Toric Geometry
Math 615A - Fall 2011
Kalle Karu
Lecture notes by Pooya Ronagh
Department of mathematics, University of British Columbia
Room 121 - 1984 Mathematics Road, BC, Canada V6T 1Z2
E-mail address: pooya@math.ubc.ca
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
Introduction
Convex polyhedral cones
Dual cone
Rational cones
Torus action
Characters
1-parameter subgroups
Completeness
Blow-ups
Singularities
Toric varieties as quotients
Finite toric morphisms
Resolution of singularities
Resolution graph and exceptional divisors
T-orbits
Homological properties
3
4
5
5
11
12
13
13
15
15
16
17
18
19
20
21
Chapter 1. Divisors and line bundles
1. Divisors
2. Cartier T -divisors
3. Maps to projective spaces
4. Ample bundles
5. Cox ring
22
22
24
27
28
29
Chapter 2. Cohomology
1. Rational singularities
2. Vanishing theorems
3. Serre duality
31
32
33
35
Chapter 3. Special topics
1. Applications of vanishing
2. Gorenstein condition
3. Fano Varieties
4. Singularities
5. Torus equivariant cohomology
6. Chow groups
7. Cool facts to recall from intersection theory [Fulton]
8. Toric Riemann-Roch theory
36
36
36
37
38
40
43
43
45
Contents
1. INTRODUCTION
3
1. Introduction
We have a fixed field k and look at commutative rings over k or often k-algebras. We take
a semigroup S and form the semigroup ring k[S] from it in the obvious way. Then S is
commutative iff k[S] is and S is a monoid iff k[S] is unital. So we would require S to
further be a monoid. Examples include (Z≥0 , +) and (N, .). As a piece of formal notation
note that if S is an additive semigroup we can write
C
k[S] = { ∑ ai χsi }.
finite
So for example if S = Z≥0 × Z≥0 then Spec k[S] = A2k . We will use the notation χ(1,0) =∶ x
and χ(0,1) =∶ y to indicate this for concreteness. In general S is a k-basis for k[S] and χs
are called monomials of k[S]
Example 1.1. Take the cone C and let S = C ∩ Z2 . Then k[S] = k[x, y, z]/(xz − y 2 )
as it is generated by the three points u, v and w but the have the relation u + w = 2v.
X
2
Y
Z
0
−2
B
2
−2
⌟
The general procedure for making affine toric varieties is likewise. We take a cone C ⊂ Rn
and consider Spec (k[C ∩ Zn ]). For the projective case we take a polytope in P ⊂ Rn and
embed it as P × {1} ⊂ Rn+1 . This gives a cone C in Rn+1 and we consider Proj k[C ∩ Zn+1 ].
The grading of C ∩ Zn+1 comes from the ‘height’ component in Rn+1 .
Example 1.2. The square [0, 1]×[0, 1] gives Proj k[x, y, z, w]/(xz −yw) = V (xz −yw) ⊂
P . Note that the combinatorial object P can be used to read of the variety in many cases:
3
2. CONVEX POLYHEDRAL CONES
4
for instance since P = I × I in this case, we get that Proj k[S] = P1 × P1 .
X
Y
W
Z
A
⌟
2. Convex polyhedral cones
What kind of cones are allowed in this construction? These are basically the convex polyhedral cones that are cones we construct from a polytope. The reason we stick to polytope
sections is getting finite generation. A circular section does not give finitely many relations.
Recall that a subset C ⊂ V ≅ Rn is convex if whenever x1 , x2 ∈ C then λx1 + (1 − λ)x2 ∈ C.
C is a cone if whenever x ∈ C then ax ∈ C for all a ≥ 0. So C is a convex cone if for all
x1 , x2 ∈ C and a1 , a2 ≥ 0 then
a1 x1 + a2 x2 ∈ C.
Remark. If C ⊂ Rn is a convex cone, then C ∩ Zn is a semigroup (monoid).
x1 , x2 ∈ C ∩ Zn ⇒ x1 + x2 ∈ C ∩ Zn .
For x1 , ⋯, xm ∈ V the convex cone generated by x1 , ⋯, xm is
C = ⟨x1 , ⋯, xm ⟩ = {a1 x1 + ⋯ + am xm ∶ ai ≥ 0}.
Here C is the smallest convex cone containing x.
Definition 1. A convex cone C is polyhedral if it can be generated by a finite set.
Definition 2. A cone C is strongly convex (or pointed) if it does not contain a nonzero
linear subspace.
Definition 3. A cone C is simplicial if it can be generated by linearly independent vector.
The picture to keep in mind is that a simplicial cone is generated by a cross section that
is a simplex. From now on when we say a ‘cone’ we mean a convex polyhedral one. Let
C ⊂ V be a cone. Let y ∈ V ∗ be a linear function on V .
4. RATIONAL CONES
5
Definition 4. y ∈ V ∗ is a support function of C if y(x) ≥ 0 for all x ∈ C, written as
y∣C ≥ 0.
Then Hy = {y = 0} is a support hyperplane of C. Then Hy ∩ C is a face of C.
The set of all face of C is denoted by [C]. Faces form a poset under inclusion. A face of
C is again a convex polyhedral cone. Note also that if σ, τ are two faces of C then σ ∩ τ is
also a face. A facet is a face of codimension one. A ray is a face of dimension one.
Lemma 1. Every proper face is contained in a facet.
3. Dual cone
For a cone C ⊂ C. Let C ∨ = {y ∈ V ∗ ∶ y∣C ≥ 0}.
Lemma 2. C ∨ is again a convex polyhedral cone and that (C ∨ )∨ = C.
In general there are tow ways to give a cone. Either by (1) generators: C = ⟨x1 , ⋯, xm ⟩ in
which case if y1 , ⋯, y` are the normals to the facets then C ∨ = ⟨y1 , ⋯, y` ⟩. (2) By intersections
of half spaces
C = {x ∶ y1 (x) ≥ 0, ⋯, y` (x) ≥ 0}
(which gives a description of type (1) for the dual cone). Note that it is computationally
hard to get from one description to the other one (convex hull is hard to compute)!
Lemma 3. C ⊂ V is strongly convex iff C ∨ has full dimension (by which we mean the span
of its elements it the ambient vector space).
Faces of C and C ∨ are related as follows:
d-dimensional faces of C ↔ n − d-dimensional faces of C ∨
via τ ↦ τ ⊥ ∩ C ∨ = τ ∗ .
One more piece of notation is that of the polar dual of a polytipe P . If P is put at height
1 in a cone C then the height one polytope of C ∨ is the polar dual of P denoted by P ○ . In
dimension three examples of such dualities are that of the classical Platonic solids.
4. Rational cones
For C ⊂ Rn , C is called rational if we can find generators x1 , ⋯, xm ∈ Qn for it. This is
equivalent to getting m generators in Zn . If C is strongly convex and rational then it is
generated by the first lattice points on the rays.
In general for M ≅ Zn and MR = M ⊗ R ≅ Rn , the rational cone C ⊂ MR is generated
as
C = ⟨x1 , ⋯, xm ⟩
4. RATIONAL CONES
6
for some xi ∈ M .
Lemma 4 (Gordon). If C ⊂ Rn is a rational cone then the semigroup S = C ∩ Zm is finitely
generated.
Corollary 1. k[S] is finitely generated k-algebra and we get Spec k[S] ⊂ AN .
√
Example 4.1. Take the line passing through (1, 2) and create a cone in R2 . Then if
A = k[S] is the coordinate ring of the corresponding affine toric variety, the fraction field
ff(A) = k(x, y) implying that we don’t have a noetherian ring.
⌟
4.1. Functoriality. Let L ∶ Rn Ð
→ Rm be a linear map that maps lattices Zn Ð
→ Zm
n
and cone to cone C Ð
→ D. Then we get a homomorphism of semigroups C ∩ Z Ð
→ D ∩ Zm
and thus a ring homomorphism k[C ∩ Zn ] Ð
→ k[D ∩ Zm ] and a morphism of corresponding
varieties in the contravariant direction.
Notation: N ≅ Zn and M = Hom(N, Z) ≅ Zn . Let σ ⊂ NR be a strongly convex rational cone.
Then the dual cone is aof full dimension, i.e. rank(M ). Let Sσ = σ ∨ ∩ M the corresponding
semigroup and A = k[Sσ ] the semi-group ring. Then we have the notation
Uσ = Spec Aσ .
Remark. For a cone in dimension 2, the minimal set of generators of the points joining
that create
Conv(Sσ ∖ {0})
together with the faces. To find the generators one need computational methods using
Groebner basis for the ideal given as follows: Let ϕ ∶ Zk Ð
→ M be the homomorphism
sending generators ei Ð
→ vi to generators we found in previous step. THen we need to find
a set of generators for this Z-module as
Lemma 5. Aσ is a domain.
Proof. Sσ ⊂ M and
±1
Aσ = k[Sσ ⊂ k[M ] = k[x±1
1 , ⋯, xn ]
as a subring. But the latter is a domain.
Lemma 6. If τ is a face of σ then Uτ ⊂ Uσ as an open set.
Proof. In fact we show that Uτ is a principal open set. If τ is defind by σ ∩ u⊥ then
τ ∨ = σ ∨ + (−u.R).
In fact ⊇ is easy and for ⊆ let v ∈ τ ∨ then v∣τ ≥ 0 and u∣σ ≥ 0 thus for large enough p we have
v + p.u ≽ σ.
So v = w − pu ∈ σ ∨ − R.u.
And therefore Sτ = Sσ + (−u)Z hence Aτ = Aσ [ χ1u ].
E
4. RATIONAL CONES
7
Example 4.2. Given a configuration of cones σ1 and σ2 as in
σ1
σ2
B
we get two affine opens Uσ and Uσ2 that glue along Uσ1 ∩σ2 .
⌟
Definition 5. A fan ∆ in NR is a finite, nonempty set of rational strongly convex cones
in NR , such that
(1) IF σ ∈ ∆ then any τ ≼ σ is in ∆.
(2) If σ, τ ∈ ∆ then σ ∩ τ is a face of each cone.
Definition 6. The toric variety X(∆) associated to the fan ∆ is
∐ Uσ /{Uσ1 , Uσ2 are glues along Uσ1 ∩σ2 }.
σ∈∆
Example 4.3. The only 1-dimensional toric variety is P1 contructed from the fan
ρ2
ρ1
O
⌟
Example 4.4. Here is a 2-dimensional
example:
F
E
1
τ2
σ2
−2
σ1
τ1
A
τ3
−1
C
1
0
−1
D
σ3
−2
G
H
4. RATIONAL CONES
8
We have
Sσ1 = Z≥0 u ⊕ Z≥0 v, Aσ1 = k[χu , χv ],
Uσ1 = A2
Sσ2 = Z≥0 (−u) ⊕ Z≥0 (−u + v), Aσ1 = k[χ−u , χ−u+v ],
Uσ2 = A2
Sσ3 = Z≥0 (−v) ⊕ Z≥0 (−v + u), Aσ1 = k[χ−v , χ−v+u ],
Uσ3 = A2
giving us P2 .
