Topic: Preliminary material
Mitchell Faulk
February 7, 2016
The goal of these notes is to introduce the topic of toric varieties in a “friendly” way. If these notes were
optimal, they would include pictures, but those will need to be supplemented by the reader.
We begin with some examples that illustrate the basic procedure.
Example 0.1. Let e1 and e2 denote the standard generators of R2 , meaning that e1 = (1, 0) and e2 = (0, 1).
Consider the “cone” σ in R2 generated by e2 and 2e1 − e2 . This means that σ is the subset of R2 given by
σ = {λe2 + µ(2e1 − e2 ) : λ, µ > 0}.
The “dual cone” σ ∨ consists of all vectors which pair non-negatively with all vectors in σ. This means
that σ ∨ is the subset of R2 given by
σ ∨ = {v ∈ R2 : hv, ui > 0 for each u ∈ σ}.
It is not too difficult to show that σ ∨ is generated by e1 and e1 + 2e2 .
Now consider the subset of Z2 given by Sσ = σ ∨ ∩ Z2 . This means that Sσ consists of all lattice points in
σ ∨ . We remark that Sσ is actually a “semi-group,” which is exactly like a group, except that inverses need
not exist. Generators for this particular Sσ are given by e1 , e1 + e2 , and e1 + 2e2 .
We now associate to the semigroup Sσ a type of “C-algebra” in the following manner. Each generator
of R2 becomes a variable so that e1 corresponds to x1 and e2 corresponds to x2 . Then the generators of Sσ
correspond to monomials whose powers are determined by the coefficients of the generators in Sσ , so that,
in our example, we obtain the following three monomials x1 , x1 x2 , x1 x22 . These monomials then become
the generators for a C-algebra which we denote C[Sσ ] = C[x1 , x1 x2 , x1 x22 ]. The notation means that C[Sσ ]
consists of all polynomials in the monomials x1 , x1 x2 , and x1 x22 with coefficients in C. So for our example,
the following is a typical element of C[Sσ ]
λxk1 + µ(x1 x2 )` + ν(x1 x22 )m ,
for some λ, µ, ν ∈ C and k, `, m ∈ N.
It turns out that the C-algebra C[Sσ ] can also be expressed as
C[Sσ ] = C[u, v, w]/(uv − w2 )
when we write u = x1 , v = x1 x22 and w = x1 x2 . This C-algebra is the “coordinate ring” of the so-called
“quadric cone,” which is a “toric variety.” The quadric cone is the subset of C3 given by the zero locus of
the polynomial uv − w2 , where u, v, w denote the three coordinate directions of C3 .
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This example demonstrates how we can take translate the combinatorial data of a “cone” into a geometric
object called a “variety.” The next example explores what happens when we consider multiple cones put
together into a “fan.”
Example 0.2. Consider the following “fan” ∆ of cones in R1 consisting of
∆ = {R>0 , R60 , {0}}.
To say that ∆ is a fan means that ∆ consists of a collection of cones such that each intersection of cones is
another cone in ∆.
Using the previous construction, we get dual cones whose corresponding list of semigroups is {Z>0 , Z60 , Z}.
−1
We also obtain corresponding C-algebras {C[x1 ], C[x−1
1 ], C[x1 , x1 ]}. The corresponding toric varieties are
{C, C, C∗ }.
The inclusion of semigroups Z>0 ,→ Z leads to an inclusion of C-algebras C[x1 ] ,→ C[x1 , x−1
1 ] satisfying
−1
x1 7→ x1 . On the other hand, the other inclusion Z60 ,→ Z leads to the inclusion C[x−1
1 ] ,→ C[x1 , x1 ]
−1
satisfying x−1
1 7→ x1 . In this way, we can view ∆ as giving two copies of C glued together via the attaching
1
∗
isomorphism x1 7→ x−1
1 on the overlap C . This is also known as “complex projective space” CP .
The examples show that there are many terms which we need to define. Many of the terms will be defined
later. Let us begin by defining complex projective space.
