≤ i, j ≤ k, i 6= j}. As before, we get a
Blowing up with respect to a subvariety.
Let
closed non-singular subvariety,
be homogenous coordinates
birational map
BlZ (A
n
)

/ An × Pk−1
φ
projection
& n
A
p = (a1 , . . . , an ) 7→ (a1 , . . . , an ) × (f1 (p) : · · · : fk (p)).
It has a birational inverse,
Exercise 1.8.
[Note:
0
Compare blowing up
y 2 = x2 (x + 1)
in
A3[x:y:z]
Also dene BlZ (X)
with respect to the
the subvariety dened by the vanishing of polynomials
= φ−1 (X \ 0).
z -axis.
fi = xi ]
For most purposes/classifying all surfaces only need to know about blowing up a point.
Example 1.9.
X be the double cone dened by x2 + y 2 = z 2 ⊂ A3[x:y:z] and let Z be the line dened by
{y = z, x = 0}. Let t, u be coordinates for P1 , hence BlZ X = {x2 + y 2 = z 2 , xt = (y − z)u}. So on the ane piece
u 6= 0, we get xt = y − z hence x2 = xt(y + z). This factorises, so we get two pieces: {x = 0, y = z, t arbitrary} = E
e (this should be nonsingular)
(the exceptional curve); {xt = y − z} := X
2
Let
Let
Invertible Sheaves
X
be a variety.
Denition 2.1.
•
F on X is a locally
F(Ui ) ∼
= OX (Ui ))
An invertible sheaf
{Ui }
of
X
so that
free
OX -module
of rank
1.
(That is, there exists an open covering
•
We will see soon that the Picard group is the group of isomorphism classes of invertible sheaves on
•
On varieties: Weil divisors are the same as Cartier divisors. A Cartier Divisor
open covering of
have
X,
and
fi
on
Ui
is an element of
OX (Ui )
fi
fj is invertible.
2
D = {(Ui , fi )}
X.
{Ui } an
Ui ∩ Uj , we
with
(think of a rational function). Also on
D
Notation 2.2. Let
L(D) to be the sub-OX -module which is generated by fi−1
Ui ∩ Uj , so fi−1 and fj−1 diers by a unit. This L(D) is called
be a divisors (Weil/Cartier), dene
fi
on Ui . This is well dened since
fj is invertible on
the sheaf associated to D = {(Ui , fi )}.
Proposition 2.3.
D, L(D) is an invertible
↔Invertible sheaves on X .
1. For any divisors
Pic(X)
2.
L(D1 − D2 ) ∼
= L(D1 ) ⊗ L(D2 )−1
3.
D1 ∼ D2
sheaf on
(linearly equivalence) if and only if
X
and the map
D 7→ L(D)
gives a
1-1
correspondence
L(D1 ) ∼
= L(D2 )
Proof.
1. The map
D
OUi → L(D)|Ui
can be recovered from
1 7→ fi−1 is the isomorphism, so L(D) is an invertible
fi on Ui to be the inverse of a generator for L(D)(Ui ).
dened by
L(D)
by
D1 = {(Ui , fi )} and D2 = {(Vi , gi )},
L(D1 ) ⊗ L(D2 )−1 .
2. If
3. By part 2. it is sucient to show that
dened by
f∈
∗
), then
Γ(X, OX
then
D = D1 − D2
is generated by
is principal if and only if
→isomorphism
−1
, so
1→f
−1
fi−1 gj .
L(D) ∼
= OX .
So
If
L(D1 − D2 ) ∼
=
D is principal,
OX ∼
= L(D).
is the isomorphism
classes of invertible sheaves.
Morphisms to Pn
On
Pn ,
{Ui := (xi 6= 0)} and on Ui , x−1
is a local
i
n
∗
sheaf O(1). For any (projective) variety X , let φ : X → P . Then L = φ (O(1)) is an invertible
∗
global sections s0 , . . . , sn (si := φ (xi )), si ∈ Γ(X, L) generate the sheaf L. Conversely, L and
the homogeneous coordinates
generator for the
sheaf on
si
Ui ∩ Vj
on
L(D) is globally generated by f
So we have a 1-1 correspondence from Pic(X)
3
L(D1 − D2 )
sheaf. Conversely,
X.