⌟
Recall that
Definition 7. A morphism (of cones) is
ϕ ∶ (N1 , σ1 ) Ð
→ (N2 , σ2 )
where ϕ ∶ N1 Ð
→ N2 is a homomorphism and ϕ ⊗ R ∶ N1 ⊗ R Ð
→ N2 ⊗ R maps σ1 to σ2 .
From such ϕ we get induced ψ ∶ Uσ1 Ð
→ Uσ2 called a toric morphism.
→ Z2 be given by
Example 4.5. Let N1 = N2 = Z2 and σ1 = σ2 = R2≥0 . Let ϕ ∶ Z2 Ð
1 1
(
) in some basis. THen the dual map ϕ∗ is given by transposition of the former
0 1
matrix. Consequently
Aσ2 Ð
→ Aσ1
is given via x = χe1 ↦ χe1 +e2 = x.y and y = χe2 ↦ χe2 = y.
∗
∗
∗
∗
∗
⌟
Definition 8. Let (N1 , ∆1 ) Ð
→ (N2 , ∆2 ) be fans. A morphism ∆1 Ð
→ ∆2 is a group
homomorphism ϕ ∶ N1 Ð
→ N2 such that ϕR take every cone σ ∈ ∆1 into some cone of τ ∈ ∆2 .
Given (N1 , ∆1 ) and (N2 , ∆2 ) the product (N1 × N2 , ∆1 × ∆2 ) correspond to the product of
varieties
X(∆1 × ∆2 ) = X(∆1 ) × X(∆2 ).
To check this it suffices to see that
Uσ×τ = Uσ × Uτ ,
which is the case if Aσ×τ = Aσ ⊗k Aτ and this also follows if
Sσ×τ = Sσ × Sτ
as product of semigroups, hence following from our construction.
4. RATIONAL CONES
9
Example 4.6. Hirzebruch surface Fa .
H
E
F
G
D
C
(-1, a)
B
−1
0
1
In fact Fa = (Uσ1 ∪ Uσ2 ) ∪ (Uσ3 ∪ Uσ4 ) and π in
H
E
F
G
D
C
(-1, a)
B
−1
0
1
π
ρ2
O
ρ1
maps the first piece to Uτ1 and the second to Uτ2 . In fact each open is ψ −1(Uτ1 ) and
ψ −1(Uτ2 ). One interesting fact is that Fa Ð
→ Fa+2 is a homeomorphism. Let O(a) ⊕ O be
the locally free sheaf on P1 . Let E Ð
→ P1 be the vector bundle of rank 2 it corresponds to.
Then we form
P(E) = Fa Ð
→ P1 .
⌟
Remark. Fa and Fa+2 are not algebraically isomorphic. Fa ≅ F−a by toric morphisms.
Another nice toric fact is that any toric variety given by a fan as in the following figure is
a P1 -bundle over P1 and in fact isomorphic to Fa for some a.
E
D CONES
4. RATIONAL
C
10
B
0
Example 4.7. Let N = Z3 . Let σ = R3≥0 and [σ] be the set of all faces of σ. Let
∆ = [σ] ∖ {σ}.
F
G
H
Let π be the projection from Z(1, 1, 1). The image is a P2 .
⌟
Why is X(∆) a veriety?
Aσ = k[Sσ ]
is a domain. Therefore Uσ is a variety (irreducible and reduced) and thus X(∆) is irreducible and reduced.
Proposition 1. X(∆) is separated.
Proof. See Fulton. This follows if one shows that X(∆) ↪ X(∆) × X(∆) is closed.
This amounts to noting that any two different cones σ, τ ∈ ∆ can be separated by a hyperplane.
If we allowed multisets instead of sets, say for instance ∆ = {σ, τ, 0} where both σ and
τ are rays from the origin in R heading the positive direction of the x-axis we get the
nonseparated toric variety
O
5. TORUS ACTION
11
5. Torus action
Since 0 ≼ σ for all σ in the fan (where 0 is the zero cone), U0 ⊆ Uσ and so U0 is an open
subset of X(∆). Note that
U0 = Spec A0 = An ∖ {x0 ⋯xn = 0} ≅ Gnm
±1
as A0 = k[M ] = k[x±1
1 , ⋯, xn ].
Corollary 2. X(∆) is a rational variety.
Recall that a rational variety is one that has k(x1 , ⋯, xn ) ≅ k(An ) as the field of rational
functions on it. This is equivalent to the condition that the variety if birational to An .
T acts on itself by the following mapping of algebras
k[M ] Ð
→ k[M ] ⊗k k[M ],
χ m ↦ χm ⊗ χm .
k[M ] Ð
→ k[M ] ⊗k k[Sσ ],
chim ↦ χm ⊗ χm
The same morphism
extends the action of T on itself by multiplication. Of course one shall show comptability
with inclusions Uτ ⊂ Uσ :
and this is obvious from
T × Uσ
/ Uσ
∪
∪
T × Uτ
/ Uτ
k[M ] ⊗ k[Sσ ] o
_
k[M ] ⊗ k[Sτ ] o
k[S σ ]
_
k[Sτ ]
The k-valued points of X(∆) are in one-to-one correspondence with maps Spec k Ð
→ Uσ or
just k-algebra morphisms k[Sσ ] Ð
→ k. If R is a k-algebra, then
Homk−alg (k[Sσ ], R) = Homsemi−gr (Sσ , (R, .)).
The algebra homomorphism is determined by where the monomials go.
 Note 5.1. If R = k[S] for a cone S, in general Homs−g (Sσ , (R, .)) is much larger
than Homs.g (Sσ , S).
⌟
So we can rewrite the action on the k-points via
Hom(Spec k, T × Uσ ) Ð
→ Hom(Spec k, Uσ )
via (ψ, ϕ) ↦ (ψ.ϕ) where ψ.ϕ ∶ Sσ Ð
→ k is the mapping m ↦ ψ(m)ϕ(m).
6. CHARACTERS
12
A toric morphism f ∶ X(∆1 ) Ð
→ X(∆2 ) is equivarient in the following sense: Say f is
induced by
ϕ ∶ (N1 , ∆1 ) Ð
→ (N2 , ∆2 ).
Then T1 = Spec k[M1 ] mapsto to T2 = Spec k[M2 ] by restriction of f to U0 ’s. One can see
that this is a group homomorphism. Note that by equivarience we only require f (t.x) =
f (t).f (x).
Example 5.2. for A1 Ð
→ A2 given by (0, 1) Ð
→ (s1 , s2 ) for (s1 , s2 ) ∈ Z2≥0 , we are not
mapping the standard tori to each other.
⌟
6. Characters
Let T = U0 given by (N, 0) (where 0 means the zero cone), and k × = V0 considered as a
toric variety generated by (Z, 0). The set of characters of T is by definition
Homgr (N, Z) = M.
An element of this set is χm ∶ T Ð
→ k induced by m ∈ M . Here χm is a regular function on
T = Spec k[M ].
Linear representations of T : If T acts on V , we get a representation T Ð
→ GL(V ), since T
is an abelian group, we have simultaneous digaonalization:
V = ⊕i Vi
where Vi is an eigenspace for all t ∈ T . So fixing v ∈ Vi , we have
t.v = λi (t).v
and λi is a character of T . So we get a decomposition by characters:
V = ⊕m Vm .
Example 6.1. If T acts on Uσ , we get an action on Aσ = k[Sσ ] which is an infinitedimensional k-vector space. So
Aσ = ⊕m∈M (Aσ )m
where
(Aσ )m = {
k.χm m ∈ σ ∨
0
otherwise.
⌟
Example 6.2. If F is a T -equivariant linear bundle on Uσ . Then T acts on
F(Uσ ) = ⊕m F(Uσ )m
we also get an action of T on the cohomology theory H i ((X(∆), F).
⌟
8. COMPLETENESS
13
7. 1-parameter subgroups
A one-parameter subgroup is a toric map k × Ð
→ T , i.e. (Z, 0) Ð
→ (N, 0). So the set of all
one-parameter subgroups is
Homgr (Z, N )N.
We get a duality pairing N × M Ð
→ Z in the obvious way.
Lemma 7. Let v ∈ N and
C
D
× ϕv
k Ð→ T
be a one-parameter subgroup. Let Uσ be an affine toric variety defined by (N, σ). Then ϕv
extends to k Ð
→ Uσ iff v ∈ σ.
Proof. If v ∈ σ we get a map of cones
σ
v
ϕv
O
1
B
O
ϕv
→ N via 1 ↦ v, we get a dual mapping
For the other direction given k × Ð→ T defined by Z Ð
v∗ ∶ M Ð
→ Z.
k[t, t−1] = k[Z] o
k[M ]
∪
∪
k[t] o_ _ _ _ _ k[Sσ ]
Image of Sσ lies in Z≥0 iff v ∗ is ≥ 0 on Sσ iff v ∈ σ.
8. Completeness
We recall the valuative criterion first. For a field L a discrete valuation is a ring morphism
ν∶KÐ
→ Z ∪ {∞}
we use the notation for the DVR Rν = ν −1(Z≥0 ∪ {∞}).
8. COMPLETENESS
14
Theorem 8.1 (Valuative Criterion). If X is a separated variety, X is complete if for
any K with discrete valuation, the diagram below always completes (uniquely because of
separatedness):
Spec K
_
x
x
x
x
/X
x<
Spec Rν
and a map X Ð
→ Y is more generally proper if it is separated and
Spec K
_
x
x
x
x
/X
x<
/Y
Spec Rν
Now let ∆ be a fan. The suppose of ∆ is
∣∆∣ = ⋃ σ ⊂ NR .
σ∈∆
The fan ∆ is complete if ∣∆∣ = NR .
Theorem 8.2. X(∆) is complete iff ∆ is complete.
v
Proof. Suppose X(∆) is complete but ∆ is not. Take v ∈ N ∖ ∣∆∣. Then k × Ð
→ T does
not extend to k Ð
→ Uσ for any σ ∈ ∆. Then it does not extend to k Ð
→ X(∆) so X(∆) is
not complete. For the other direction if ∆ is complete, we check the valuative criterion for
field K and discrete valuation ν. Assume Spec K Ð
→ T ⊂ X(∆) without loss of generality.
We also exploit the assumption that K is a k-algebra. So Spec K Ð
→ X(∆) gives
MÐ
→ K× Ð
→Z
by some v ∈ N . We choose σ such that it contains v. Then we get a diagram
v
z
Zo
ν
s.g
KO o
?
Rν
~|
|
|
|
Sσ
The dotted map gives a ring homomorphism and therefore Spec Rν Ð
→ Uσ as needed.