Definition 0.3. Define a relation on Cn+1 \ {0} by the rule x ∼ y if there is a scalar λ ∈ C such that
x = λy. Let CPn denote the set of equivalence classes under this equivalence relation and call CPn complex
projective space of dimension n. Informally, CPn consists of all lines through the origin in Cn+1 .
Remark 0.4. To equip CPn with the structure of a topological space, we can equip it with the quotient
topology coming from the standard topology on Cn+1 . There is another topology on CPn which we will
consider later.
If (x0 , . . . , xn ) is a point of Cn+1 \ {0}, we let [x0 , . . . , xn ] denote point of CPn it determines. Note that
if λ is a nonzero scalar, then [x0 , . . . , xn ] = [λx0 , . . . , λxn ].
For j satisfying 0 6 j 6 n, let Uj denote the subset of CPn given by
Uj = {[x0 , . . . , xn ] : xj 6= 0}.
Note that the collection {Uj } forms a cover of CPn .
We claim that each Uj can be identified with Cn . Indeed, if [x0 , . . . , xn ] is a point of Uj , then this point
is equal to the point [ xx0j , . . . , 1, . . . , xxnj ]. Hence we have a correspondence ϕj : Uj → Cn given by
[x0 , . . . , xn ] 7→
x0
xj−1 xj+1
xn
,...,
,
,...,
xj
xj
xj
xj
2
.
An inverse for ϕj is given by
(y1 , . . . , yn ) 7→ [y1 , . . . , yj−1 , 1, yj , . . . , yn ].
Example 0.5. In particular, let us consider the case when n = 1. We then have two sets U0 and U1 in our
cover. There is a correspondence ϕ0 : U0 → C given by
[x0 , x1 ] 7→
x1
x0
[x0 , x1 ] 7→
x0
.
x1
and a correspondence ϕ1 : U1 → C given by
The “transition map” ϕ1 ◦ ϕ−1
: C∗ → C∗ is given by x 7→ x−1 . This is the description of CP1 given in
0
Example 0.2.
Example 0.6. When n = 2, we have three elements of the cover U0 , U1 , U2 . The coordinates on each are
x1 x2
,
ϕ0 : [x0 , x1 , x2 ] 7→
x0 x0
x0 x2
ϕ1 : [x0 , x1 , x2 ] 7→
,
x1 x1
x0 x1
ϕ2 : [x0 , x1 , x2 ] 7→
,
.
x2 x2
We conclude by discussing a fan which gives rise to CP2 .
Example 0.7. Consider the fan ∆ in R2 generated by the following three cones
σ0 = {λe1 + µe2 : λ, µ > 0}
σ1 = {λe2 + µ(−e1 − e2 ) : λ, µ > 0}
σ2 = {λe1 + µ(−e1 − e2 ) : λ, µ > 0}.
To say that ∆ is a fan means that we also take all intersections of these cones to be members of ∆ as well.
Then the corresponding dual cones are
σ0∨ = σ0
σ1∨ = {λ(−e1 ) + µ(e2 − e1 ) : λ, µ > 0}
σ2 = {λ(−e2 ) + µ(e1 − e2 ) : λ, µ > 0}
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with corresponding C-algebras
C[Sσ0 ] = C[x, y]
C[Sσ1 ] = C[x−1 , yx−1 ]
C[Sσ2 ] = C[y −1 , xy −1 ].
All three of these C-algebras can be viewed as the coordinate rings of C2 . In particular, if we let x = x1 /x0
and y = x2 /x0 , where (x0 , x1 , x2 ) are coordinates on C3 , then we find that these coordinate rings become
x1 x2
,
C[Sσ0 ] = C
x0 x0
x0 x2
C[Sσ1 ] = C
,
x1 x1
x0 x1
C[Sσ2 ] = C
,
.
x2 x2
These are the coordinates used for U0 , U1 , U2 in Example 0.6 of CP2 . Hence this fan gives rise to CP2 .
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