The
determines
x0 : · · · : xn
give the standard cover
φ.
Theorem 3.1.
1. If
φ : X → Pn
is a morphism, then
2. Any invertible sheaf
L
on
X
φ∗ (O(1))
is an invertible sheaf generated by global sections
determines a unique morphism
si = φ∗ (xi )
φ : X → Pn
Proof.
1. From Above
2. Lengthy argument in Hartshorne, pg 150
Proposition 3.2.
Let
L
and
corresponding to
k be an algebraically closed eld. Let X be a variety, and φ : X → Pn be
s0 , . . . , sn be as above. Let V ⊂ Γ(X, L) be a subspace spanned by si = φ∗ (xi ).
a morphism
Then
φ
is a
closed immersion if and only if:
P 6= Q
1. Elements of
V
separate points, i.e., for any
2. Elements of
V
separate tangent vectors, i.e., for each points
mp Lp /m2p Lp .
the vector space
3
on
X,
exists
s∈V
P ∈ X,
with
s ∈ mP LP
the set of
but
s∈
/ mQ LQ .
{s ∈ V : sP ∈ mp Lp }
span
⇒) If φ is a closed immersion, think of X as a closed subvariety of Pn . So L = OX (1) and the
n
vector space V ⊂ Γ(X, OX (1)) is spanned by the images of x0 , . . . , xn ∈ Γ(P , O(1)). Given P 6= Q in X , we can
P
P
nd a hyperplane H containing P and not Q. If H = (
ai xi = 0) for ai ∈ k , then s =
ai xi |X satises the rst
2
property. For the second, each hyperplane passing through P gives rise to sections which generate mP LP /mp Lp .
Proof. (Only proving
Example.
P = (1 : 0 : · · · : 0),
spanned by yi .
If
vector space
1-1
So we have a
4
then
U0
has local coordinates
↔isomorphism
correspondences Pic(X)
yi =
xi
x0 , so
P = (0, . . . , 0) ∈ Ui
classes of invertible sheaves
↔
and
mp /m2p
morphisms to
is the
Pn .
Linear systems of Divisors
Denition 4.1.
A complete linear system
That is
|D0 |
is in
Denition 4.2.
|D0 | on a non-singular projective variety is the set of all eective divisors
D0 .
linearly equivalent to
1-1
Γ(X, L(D0 )) \ {0}/k ∗ ,
i.e.,
|D0 |
is a projective space.
is a subset of a complete linear system
|D0 |
which is a linear subspace for
correspondence to this set:
A linear system
δ
on
X
|D0 |
That is,
δ
is a sub-vector space of
Denition 4.3.
A point
P ∈X
Γ(X, L(D0 ))
is a base point for a linear subsystem
δ
is
P ∈ Supp(D)
for every
D ∈ δ.
(Where
Supp(D) is the set of all prime divisors whose coecient is non-zero)
Lemma 4.4.
δ be a linear system on X
δ if and only if sp ∈ mP Lp
global sections in V .
Let
is a base point of
generated by
corresponding to the subspace
for all
Proof. This follows from the fact that for every
OX (Ui )) and D is an
X|sx ∈
/ mp L(D0 )}.
eective divisors on
X.
So
s∈V.
In particular,
δ
V ⊂ Γ(X, L(D0 )).
P ∈X
L(D0 ) is
Then a point
is base point-fee if and only if
∼
=
s ∈ Γ(X, L(D0 )), s 7→ D(Ui , φi (s)), (where φi : L(D0 )(Ui ) →
the support of D is the complement of the open set Xs := {x ∈
Remark. We can use this to rephrase Prop 3.2 in terms of linear systems (without base points).
φ : X → Pn
is a
closed immersion if and only if
P 6= Q, ∃D ∈ δ
1.
δ
separates points, i.e., for all
2.
δ
separates tangent vectors, i.e., if
that
P ∈ SuppD
but
t∈
with
P ∈ SuppD
and
Q∈
/ SuppD
P ∈ X and t ∈ mp /m2p (is a tangent vector) then there exists
2
(mp,D /mp,D ) where we consider D ⊂ X as a closed subvariety.
4
D∈δ
such