Likewise one can show
ϕ
Theorem 8.3. X(∆1 ) Ð
→ X(∆2 ) given by (N1 , ∆1 ) Ð
→ (N2 , ∆2 ) is proper iff ϕ−1∣∆2 ∣ = ∣∆1 ∣.
10. SINGULARITIES
15
9. Blow-ups
A subdivision of ∆1 is another cone in the same vector space, such that ∣∆2 ∣ = ∣∆1 ∣ and
the identity mapping id ∶ ∆1 Ð
→ ∆2 is a morphism of fans. The morphism induced by id is
proper and a birational morphism.
Example
of a subdivision giving
the blow-up of A2C at the
J 9.1. Here is an example
I
D
origin.
(1,1)
ϕv
σ
O
O
⌟
10. Singularities
Let σ ⊂ N be generated by ⟨v1 , ⋯, vn ⟩.
Definition 9. σ is non-singular if v1 , ⋯, vm can be extended to a basis v1 , ⋯, vn of N .
For instance if σ is simplicial and dim σ = n = rk N , then it is non-singular iff det(v1 , ⋯, vm ) =
±1.
Theorem 10.1. Uσ is nonsingular if and only if σ is non-singular.
Proof. ⇐: let σ = ⟨v1 , ⋯, vm ⟩ be non-singular. Complete this to ⟨v1 , ⋯, vn ⟩, and let
their duals be e∗1 , ⋯, e∗n . Then
±1
Aσ = k[x1 , ⋯, xm , x±1
m+1 , ⋯, xn ]
therefore Uσ = Am × (k × )n−m .
⇒: Assume Uσ is non-singular and dim σ = m < n = rk N . Then
(σ, N ) = (σ, N1 ) × (0, N2 ).
11. TORIC VARIETIES AS QUOTIENTS
16
From that Uσ = Uσ′ × (k × )n−m . So we have that Uσ is non-singular iff Uσ′ is nonsingular.
So without loss of generality, dim σ = rk N so then σ ∨ is strictly convex. Let I ⊂ Aσ be
generated by all χm for all m ∈ σ ∨ . So Aσ /I ≅ k so I is a maximal ideal and should
correspond to some x ∈ Uσ .
If Uσ is nonsingular, then dim mx / dim m2x = n. Note that m2x is generated by (χm1 +m2 )m1 ,m2 ≠0 .
But mx /m2x has a basis consisting of χmi over a set of {mi } = B of minimal generators of
Sσ . In fact, B contains the first element on each ray. So then the first elements on rays
generate Sσ . Therefore σ ∨ has n rays, so it is simplicial. That is to say,
σ ∨ = ⟨u1 , ⋯, un ⟩
with ui ’s generating Sσ . So u1 , ⋯, un is a basis for M . So σ ∨ is nonsingular and consequently
σ is nonsingular.
Theorem 10.2. The ring Aσ is integrally closed.
Proof. Write
σ ∨ = ⋂ half-spaces H
therefore Sσ = ⋂ H ∩ M . So
Aσ = ⋂
k[H ∩ M ]
.
´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¸¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¶
integrally closed
So Aσ is integrally closed. (In fact Aσ is Cohen-Macauley).
11. Toric varieties as quotients
So far we have seen that if X(∆) is a toric variety, then it is normal, and has a toric T
embedded that has an action extending the action of the torus on itself.
If Uσ is smooth then σ is simplicial. The claim is now that if σ is simplicial, then Uσ has
finite quotient singularities, i.e. Uσ = An /G with G a finite group. Here we are assuming
char k = 0. First we see an example:
Example 11.1. Let µ3 be the group of third roots of unity. This acts on A2 by
ξ(x, y) = (ξ.x, ξ 2 .y).
Then A2 /G = Spec k[x, y]G . The induced action on k[x, y] is the obvious one
ξ ∶ x ↦ ξ.x, y ↦ ξ 2 .y.
Let M ′ ⊂ M be the lattice of invariant monomials as in the picture
f ig22
a b
In fact x y is fixed by G iff a + 2b = 0 mod 3. Then observe that for
Sσ′ = σ ∨ ∩ M ′
12. FINITE TORIC MORPHISMS
17
we have A2 /G = Spec k[Sσ ]G = Spec k[Sσ′ ]. For computational purposes, we consider the
exact sequence
0Ð
→ M′ Ð
→M Ð
→ M /M ′ Ð
→0
´¹¹ ¹ ¹ ¸¹ ¹ ¹ ¹¶
Q
of abelian groups, right deriving gives
0Ð
→N Ð
→ N′ Ð
→ Ext1 (Q, Z) Ð
→0
so
N ′ = {(α, β); ⟨(α, β), m′ ⟩ ∈ Z, ∀m′ ∈ M ′ } ⊂ NR .
It follows that if we look at u1 , u2 , u3 as follows:
f ig23
then N ′ is realized as
N ′ = {(α, β) ∶ (α, β).ui = 0 mod 3(i = 1, 3), (α, β).u2 ∈ Z → α + β ∈ Z}.
⌟
Theorem 11.1. Let σ be a simplicial cone σ = ⟨v1 , ⋯, vn ⟩. Let N ′ ⊂ N be generated as
Zv1 + ⋯ + Zvn then (σ, N ′ ) is nonsingular. In fact Uσ′ ≅ An . (σ, N ) is singular. Let
G = N /N ′ then the assertion is that G act An and Uσ = An /G.
Proof. We have an action of G on An with
χm ↦ e2πi⟨v,m ⟩ χm .
′
′
′
Now we want to find the invariants. Take χm ∈ A′σ then
′
′
′
v.χm = χm , ∀v ∈ G
iff e2πi,⟨v,m ⟩ = 1, if and only if ⟨v, m′ ⟩ ∈ Z and consequently m′ ∈ M , proving the assertion.
′
Theorem 11.2. If ∆ is simplicial, then X(∆) has finite quotient singularities.
In this case X(∆) is an orbifold.
12. Finite toric morphisms
Sublattices of finite index N ′ ⊂ N given finite toric morphisms: So A′σ is a finite Aσ module. What we proved shows that if Uσ is singular (say simplicial), we can find finite
toric morphism Uσ′ Ð
→ Uσ where Uσ′ is nonsingular.
Here is an example
f ig24
13. RESOLUTION OF SINGULARITIES
18
13. Resolution of singularities
Let X be a singular variety. By resolution of singularity we mean f ∶ Y Ð
→ X, proper
birational and such that Y is non-singular. In the toric case, when ∆ is a singular fan, we
only have to find a subdivision ∆′ of ∆ such that ∆′ is non-singular:
Theorem 13.1. Every ∆ has a nonsingular subdivision.
Star subdivision: Let σ be a strongly convex cone, and ρ ⊂ σ a ray. Then Stρ σ is the star
subdivision which has maximal cones generate by the ray and a facet of σ, ⟨F, ρ⟩ for any
facet F .
First barycentric subdivision: Take a rational ρσ ⊂ intσ for every cone σ ∈ ∆. And then
do iterative start subdivisions to get bs1 (∆). So maximal dimension cones correspond to
chains of rays we have chosen:
ρ1 ≽ ρ2 ≽ ⋯.
Then the observation is that bs1 (∆) is simplicial.
Definition 10. Let σ = ⟨v1 , ⋯, vn ⟩ be of maximal dimension rk N and let
N ′ = Zv1 ⊕ Zv2 ⊕ ⋯Zvn ⊂ N.
The the multiplicity of σ is
mult(σ) = ∣N ∶ N ′ ∣.
Another description is the cardinality of the set
{v ∈ N ∶ v = ∑ αi vi , 0 ≤ αi < 1}
for obvious reasons. And it is also immediate that σ is nonsingular if and only if its
multiplicity is one.
Let σ be simplicial and singular and 0 ≠ v = ∑ αi vi an element in the above set. Now star
subdivide at ⟨v⟩. Then the multiplicity of each maximal cone
σi = ⟨v1 , ⋯, ̂
vi , ⋮, vn , v⟩
is of multiplicity strictly less that multiplicity of σ:
multσi = ∣ det(v1 , ⋯, ̂
vi , ⋯, vn , v)∣ = ∣αi det(v1 , ⋯, vi , ⋯, vn )∣ = αi multσ.
Remark. If σ is nonsingular, then for ρ = ⟨v1 + ⋯ + vn ⟩, Stρ σ is the blow-up at the local
0 ∈ An , and if ρ = ⟨v1 + ⋯ + vm ⟩ the it is the blow-up of
An−m ⊂ An .
By this algorithm we have a resolution which is not canonical in any sense. Is there a
minimal resolution then? Recall that the minimal resolution is unique up to a sequence of
flip-flops.
f ig25
14. RESOLUTION GRAPH AND EXCEPTIONAL DIVISORS
19
It is however easy to see that if X is a toric surface then it has a unique minimal (toric)
resolution.
f ig26
In general if a (non-toric) surface has a G-action, the G action lifts to its resolutions. This
will show that the minimal resolution of a toric surface is unique.
14. Resolution graph and exceptional divisors
The resolution graph has vertices E1 , ⋯, En for the exceptional divisors, and the edges are
the intersection points as in the following picture:
f ig27
The self-intersection numbers of the exceptional divisors can be read off in the toric case
via
vi−1 + vi+1 = −ai vi
where vi ’s are the generators of Ei ’s.
Example 14.1 (ADE-singularities). All self-intersection numbers are −2.
f ig27
⌟
This picture generalizes as far as finding the resolution graph of the exceptional divisors
If σ is a singular 2-dimensional cone up to isomorphism. It has the form
f ig28
up to isomorphism, where k < m. For this just add or subtract enough (m, ` ± m) by lattice
isomorphisms
f ig29
And then the resolution data can be found from this canonical cones:
f ig30
0 m
1 m
for instance multσ = ∣ det (
) ∣ = m and multσ2 = ∣ det (
) ∣ = k.
1 −k
0 −k
Example 14.2. Hirzebruch Yung continuous fractions
Ei .Ei = −ai
then
m
1
= a1 −
.
k
a2 − a31−⋯
15. T-ORBITS
20
⌟
15. T-orbits
For the action of T on Uσ , the claim is that orbits in Uσ are in one-to-one correspondence
with faces of σ.
If ρ ≼ τ then Uρ ⊂ Uτ and Uτ contains smaller orbits: Oτ is smaller than Oρ . Here is the
definition of orbit Oτ ⊂ Uσ .
Let N ≅ Zn and suppose dim τ = m. If m = n let I ⊂ Aσ be the ideal generated by χu for
all u ≠ 0, this cuts Oτ = V (I) which is a point in this case. If dim τ ≤ n then Let
M (τ ) = M ∩ τ
then I ⊂ Aτ will be the ideal generated by all (χu )u/∈τ ⊥ in which case
Spec Aτ /I = Oτ = (k × )rk M (τ ) = (k × )n−m .
What are the points in Oτ ? If x ∈ Uτ correspond to the semigroup homomorphism x ∶ Sτ Ð
→
(k, .), then x ∈ Oτ if and only if x ∶ u ↦ 0 if u ∈/ τ ⊥ . Giving x ∈ Oτ is hence the same data
as a semigroup morphism M (τ ) Ð
→ (k, .).
Claim: Oτ is a T -orbit. Fix a special point xτ ∈ Oτ . Then T.xτ = Oτ .
Lemma 8. Uσ = ⊔τ ≼σ Oτ .
Proof. Show that any x ∈ Uσ lies in a unique Oτ , τ ≼ σ. Say x ∶ Sσ Ð
→ k is a semigorup
morphism. Then x−1(0) ⊂ Sσ is a prime deal in Sσ . Then by the homework problem,
x−1(0) = Sσ ∖ {u ∈ τ ⊥ } for some face τ ≼ σ. We extend this to Sτ and this completes the
proof.
Example 15.1.
f ig31
⌟
Let V (τ ) = Oτ be the closure of Oτ in X(∆). Check that
V (τ ) = ⊔T − orbits.
Example 15.2.
f ig32
V (ρi ) ∩ V (ρi+1 ) is a point and V (ρ1 ) ∩ V (ρ3 ) is empty.
⌟
16. HOMOLOGICAL PROPERTIES
21
The claim is that V (τ ) is itself a toric variety of dimension n − dim τ . To prove this one
needs to observe that Vτ ⊂ Uσ is V (J) for the ideal generated by
f ig33
 Note 15.3. V (τ ) ↪ X(∆) is not a toric morphism.
⌟
Morphisms:
ρ ∶ (N1 , ∆1 ) Ð
→ (N2 , ∆2 )
f is equivariant, i.e. it takes orbits to orbits, Oτ ↦ Oσ . σ is the smallest cone containing
ϕ(τ ). If ϕ ∶ N1 Ð
→ N2 is surjective or has a finite cokernel (i.e. ϕR ∶ N1 , R Ð
→ N2 , R is
surjective) then f ∶ T1 Ð
→ T2 is surjective and orbits maps onto orbits.
16. Homological properties
We know that the Euler characteristic satisfies the science relation and trivial fibration
axioms:
(1) χ(X) = χ(U ) + χ(X ∖ U ) for open U ⊂ X.
(2) χ(X × Y ) = χ(X)χ(Y ).
So we can find the Euler characteristic of X(∆) of a fan ∆ from its toric orbits:
X(∆) = ∐ Oσ .
σ∈∆
So
χ(X(∆)) = ∑ χ(Oσ ).
σ
∗ n−dim σ
But we know that Oσ = (C )
. Therefore
χ(Oσ ) = {
0 dim σ < n
.
1 dim σ = n
We conclude that
χ(X(∆)) = number of n-dimensional cones in ∆ = number of T -fixed points .
For further topological information one should study the Mixed Hodge theory of these
varieties. When X is nonsingular and is stratified into locally closed subsets, then one can
recover the cohomology ring H ∗ (X, C) from the cohomology of the strata.
CHAPTER 1
Divisors and line bundles
1. Divisors
1.1. Review. Let X be a normal variety of dimension n. A divisor on X is a formal
finite sum
D = ∑ ai Di
where ai ∈ Z and Di are irreducible (n − 1)-dimensional subvarieties. Div(X) is the set of
divisors on X. This is an abelian group. We call D effective if all ai ≥ 0 and write this as
D ≥ 0.
For any f ∈ k(X)∗ we have div(f ) = ∑ ai Di as a finite sum of divisors. In that case it is
called a principal divisor. Here
ai = ordDi f
which is positive if f vanishes along Di with multiplicity ai and is negative if f has a pole
of order ai . We have div(f.g) = divf + divg giving a homomorphism from the multiplicative
group to the additive group:
k(X)× Ð
→ Div(X).
If for a covering Ui we have D∣Ui = div(fi ) for some fi ∈ k(Ui )× = k(X)× we call D a Cartier
(or locally principal) divisor. The subgroup of these divisors is denoted by CDiv(X) ⊂
Div(X).
We can recover the codimension 1 Chow group from this:
An−1 = Div(X)/{ principal divisors } = Div(X)/ linear equivalence
The linear equivalence reads D ∼ E if D − E = div(f ).
Recall the sheaves OX (D) for a divisor D:
OX (D)(U ) = {f ∈ k(X) ∶ D + div(f ) ≥ 0}
which is the set of f ∈ k(X) such that f can have a pole of order ai along Di if ai ≥ 0
and f must vanish at least to order ai along Di if ai < 0. For instance O(−Di ) = IDi for
an irreducible component Di , i.e. the ideal sheaf of Di . And O(Di ) are the functions that
have at most one pole along Di .
22
1. DIVISORS
23
If D is Cartier, O(D) is invertible. In fact, in that case on each Ui we have
O(D)∣Ui =
1
OU ⊂ k(X).
fi i
We have
Pic(X) = CDiv(X)/ linear equivalence
which is the group of isomorphism classes of invertible sheaves, with the operation given
by tensor product of sheaves. One important property to recall is that
Proposition 2. OX (D) ≅ OX (E) if and only if D ∼ E.
If X is smooth, then CDiv(X) = Div(X). If CDiv(X) = Div(X), then X is called fractional.
If R is a UFD then every divisor on X is principal.
1.2. Toric divisors. If ∆ is a fan, and ρ1 , ⋯, ρd are rays of ∆. Let Di be the irreducible
divisor that is induced by ρi .
Let
d
DivT (X(∆)) = {∑ ai Di ∶ ai ∈ Z} = Zd .
i=1
Lemma 9. For any D ∈ Div(X(∆)) there exists E ∈ DivT (X(∆)) such that D ∼ E.
±1
Proof. T ⊂ X(∆) is just Spec k[x±1
1 , ⋯, xn ] which is a UFD. Then D∣T = div(f ). So
let E = D − div(f ). E has no component on T therefore
E = ∑ ai Di .
Corollary 3.
An−1 (X(∆)) = Div/ ∼= DivT (X(∆))/ ∼ .
Lemma 10. For any m ∈ M ,
div(χm ) = ∑ ai Di
where ai = ⟨m, vi ⟩.
Proof. Compute ordDi χm . Do this in Uρi .
±1
Aρi = k[x1 , x±1
2 , ⋯, xn ].
Then Uρi = A1 × (k × )n−1 . So Di = {0} × (k × )n−1 . Therefore ordDi x1 = 1.
Lemma 11.
DivT (X(∆)) ∩ {principal divisors} = {div(χm ) ∶ m ∈ M }.
2. CARTIER T -DIVISORS
24
Proof. D = div(f ) = ∑ ai Di such that D∣T = 0, then f in invertible on T . But
±1 ×
m
×
k[x±1
1 , ⋯, xn ] = {a.χ ∶ a ∈ k , m ∈ M }.
Hence div(f ) = div(aχm ) = div(χm ).
Corollary 4. An−1 (X(∆)) = DivT (X(∆))/M .
Example 1.1. For P2 , DivT P2 ≅ Z3 and
e1 ↦ D1 − D3 ,
2
e 2 ↦ D2 − D3 .
2
Hence Pic(P ) = A1 (P ) = Z.
⌟
Example 1.2. In
f ig34
DivT Uσ = Z2 and M ↦ DivT Uσ via (
1 0
). So
1 2
A1 (Uσ ) = Z/2Z.
In fact in this case
{principal divisors} = CDiv(X) ⊊ Pic(X).
⌟
2. Cartier T -divisors
If D = ∑ D(ρi ) (locally) we call it a Cartier T -divisor.
Lemma 12. Every T -Cartier divisor on Uσ is principal.
Proof. D ∈ DivT (Uσ ). Assume D is Cartier. The invertible sheaf O(D) is coherent
so let
P = Γ(Uσ , O(D))
be an Aσ -module. We need to prove that P is a free Aσ -module. Then P ≅ Aσ and O(D)
is trivial. Then D is principal.
Now T acts on P and we have a grading P = ⊕m Pm . Assume σ is full-dimensional. For
the special point xσ ∈ Uσ we have P /mxσ = k. Choose f ∈ Pm such that f ≠ 0. Then f is a
generator of O(D) near xσ . Then by the action of T we have f = c.χm for a character χm .
Near xσ , D is defined by f1 = cχ−m . Thus χ−m defined D everywhere.
Let D ∈ CDivT (Uσ ). The above theorem shows that D = divχu for some u ∈ M . We observe
that if
divχu1 = divχu2
then
χu1
χu2
= χu1 −u2 which is invertible on Uσ . Then u1 − u2 ∈ σ ⊥ .
2. CARTIER T -DIVISORS
25
For any u ∈ M we can view u as a linear functional on σ. u is integral if u(v) ∈ Z for any
v ∈ N ∩ σ. Then
CDivT (Uσ ) = { integral linear functions on σ }.
For the ⊇ direction if u(σ) ∶ σ Ð
→ R maps N ∩ σ to Z then extend this to a linear function:
u(σ) ∈ MR
and also conclude that u(σ) ∈ M .
If D ∈ CDivT X(∆) then for every σ ∈ ∆ u( σ) is an integral linear function on σ:
D∣Uσ = divχu(σ) .
If τ ≤ σ then u(σ)∣τ = u(τ ). So {u(σ)}σ∈∆ gives a piecewise linear (linear on every cone)
integral (i.e. taking ∆ to Z) on ∆. So this classifies all Cartier T -divisors on X(∆):
CDivT X(∆) = {ϕD ∶ piece-wise linear integral functions on ∆}.
So the Picard group is realized as the quotient
{ piecewise linear integral functions }/{ global linear integral functions }.
This is becuase we have
CDivT (X(∆)) = π −1Pic(X)
Pic(X) 

/ DivT (X)
π
/ An−1 (X)
Example 2.1. For X = P1 we have CDivT (P1 ) = Z ⊕ Z and Pic(P1 ) = Z easily!
⌟
Lemma 13. Pic(X(∆)) is a finitely generated free group. For this we need a completeness
assumption or at least that ∆ has a cone of full dimension.
Proof. We show that PicX(∆) has no torsion: If [ϕD ] ∈ PicX(∆) satisfies
[`ϕD ] = 0
then `.ϕD is globally linear, so ϕD is globally linear. Thus D ∼ 0.
Example 2.2.
PicUσ = 0
and A1 Uσ = Z/2Z. Here D1 is not Cartier and
1
ϕD (v) =
2
so it is not integral.
f ig40
⌟
2. CARTIER T -DIVISORS
26
Example 2.3. Here D1 is a Cartier divisor but Z/2Z = Pic(X) as we do not have the
full-dimensionality condition satisfied.
f ig41
⌟
If σ is simplicial, for any D ∈ DivT (Uσ ) there is ` > 0 such that `.D is Cartier. In this case
we say D is Q-Cartier. If σ is not simplicial however, for any choice of numbers a1 , ⋯, ak
over the rays, a linear functional ϕD has to satisfy some extra relations to be Q-Cartier.
This motivated the following notion and theorem:
Definition 11. X is Q-fractional if for any D ∈ Div(X), we have integer ` > 0 such that
`D ∈ CDiv(X).
Theorem 2.1. ∆ is simplicial if and only if X(∆) is Q-fractional.
Notation: In [F] the notation for u(σ) is actually −u(σ) and ψD = −ϕD . So for ψD a
piecewise linear function, D = ∑ ai Di , with ai = −ψD (vi ) if the corresponding divisor and if
ψD ∣σ = u(σ)∣σ then
D = divχ−u(σ)
and the linear bundle is trivialized via
L∣uσ = χu(t) OX ∣uσ .
Example 2.4. Let D = −2D1 + D2 − D3 in the following picture we have all u(σi )’s and
all L(Uσi ) = χu(σi ) Aσi ’s illustrated.
f ig46
⌟
The global sections Γ(X(∆), L) can be read off from such pictures: Let
PD = ⋂ (u(σ) + σ ∨ ).
σ∈∆
u
If u ∈ PD ∩ M , then χ is a rational function on X(∆) so that χu is a section of L on each
Uσ :
χu ∈ Γ(X, L).
So {χm ∶ m ∈ PD ∩ M } is a basis for Γ(X(∆), L).
Remark. If D is T -invariant (in particular true for all T -divisors), L = O(D) is T equivariant. Also L(Uσ ) has an induced T action so the space of global sections Γ(X(∆), L)
has a T -action according to which we have a grading
Γ(X(∆), L) = ⊕ Γ(X(∆), L)m .
m∈M
3. MAPS TO PROJECTIVE SPACES
27
Example 2.5. F1 embeds in P4 by D = D1 + D2 + D3
f ig47
⌟
Remark. If D is effective then 1 ∈ O(D)(X) reconstructs D via div(1) = D.
Note that when ∆ is complete, PD is compact. This gives a proof of finite-dimensionality
of the space of global sections. In general PD is described by the condition
u ∈ PD if and only if u ≥ ψD .
In general the conditions are
u ≥ ψD on every vi i.e. u(vi ) ≥ −ai .
So
PD = {u ∈ MR ∶ ⟨u, vi ⟩ ≥ −ai }.
The relations in linear equivalence translate to
PD+divχm = PD + m
P`.D = `.PD
PD+E = PD + PE
where in right-hand side of the last relation is the Miskowski sum and in general one needs
non-emptiness condition on all polytopes above.
3. Maps to projective spaces
Recall that L is said to be globally generated if for any x ∈ X, there is a section s ∈ Γ(X, L)
such that s(x) ≠ 0. If L is globally generated and s0 , ⋯, sn is a basis for Γ(X, L) we get a
well-defined map to Pn .
Theorem 3.1. If ∆ is complete, O(D) is globally generated if and only if ψD is convex.
Proof. If O(D) is globally generated, then u(σ) is a global section for every cone. So
u(σ) ≥ ψD for all σ, therefore ψD is convex. Conversely if ψD is convex, then u(σ) ≥ ψD for
all maximal dimensional σ. So there is a section s ∈ Γ(X(∆), L) such that s(xσ ) ≠ 0. So
V ( global sections of L ) = {x ∈ X(∆); L is not globally generated }
which is a closed set and T -invariant so it is a union of T -orbits containing some xσ . This
set is ϕ.
one section missin
4. AMPLE BUNDLES
28
4. Ample bundles
Let X be a complete variety and O(D) an invertible sheaf on X. The section ring of
O(D)
R(O(D)) = ⊕∞
m=0 Γ(X, O(mD))
is a graded ring. When O(D) is ample then R(O(D)) is finitely generated and X =
Proj (R(O(D)). Note that it is not in general the case that R(O(D)) is finitely generated.
It is in fact a question of the minimal model program to investigate whether R(O(D)) is
finitely generated when D = KX .
In the toric case the situation is much better. Given D we have O(D) and PD as before
and then Γ(X(∆), O(mD)) has a basis M ∩ mPD .. Then R(O(D)) = k[CD ∩ M × Z] which
is always finitely generated, normal, and also Cohen-Macauly.
We already know that if O(D) is ample then we recover X(∆) from PD via
X(∆) = Proj R(O(D)).
We now show that we can recover ∆ and ψD form PD when D is ample.
f ig50
∨
Claim: ∂CD
is the graph of a piecewise linear function on NR . In fact it is the graph of
−ψD . Also ∆ is the image of the fan ∂CD under projection from (0, 1).
Proof. Start with ∆ and ψD strictly convex and construct
∨
= {(v, z) ∶ z ≥ −ψD (v)}.
CD
∨ ∨
) is the cone over PD . For this note that
Then check that (CD
∨
∨ ∨
}
) = {(u, y) ∶ (u, y).(v, z) ≥ 0, ∀(v, z) ∈ CD
(CD
and hence
∨ ∨
) ∩ {y = 1} = {(u, a) ∶ u.v + z ≥ 0}.
(CD
So the general picture is that
∨
{faces of CD
}
´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¸¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¶ o
F,dim d
O
{ faces of CD }
/ ´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¸¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹¶
F ∗ . dim n+1−d
O
{ cones of ∆ }
´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¸¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¶
{ faces of PD }
´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¸¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¶
π(F )=σ,dim d
Pσ =F ∗ ∩{y=1},dim=n−d
5. COX RING
29
then σ ⊥ Pσ and SpanF ⊥ SpanF ∗ . PD is a lattice polytope with vertices being the lattice
points u( σ) for maximal cones σ. If D is very ample, then PD induces an embedding
X ↪ P#{M ∩PD }−1 .
Example 4.1. Here is the embedding F1 ↪ P4 given via x ↦ (χu1 (x) ∶ ⋯ ∶ χu5 (x)) =
(x1 ∶ ⋯ ∶ x5 ) then since u1 + u5 = u2 + u4 the embedding is cut out by
x1 x5 = x2 x4 ,
x 3 x4 = x2 x5 ,
x1 x3 = x22 .
f ig51
⌟
Start with any lattice polytope P ⊂ MR and apply the construction to get ψD , a piecewiselinear function, and ∆, the projection of ψD . Thus from P we get a polarized toric variety
(X(∆), O(D)). Note that if we start from (∆, ψD ) with ψD convex, and get PD and then
′
′
go backwards to some (∆′ , ψD
) we get a strictly convex function ψD
and ∆ is a subdivision
′
of ∆ .
Example 4.2. Here let D = D4 then O(D4 ) is globally generated but not ample.
f ig52
Note that O(D) is just the pull back of O(D′ ) which is ample. One thing we have not
mentioned before though is that to pull back a bundle we may just pull-back the piecewise
linear function it corresponds to.
X(∆)
FF
FF
FF
FF
F"
′
/ P2
X(∆ )
f
⌟
If ψD is a convex function on ∆, then there is a unique fan ∆′ , and a strictly convex ψD′
on ∆′ such that f ∶ ∆ Ð
→ ∆′ satisfies ψD = f ∗ ψD′ .
5. Cox ring
When X is a complete variety, the Cox ring of X is
Cox(X) = ⊕O(D)∈Pic(X) Γ(X, O(D))
which is graded by the Pic(X) (in a meaningful way if the Picard group is a lattice):
Γ(X, O(D)) × Γ(X, O(E)) Ð
→ Γ(X, O(D + E)).
5. COX RING
30
In the toric case, when ∆ is a complete, nonsingular fan with rays ρ1 , ⋯, ρd we have a
projection
ei ↦vi
0Ð
→KÐ
→ Zd ÐÐÐ→ N Ð
→0
which dualizes to
π
0Ð
→M Ð
→ Zd Ð
→ Pic(X(∆)) Ð
→ 0.
d
Note that Z = Div(X(∆)) having the cone of effective divisors inside it. So π restricted to
the effective locus, can be thought of as the projection from a real cone in Rd to Pic(X(∆))⊗
R.
Proposition 3. π −1(O(D)) ∩ Rd≥0 = PD .
Proof.
π −1O(D) ∩ Zd≥0 = {D′ ∶ D′ is effective , D′ ∼ D}.
Assume D is effective then π −1(O(D)) − D ⊂ M is then
{D′ − D ∶ D′ is effective , D′ − D ≠ 0} = {m ∈ M ∩ PD }.
Corollary 5. In the above case
Cox(X(∆)) = k[Zd≥0 ] = k[X1 , ⋯, Xd ].
CHAPTER 2
Cohomology
We use our basis toric affine patching for X(∆) to compute H ∗ (X, F) for the line bundle
F = O(D).
Example 0.1. Let us choose −1, 0, −1 for the coefficients of ρ1 , ρ2 and ρ3 is the usual
picture of P2 . Then D = D1 + D3 and we have u(σ1 ) = (−1, 0), u(σ2 ) = (1, 0), u(σ3 ) = (−1, 2)
giving our piecewise-linear function. Then it is easy to form the regions of the 2-space
corresponding to F(Uσi ) and F(Uρj )’s. So we have
0Ð
→ F(Uσ1 ) × F(Uσ2 ) × F(Uσ3 ) Ð
→ F(Uρ1 ) × F(Uρ2 ) × F(Uρ3 ) Ð
→ F(U0 ) Ð
→ 0.
and since we have
H i (X(∆), O(D)) = ⊕ H i (X(∆), O(D))u
u∈M
we form this sequence for any u. For instance u = (0, −1) gives
⎛1⎞
⎝1⎠
(1 1)
→ 0.
0Ð
→ k ÐÐ→ k 2 ÐÐÐÐ→ k Ð
⌟
Fix u ∈ M and let ∣∆∣ be the support of ∆. Let
Z(u) = {v ∈ ∣∆∣ ∶ u(v) ≥ ψD (v)} ⊂ ∣∆∣
be a closed subset. The property of Z(u) is that Z(u) = ∣∆∣ if and only if χu ∈ Γ(X, O(D)).
Theorem 0.1. We have
i
H i (X(∆), O(D))u = HZ(u)
(∣∆∣; k)
where the second term si the singular cohomology of ∆ with support in Z(u). For U =
∣∆∣ ∖ Z(u) we have
̃ i−1 (U ) (i > 0)
H i (X(∆), O(D))u = H
and that
H 0 (X(∆), O(D))u = {
31
k u ∈ PD
.
0 otherwise
1. RATIONAL SINGULARITIES
32
The second part follows from the first part by looking at the long exact sequence of the relative cohomology of the pair (∆, Z(u)) by noting that ∣∆∣ is always a contractible subspace
of some real space.
Corollary 6. If ∣∆∣ is convex, and ψD is a convex function (then O(D) is globally generated and) we have
H i (X(∆), O(D)) = 0 ∀i > 0.
Proof. If ψD is convex, then U = ∣∆∣ ∖ Z(u) is also convex, hence contractible. Therẽ i (U ) = 0 for all i ≥ 0.
fore H
Proof of the theorem. Cover ∣∆∣ with {σi } for all maximal cones in ∆. We compute the Cech cohomology of ∣∆∣ with coefficients in k.
K● ∶
⊕i0 ∶σi0 ⊂Z(u) k
/ ⊕i0 <i1 ∶σi ∩σi ⊂Z(u) k
0
1
/ ⊕k
/⋯
C ● (∣∆∣) ∶
⊕σi0 ∈∆ k
/ ⊕i0 <i1 ∶σi ∩σi ≠∅ k
0
1
/ ⊕i0 <i1 <i2 ∶σi ∩σi ∩σi ≠∅ k
0
1
2
/⋯
C ● (U ) ∶
⊕i0 ∶σi0 ∩U ≠∅ k
/ ⊕i0 <i1 ∶σi ∩σi ∩U ≠∅ k
0
1
/ ⊕i <i <i k
0 1 2
/⋯
By definition K ● computes HZ∨ (∣∆∣). The claim is that K ● is the degree u part of C ● (X(∆).
But this follows easily from
Cup =
i0
k u ≥ ψD on σi ∩ ⋯ ∩ σip
.
∏ O(D)(Uσi0 ∩ ⋯ ∩ Uσip ) = { 0 otherwise 0
<⋯<ip
1. Rational singularities
Recall that when f ∶ Y Ð
→ X is a morphism and F a sheaf on Y , then f∗ F is a sheaf on X
̂
−1(U )). f can be right-derived as a functor
defined by f∗ F(U ) = F(f
∗
̂
Ri f∗ F(U ) = H i (f −1(U
), F∣ −1 )
f (U )
Definition 12. Let X be a singular variety and Y Ð
→ X a resolution of singularities. Then
X has rational singularities if
R0 f∗ OY = f∗ OY = OX ,
andRi f∗ (OY ) = 0(∀i > 0).
Example 1.1. If x ∈ X is an isolated surface singularity and has rational singularities
then
p(Z) = 1 − χ(OZ ) = 0
is the arithmetic genus.
⌟
2. VANISHING THEOREMS
33
Theorem 1.1. All toric varieties have rational singularities.
Proof. Given a resolution f ∶ X(∆′ ) Ð
→ X(∆) compute Ri f∗ OY (Uσ ). This coincides
i
′
′
with H (X(σ ), OX(σ′ ) ). ∣σ ∣ is convex, and OX(σ′ ) is globally generated therefore H i = 0
for i > 0.
2. Vanishing theorems
If X is toric and complete, and O(D) is ample (or globally generated) then we have
H i (X, O(D)) = 0,
∀i > 0.
Kodaira vanishing: If X is smooth and projective, O(D) is ample, then
H i (X, O(D + KX )) = 0,
∀i > 0.
When X is a smooth variety of dim X = n, and Ω1X is the sheaf of regular 1-forms, ΩnX =
∧n Ω1X is an invertible sheaf and is by definition O(KX ) where KX is defined up to linear
equivalence. To compute it we can choose a rational section ω of ΩnX and set divω =
KX .
Theorem 2.1. Let X = X(∆) is smooth, then KX = − ∑i Di where Di ’s range over all
irreducible T -invariant divisors.
Proof. Basis v1 , ⋯, vn for N and v1∗ , ⋯, vn∗ for M . Let
ω∈
dx1
dxn
∧⋯∧
∈ ΩnX (T ).
x2
xn
ω is a rational section of ΩnX and let div(ω) = KX . Then we note that if v1 , ⋯, vn generate
Uσ (smooth implies simplicial) we have
Uσ = Spec k[x1 , ⋯, xn ]
and that div(ω) = −D1 − D2 ⋯ − Dn on Uσ and we are done.
Notice that for Q-divisors (i.e. D = ∑ ai Di with all ai ∈ Q) we get ψD which is piecewise
linear function not integral. D is ample if ψD is strongly convex.
Theorem 2.2. Let ∆ be a nonsingular complete fan (or ∣∆∣ is convex). Let D be a Q-divisor
on X(∆) such that ψD is convex. Then
H i (X, O(D)) = 0, ∀i > 0.
2. VANISHING THEOREMS
34
Proof. Fix u ∈ M and let O(E) be a Z-divisor. Let
Z(u) = {u ≥ ψE },
U (u) = {u < ψE }.
Then we know that
̃ i−1 (U (u)), i > 0.
H i (X(∆), O(E)) = H
Since 0 ∈ Z(u), U (u) ∩ S n−1 is the deformation retract of U (u). Note that ∆ ∩ S n−1 is a
simplicial complex and ∆u is a subcomplex of ∆ ∩ S n−1 of simplices in U . Then ∆u is a
deformation retract of U and we have
̃ i−1 (∆u ).
H i (X(∆), O(E)) = H
In the same fashion as in previous vanishing theorems take a convex neighborhood U2 ⊃ ∆u ,
we get
̃ i−1 (U2 ) = H
̃ i−1 (∆u ) = H
̃ i−1 (U1 ) = 0
H i (X, O(D)) = H
giving us the vanishing. Note that
vi ∈ Ui ⇔ u(vi ) < ψD (vi ) ∈ Q ⇔ u(vi ) < ⌈ψD (vi )⌉ ⇔ vi ∈ U2 .
Example 2.1. Let X = Pn . We know that Pic(X) = Z = {O(mH) ∶ m ∈ Z} fixing a
hyperplane H. If m ≥ 0 we have that O(mH) is globally generated hence H i (Pn , O(mH)) =
0. For m ∈ {−n, −n + 1, ⋯, −1} let
D = −εD1 − εD2 − ⋯ − εDn + (1 − ε)Dn+1 .
Then D is ample for small values of ε > 0 and we have ⌊D⌋ = −D1 − D2 − ⋯ − Dn ≅ −nH. It
follows that H i (Pn , O(mH)) = 0 by convexity of ψD .
⌟
Proof of Kodaira’s vanishing theorem. If X is complete and D is ample, consider the Q-divisor E = D + εKX . Then E is ample if and only if ψD is strongly convex.
This proves our claim since ⌊E⌋ = D + KX .
Theorem 2.3 (Kawamata-Viehweg vanishing). Let D be an ample Q-divisor and X a
complete smooth variety. Then
H i (X, O(⌈D⌉ + KX )) = 0
∀i > 0.
Proof. E = D + εK is ample and ⌊E⌋ =⌋D + εK⌋ = ⌈D⌉ + K.
3. SERRE DUALITY
35
3. Serre duality
Theorem 3.1 (Serre duality). Let X be n-dimensional and complete and smooth. Let E
be the locally free sheaf O(D). Then we have
H n−i (X, O(KX − D)) ≅ H i (X, O(D))∨ .
The ideal is that this isomorphism is using the fact that this isomorphism is supposed to
be preserving the grading by M . From what we know
ψK−D = ψK − ψD
and hence
U (u) = {u < ψK − ψD } ⊃ ∆u .
Thus H (X, O(KX − D)) = H n−i−1 (∆u ). On the other hand from ψD we have U (−u) =
{−u < ψD } ⊃ ∆−u and we have
̃ i−1 (∆−u ).
H i (X, O(D))∨ = H
n−i
−u
The isomorphism of the above two cohomology groups follows from
̃ i−1 (∆−u ) = H
̃n−i−1 (∆u ) = H
̃ n−i−1 (∆u )∨ .
H
The first isomorphism is Alexander duality and the second one the universal coefficients
theorem.
Remark. In the general case of the Grothendieck-Serre duality for example when X is
●
is
singular, replace KX by the dualizing complex ωx● . When X is Cohen-Macauly, then ωX
a sheaf ωX . This is the case for example for toric varieties as we know. Note that in this
case KX = KXs where Xs is the smooth locus of X.
CHAPTER 3
Special topics
1. Applications of vanishing
Example 1.1. Let P be a lattice polytope. Let f (m) denote the number of points in
mP ∩ M for all m ≥ 0. One observes that f (m) is a polynomial in m. Then the claim is
that f (−m) is the number of points of intmP ∩ M . This was a conjecture of Ehrhardt and
then proved by McDonald.
Solution: Assign a fan ∆ to P in the usual way. So we have (X(∆), O(D)) a polarized
toric variety. Then for the ring
R = ⊕m≥0 Γ(X(∆), O(mD)) = k[C ∩ M Z]
we have that f (m) = dim Rm is the Hilbert function of R. Which is a polynomial in m
for large enough m in general. In fact it is a polynomial in m for all m in the toric case.
Because
χ(P (mD)) = dim H 0 (X, O(mD))
happens for all m in the toric case and this is is a polynomial in m because of Riemann-Roch
theorem. Therefore f (−m) = χ(O(−mD)) but by Serre duality we have
χ(O(−mD)) = ∑(−1)i dim H i (X, −mD)
= (−1)−n ∑(−1)−i+n dim H n−i (X, KX + mD) = (−1)n χ(KX + mD).
Then χ(KX +mD) = dim H 0 (X, KX +mD) by Kodaira vanishing (otherwise we may resolve
∆ first and then pull-back D to f ∗ (D) and do everything for X(∆′ )). Our final observation
will be that
dim H 0 (KX + mD) = #(int mP ∩ M ).
⌟
2. Gorenstein condition
In general KX is defined as the dualizing complex of the Serre duality. However when X
is normal (and of course singular) we observe that this dualizing element is a sheaf and in
fact constructed as the closure of KX sm . The next problem is that KX constructed this
way may not be Cartier. But X is called Gorenstein if KX is Cartier.
36
3. FANO VARIETIES
37
We have KX sm = ∑ ai Di . This is the case because Uρi is completely in the smooth locus of
the toric variety therefore Di ∣X sm is all of Di . So we have
K X sm = ∑ ai Di .
So the cases in which this may fail to be Cartier are
(1) There is no plane through vi ,
(2) There is a plane but ψ is not integral and m.ψ is integral m > 0.
X is Q-Gorenstein if mKX is Cartier for some m > 0.
Example 2.1. If ∆ is simplicial then X(∆) is Q-Gorenstein.
⌟
Example 2.2. The fan of the cube centered at the origin is Gorenstein.
⌟
In general we have
Proposition 4. X(∆) is Q-Gorenstein if and only if there is a polytope (not necessarily
convex) passing through vi ’s.
3. Fano Varieties
Let X be Gorenstein and complete. Then X is Fano if −KX is ample.
Example 3.1. Pn is fano since KPn = −(n + 1)H.
⌟
Example 3.2. For a hypersufrace X ⊂ Pn of degree d, we have
KX = KPn + X∣X = −(n + 1)H + dH = (−n − 1 + d)H.
So X is Fano if and only if −n − 1 + d < 0 i.e. n ≥ d.
⌟
If KX is Q-Cartier and −KX is ample then X is called Q-Fano.
3.1. Toric Fano. If X(∆) is Gorenstein, then the condition of being fano is equivalent
to −ψKX being strictly convex which is equivalent to having Q being strictly convex for any
polytope Q constructed from vi ’s of each of the maximal cones. Recall that here Q ⊂ N is
the polar dual of P where P is the polytope in MR associated to −KX . So we have
Proposition 5. The is a one-to-one correspondence between Fano X(∆) and pairs of
polar dual lattice polytopes (P, Q). The smooth Fano’s are in one-to-one correspondence
with those pairs (P, Q) as above such that the fan over Q is nonsingular.
Remark. Usually if P is a lattice polytope the the dual P ∨ sis not a lattice polytope. P
is called reflexive if P ∨ is again lattice a polytope.
4. SINGULARITIES
38
For each dimension d, there is a finite number of nonsiomorphic smooth relfexive polytopes
of dimension d. So there is a finite number of isomorphic classes of smooth toric Fano’s od
fimension d.
If d = 2 the Fano variety is also called del Pezzo surface which is either X = P1 × P1 or P2
blown-up at ≤ 8 points. Among these there are five toric Fanos
f ig60
If d = 3 there are 18 smooth toric fanos and if d = 4 there are 123 of them.
4. Singularities
Recall that given f ∶ Y Ð
→ X surjective we can pullback the Cartier divisors since if D ⊂
CDiv(X) is locally div(g) then f ∗ (D) is locally div(g ○ f ). If D is Q-Cartier then f ∗ (D)
is also Q-Cartier. But if D is not Cartier, then f ∗ (D) is not defined.
Example 4.1. Look at Bl0 A2 Ð
→ A2 . Then for any D we have
f ∗ D = D′ + mE
where D′ is the strict transform of D and m is the multiplicity of D at zero.
⌟
Let X be a Q-Gorenstein singular variety with KX being Q-Cartier, and f ∶ Y Ð
→ X be a
resolution of singularities. The discrepancy of f is
KY /X = KY − f ∗ KX = ∑ ai Ei .
Here Ei ’s are the exceptional divisors.
Example 4.2. For f ∶ Bl0 A2 Ð
→ A2 we have KY /X = E. If X is smooth then f ∶
BlZ X Ð
→ X has discrepancy
KY /X = (c − 1)E
where c is the codimension of Z. For this we assume that Z is a nonsingular subvariety. ⌟
We say that X has terminal, canonical, log-terminal and log-canonical singularities if all ai
are respectively positive, non-negative, > −1 and ≥ −1. The key observation is that min{ai }
does not depend on Y if ai ≥ −1 in general. If ai < −1 can happen then min{ai } = −∞.
In the context of the minimal model program one wants to classify the smooth complete
(projective) varieties up to birational isomorphism. In dimension 2, this has been carried
out. The result is that there is a minimal smooth model after a finite sequence of blowdowns
XÐ
→ X1 Ð
→⋯Ð
→ Xmin .
So Xmin ≅ Ymin if X and Y are birational. The only expections are P1 × P1 and P2 .
4. SINGULARITIES
39
Note that terminal singularities is equivalent to being smooth in dimension 2. In dimension
greater than two the minimal varieties have terminal singularities. And if X and Y are
birational then Xmin and Ymin are related by a sequence of flops. Another way to perform
this classification is to use the fact that if X and Y are birational (and of general type)
then
Γ(O(mKX )) ≅ Γ(O(mKY ))
hence R = ⊕m≥0 Γ(O(mKX )). Then
Xcan = Proj R = Ycan
is the canonical model of X.
Example 4.3. In
f ig61
we have X = Uσ and KX = −D1 − D2 . Then ψKX (vi ) = 1 for i = 1, 2 so ψKX = x − 31 y. We
pull this back and get
2
ϕ∗ ψKX (v3 ) =
3
and f ∗ (KX ) = −D1 − D2 − 32 D3 . In fact
1
KY /X = KY − f ∗ KX = − D3
3
so X has log-terminal singularity.
⌟
In general if X = Uσ and Q-Gorenstein then X is log-terminal. This is easy to se by
finding ψKY − ψKX one say a v3 that is between v1 and v2 in some certain ordering. So if
KY /X = ⋯ + a3 v3 + ⋯ then a3 = `(v3 ) − 1 > −1 where `(v3 ) = ψKX (v3 ). But this is the same
as ψKY − ψKX (v3 ).
If we construct triangles from vi ’s and the hyperplane ` = 1. Then X is moreover canonical
if there is no lattice points in the triangle (but maybe on the ` = 1 leg), and is terminal if
there is no lattice point on the triangle expect at the vi ’s and origin.
Remark. In the general (not necessarily toric) case, if X is Gorenstein the KY /X always
has Z-coefficients therefore canonical is equivalent to log-terminal and this is equivalent to
having rational singularities.
Definition 13. Y Ð
→ X is a crepant resolution if there is no discrepancy: KY = f ∗ KX .
Question: If X is Gorenstein
5. TORUS EQUIVARIANT COHOMOLOGY
40
5. Torus equivariant cohomology
Let ∆ be a complete nonsingular fan. Our goal is to compute H ● (X(∆), R) from the
equivariant cohomology HT● (X(∆), R). So we will rather look at the analytic topology of
X and we restrict ourselves to k = C. All the following constructions are carried out in the
analytic category.
In general let T be a group acting freely on X then we define HT● (X) = H ● (X/T ). Otherwise
replace X by X × ET . ET is the universal bundle over BT , the classifying space of T over
a point. Note that ET is contractible and T acts freely on ET . At any rate T is acting
freely on X × ET and we define
HT● (X) ∶= H ● (X × ET /T ).
For instance when X is a point then
HT● (pt) = H ● (ET /T ) = H ● (BT ).
→ HT● (X) from the projection
Then induced mapping HT● (pt) Ð
π2 ∶ X × ET /T Ð
→ ET /T
provides the structure of a
HT● (pt)-algebra
on HT● (X).
Example 5.1. Let T = C∗ acting on Cn ∖ {0}. We know that πi (Cn ∖ {0}) = 0 for all
i < 2n − 1. We have
Cn ∖ {0}/T = Pn−1
and therefore H ● (Pn−1 ) = R[x]/(xn ), with x of degree 2 is the T -equivariant cohomology
ring.
For the action of T on C∞ ∖ {0} we have πi (C∞ ∖ {0}) = 0 for all i > 0 so ET = C∞ ∖ {0}.
This implies that HT● (pt) = H(P∞ ) = R[x] and BT = P∞ .
For T = (C∗ )n we have ET = (C∞ ∖ {0})n and BT = (P∞ )n . We get
HT● (pt) = R[x1 , ⋯, xn ]
where xi ∈ HT2 (pt) are all of degree 2.
⌟
If HT● (X) has only even degree cohomology then we recover the cohomology of X from
H ● (X) = HT● (X)/(x1 , ⋯, xn ).HT● (X).
In nice cases, HT● (X) is a free graded R[x1 , ⋯, xn ]-module of the form
HT● (X) = ⊕ei k[x1 , ⋯, xn ],
ei ∈ HTdi (X).
In this case {ei } forms a basis for H ● (X) as stated.
In the toric case let T be the maximal torus acting on X(∆). Let
A(∆) = { continuous piecewise polynomial functions (real-valued) on ∆ }.
5. TORUS EQUIVARIANT COHOMOLOGY
41
By this we mean that f ∶ ∣∆∣ Ð
→ R is a continuous function such that f ∣σ is a polynomial.
Then we have
Theorem 5.1 (Goresky-Kottwitz-MacPherson). HT2i (X(∆)) = Ai (∆) when ∆ is complete.
A(∆) has a grading with Ai (∆) containing f ∈ A(∆) such that f ∣σ is homogeneous of
degree i. Over a point we have A the set of global polynomial functions on NR :
A = R[x1 , ⋯, xn ].
So we have that the T -equivariant cohomology is an A algebra via A ↪ A(∆). The real
cohomology is given by
H ● (∆) = A(∆)/(x1 , ⋯, xn ).A(∆).
Example 5.2. Let X(∆) = P1 . We have
A(∆) = {(f1 , f2 ) ∈ R[x] ∶ f1 (0) = f2 (0)} = R[x] ⊕ xR[x].
So we conclude that
H i (X(∆)) = {
R i = 0, 2
.
0 otherwise
The basis of (1, 1) and (0, x) is the case of H 0 and H 2 respectively.
⌟
To be able to do more interesting computations of the type in the above example we will
use Morse theory. Let P be a polytope. The associated fan ∆ is simplicial if and only if P
is simple (i.e. at every vertex v there are n edges). Let ` ∶ P Ð
→ R be a continuous function
that distinguishes all vertices. Then ` will order the vertices v1 , ⋯, vm and the maximal
cones of ∆ say as σ1 , ⋯, σm . The index of a vertex if
µ(vi ) = # edges going down .
Let ∆i = σ1 ∪ ⋯σi . We compute A(∆i ) be induction on i. It is a free A-module so we only
res
need to find generators. The claim is that A(∆i+1 ) ÐÐ→ A(∆i ) is surjective with a kernel
K given by
K = {f ∈ A(σi+1 ) ∶ f ∣Fj = 0}.
So K is a free A-module with generators in degree µ(vi+1 ) and it fits in the trivial extension
A(∆i+1 ) = A(∆i ) ⊕ K.
Our conclusion will be
Proposition 6. Every vertex vi contributes a generator of degree µ(vi ) to A(∆) and
dim H 2k (X(∆)) = #{vi ∶ µ(vi ) = k}.
Corollary 7. If we replace ` by −`, we get Poicare duality
dim H 2k (X(∆)) = dim H 2n−2 (X(∆)).
5. TORUS EQUIVARIANT COHOMOLOGY
42
5.1. Stanley-Reisner ring. Let ρ1 , ⋯, ρd be the rays of ∆ each generated by vector
ui . Let χi ∈ A1 (∆) be piecewise linear:
χi (uj ) = {
1 i=j
0 otherwise.
Then the claim is that χ1 , ⋯, χd generate A(∆) as an R-algebra. Let Rd Ð
→ NR be given by
̃
̃
ei ↦ ui . Then we get a mapping of ∆ Ð
→ ∆ were ∆ has cones
̃
σ = ⟨ei1 , ⋯, , eim ⟩
for any σ = ⟨ui1 , ⋯, uim ⟩. On Rd the coordinate fucntions X1 , ⋯, Xd will correspond to our
̃
χ1 , ⋯, χd . So Xi ’s generate A(∆).
0Ð
→I Ð
→ R[χ1 , ⋯, χd ] Ð
→ A(∆) Ð
→ 0.
Proposition 7. I = (χi1 ⋯χim ) were ij ’s run over all indices such that ρij do not lie in a
cone.
If f (χ1 , ⋯, χd ) = 0 in A(∆). restrict to σ = ⟨ρ1 , ⋯, ρn ⟩ we get f (χ1 , ⋯, χn , 0, ⋯, 0).
Definition 14. Stanley-Riesner ring is SR(∆) = R/I.
Example 5.3.
So
HT● (P2 ) = R[χ1 , χ2 , χ3 ]/(χ1 .χ2 .χ3 ).
H ● (P2 ) = R[χ1 , χ2 , χ3 ]/(χ1 χ2 χ3 , χ1 − χ3 , χ2 − χ3 ) = R[χ3 ]/(χ33 ).
⌟
Example 5.4 (Hard Lefschetz theorem). We can do intersection theory using the above
theory. For instance if [D] = −ψD ∈ A(∆), the Lefschetz operator is
L ∶ H ● (X(∆)) Ð
→ H ● (X(∆))
is just multiplication by −ψD .
⌟
Example 5.5 (Face numbers). If ∆ is simplicial complete let hi = dim H 2i (X) be the
Betti numbers and let fk be the number of k-dimensional cones of ∆. Then the two types
of numbers are related to each other via
n
n
i=0
k=0
P∆ (t) = ∑ hi ti = ∑ fk (t − 1)n−k .
⌟
Remark. There are conditions that fi ’s always satisfy. Stanley [1980] shows that there
is a list of relations between fi ’s or equivalently hi ’s that if guaranteed there is always a
polytope P such that X(∆P ) yields the given hi ’s.
7. COOL FACTS TO RECALL FROM INTERSECTION THEORY [FULTON]
43
6. Chow groups
Let X be any variety. We have already defined
A1 (X) = An−1 (X) = Div(X)/ ∼
with ∼ being our usual equivalence relation. In general we set
Aj (X) = {∑ ai Zi ∶ Zi ⊂ X is irreducible of codimension j }/ ∼
with the equivalence relation being generated by Z1 ∼ Z2 if there is a flat subvariety V ⊂
X × P1 over P1 of codimension j such that Z1 is the geometric fiber over 0 and Z2 is that of
∞. So except A0 (X) = Z[X] computing any other Chow group is a hard question.
Our old equivalence relation is a special case of the general one: For any f ∈ k(X) we have
a rational mapping f ∶ X Ð
→ P1 . Let V be the graph of f in X × P1 . Then (show that) V is
1
flat over P and the claim follows.
Chow groups form a ring if X is nonsingular. If we replace the equivalence relation (in the
divisors case) with algebraic equivalence (i.e. instead of P1 using any curve C), we get the
connected components of Pic(X) with is called the Neron-Severi group:
N S(X) = Div(X)/ ∼alg .
If we use homological equivalence (i.e. Z1 ∼ Z2 if [Z1 ] = [Z2 ] ∈ H ● (X)) we get a mapping
A● (X) Ð
→ H 2● (X).
This is not injective or necessarily surjective. One case also consider moding out further by
numerical equivalence Z1 ∼ Z2 if Z1 .Y = Z2 .Y for all Y .
In the toric case
A● (X) ⊗ R Ð
→ H 2● (X(∆), R)
is an isomorphism. For H ● (X(∆)) is spanned by [V (τ )] so the above is surjective. Also
the relations in cohomology among χ1 , ⋯, χd are
χi1 ⋯χim = 0
which correspond to relations in A● (X): Di1 ∩ ⋯ ∩ Dim = ∅ and div(χm ) = 0. However the
same is true in A● (X) proving injectivity.
7. Cool facts to recall from intersection theory [Fulton]
● All symmetric polynomial in the Chern roots of E has a definite expression as a polynomial
in the Chern classes of E. This includes Todd class and the Chern character. So we start
with axiomatic definition of the Chern classes. Find them in terms of the Chern roots
7. COOL FACTS TO RECALL FROM INTERSECTION THEORY [FULTON]
44
of a decomposition and then define the characteristic classes in terms of the Chern roots;
e.g.
s
αi
−αi
i=1 1 − e
td(E) = ∏
ch(E) = ∑ eαi .
● If X is a complex manifold, and K0 (X) is the K-group of vector bundles over X, then
the Chern character
ch ∶ K0 (X)Q Ð
→ H 2● (X, Q)
is a ring homomorphism which is an isomorphism.
● If X is a smooth projective scheme we have a definition for Chern classes and the Chern
character of the locally free sheaves on X: The axioms
(1) L = O(D) then ch(L) = e[D] = 1 + [D] + 21 [D][D] + ⋯;
(2) For exact 0 Ð
→ E1 Ð
→EÐ
→ E2 Ð
→ 0 then ch(E) = ch(E1 ) + ch(E2 );
(3) Over flat pullbacks f ∗ (ch(E)) = ch(f ∗ (E))
determine ch(.) uniquely and it coincide with ∑ eαi if αi = c1 (Li ) where Li ’s are the factors
of a filteration
E 0 ⊂ E 1 ⊂ ⋯ ⊂ E s−1 ⊂ E
on flat Y Ð
→ X.
● If F is a coherent sheaf on X, take a free resolution
0Ð
→ Em Ð
→⋯Ð
→ E0 Ð
→F Ð
→0
by locally free sheaves. This exists if X is smooth. Then we define
ch(F) = ch(E0 ) − ch(E1 ) + ⋯.
● If f ∶ X Ð
→ Y is proper, have the push-forwards
f∗ [V ] = {
deg(f ∶ V Ð
→ f (V )).f (V ) dim f (V ) = dim V
0
otherwise
on the Chow groups and on the Grothendieck group of coherent sheaves on X, K0 (X)Q
by
f∗ (E) = [Rf∗ E].
● The Grothendieck-Riemann-Roch says that the following diagram is commutative
ch(.).tdX ●
/
K0 (X)Q
f∗
K0 (Y )
A (X)Q
ch(.).tdY
f∗
/ A● (Y )Q
8. TORIC RIEMANN-ROCH THEORY
45
i.e.
f∗ (ch(E).tdX ) = ch(Rf∗ (E)).tdY
for any coherent sheaf E on X.
● If X is singular, E is locally free we still have a version on the Hirzebruch-Riemann-Roch.
Recall that HRR on nonsingular complete variety is the reduction of the above formula to
the category of schemes over a point:
χ(E) = ∫ ch(E).td(X).
X
For singular X let Y be a resolution of singularities. Then we have H-R-R for E locally
free. Since in that case
χ(E) = χ(f ∗ E) = f ∗ (ch(E)) ∩ tdY = ch(E) ∩ f∗ tdY = ch(E) ∩ tdX .
Remark. If D is a Cartier divisor then O(mD) is always locally free (really?) so H-R-R
generated a proof that χ(O(mD)) is a polynomial in m:
χ(O(mD)) = ∫ ch(O(mD)).tdX = ∫ em.[D] (1 + td1 (X) + td2 (X) + ⋯)
= ∫ (1 + m[D] + m2
= mn
[D]n
[D]2
+ ⋯ + md
)(1 + td1 (X) + ⋯)
2!
n!
Dn−1
Dn
+ mn−1
td1 + ⋯
n!
(n − 1)!
Note that td1 = −KX .
8. Toric Riemann-Roch theory
Let P be a lattice polytope and form (X, O(D)). Then
Dn
n! is
n
the volume of P . So
f (m) = #(mP ∩ M ) = Vol(mP ) = m Vol(P ).
Recall the sheaf of log-pole differentials associated to a divisor D = ∑ Di of normal crossing.
Thus locally D = V (x1 ⋯xm ) in some local coordinates. Then Ω1X (log D) is locally free with
dxm
1
basis dx
x1 , ⋯, xm , dxm+1 , ⋯, dxn . Recall the short exact sequence
res
0Ð
→ Ω1X Ð
→ Ω1X (log D) ÐÐ→ ⊕i ODi Ð
→ 0.
i
The projection is f dx
xi ↦ f ∣Di .
For X = X(∆) and D = ∑ Di = X ∖ T we have
⊕n
Ω1X (log D) = OX ⊗Z M ≅ OX
´¸¶
Zn
8. TORIC RIEMANN-ROCH THEORY
46
with a slight abuse of notation for constant sheaves. The isomorphism is given in the reverse
direction via
dχu
f ⊗u↦f u .
χ
In K0 (X) we therefore have
d
[Ω1X ] = [ΩX (log D)] − ∑[ODi ].
i=1
In view of the short exact sequence
0Ð
→ IDi Ð
→ OX Ð
→ ODi Ð
→0
we have
d
[Ω1X ] = (n − d)[OX ] + ∑[O(−Di )].
i=1
We dualize to
[TX ] = (n − d)[OX ] = ∑[O(Di )].
We compute the Todd genus and Chern character now. We have td(O(D)) =
tdTX = ∏(1 +
i
D
1+e−D
so
P D[Di ] P D[Di ]2
+
+ ⋯).
2
12
For Chern character we have
d
chX = ch(TX ) = ∏(1 + [P D(Di )]) = 1 + ∑[P D(Di )] + ∑ P D[Di ∩ Dj ] + ⋯ = ∑ [Vτ ].
i=1
i
i<j
τ ∈∆
´¹¹ ¹ ¹ ¹ ¹ ¸ ¹ ¹ ¹ ¹ ¹ ¹ ¶
Vτ
So we also recover TdX = ∑τ rτ [Vτ ] for some rτ ∈ Q. We have
χ(O(D) = ∫
X
Dn−1
Dn
td0 +
td1 + ⋯
n!
(n − 1)!
and we observe that for any n − i dimensional Vτ we have
Dn−i
(D∣Vτ )n−i
∩ [Vτ ] =
= VolPτ .
(n − i)!
(n − i)!
So our combinatorial Riemann-Roch theorem is
#(mP ∩ M ) = χ(O(mD)) = ∑ rτ .Vol(Pτ ) = mn Vol(P ) +
τ
mn−1
∑ Vol(Pτ ) + ⋯.
2 facets
Example 8.1 (Pick’s theorem). For surfaces we have
#(mP ∩ M ) = m2 area(P ) + m/2 ∑ length(E) + ∫ td2 (X).
edges
8. TORIC RIEMANN-ROCH THEORY
47
We do not need to compute the last term since 1 = χ(OX ) in the case m = 0 and thus
td2 (X) = 1. We conclude that
1
#(P ∩ M ) = area(P ) + perimeter(P ) + 1.
2
⌟
Example 8.2. For O(D) = O(d + 1) in P3 we get a regular polytope of edge length d
and by the above we have
d3
11
#P ∩ M =
+ d2 + d + 1.
6
6
⌟