Toric Varieties: The Path Towards Faulk
Nawaz Walter Sultani
April 27, 2016
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Chapter 2.2: The Jam’s Jam
1.1
Notation
We follow the notation set out by Fulton. In particular we have the following:
1. N s a lattice, and M is it’s dual lattice
2. Given a cone , we have that S =
_
\ M is the associated finitely generated semigroup.
3. C[S ] is the associated finitely generated commutative C-algebra
4. U = Spec C[S ]
In addition, all cones we work with will be strongly convex rational polyhedral cones.
1.2
Quotient Singularities
Consider the lattice N = Z2 and the cone which is generated by e2 and me1 e2 for m 2 Z. (See figure
1). Then it’s associated semigroup is easily verified to be generated by the dual vectors e⇤1 , e⇤1 + e⇤2 , e⇤1 +
2e⇤2 , . . . , e⇤1 , me⇤2 . This results in the associated C algebra to be of the form
C[S ] = C[X, XY, . . . , XY m ]
Setting A := C[S ], and changing variables such that X = U m and Y = V /U , we obtain that
A = C[U m , U m
1
V, . . . , V m ] ⇢ C[U, V ]
However, observe that the variables of A are precisely in the form associated to the Veronese embedding of
degree m from P1 , and thus we have that U is the cone over the rational normal curve of degree m. The
above inclusion then implies that we have a mapping C2 ! U .
Now consider the group G ⇠
= Z/mZ where G is comprised of the m-th roots of unity. This group has a
natural action on C2 as follows. For ⇠ 2 G and (u, v) 2 C2 , we have
⇠ · (u, v) = (⇠u, ⇠v)
Then we can take the quotient of C2 by this action to get the quotient variety C2 /G. Of course, this
corresponds to an action on the corresponding rings, given as follows. For a function f 2 C[U, V ], we have
that ⇠ acts through the action
⇠·f =f ⇠
where on the right, ⇠ acts by its topological action.
1
Claim. C2 /G ⇠
= U , the quotient singularity, and A = C[U, V ]G , the ring of invariants under this G action.
In this scenario, we call U the cyclic quotient singularity.
claim in di↵erent light. To prove the claim, we observe it in a more natural setting using the toric structure.
Consider the following sub lattice N 0 ⇢ N generated by e2 and me1 e2 (Hella dumb lattice). Then we can
regard again as a cone corresponding to N 0 . Call this interpretation of the cone 0 . Then by construction,
its obvious that 0 is generated by two generators for N 0 , so its clear that U 0 has two generators and is thus
equivalent to C2 . Then the inclusion map N 0 ! N gives a corresponding map on the varieties U 0 ! U .
Now note that while we chose our generators of N 0 as above, in fact it’s clear that N is generated solely by
the vectors e2 and me1 . Since N 0 ⇢ N , we have the reverse for the dual cones, namely M 0 M . Morevoer,
1 ⇤
M 0 is generated by m
e1 and e⇤2 , and these generators correspond to U and Y respectively, where U is defined
as above (Here, X and Y are the standard coordinate basis).
Now the generators for S 0 is given by
N 0 . Therefore, we have that
1 ⇤
m e1
A
0
and
1 ⇤
m e1
+ e⇤2 , again an easy check by our construction of
= C[U, U Y ] = C[U, V ]
and thus, we get the same map as before, namely C[U, V ] ! A .
To see that A is the quotient singularity, one just needs to observe that the invariant coordinate ring
under the action of G is precisely the homogenous degree m polynomials, which is what we described above.
While the above quotient singularity construction was ballin, it was really for a specific cone we chose.
However, as our gracious lord Faulk allowed, we can extend this procedure to any two-dimensional affine
toric variety.
Proposition 1.1. For an appropriate choice of basis for N , we can assume that the cone
the vectors e2 and me1 + ke2 , where k and m are relatively prime integers.
is generated by
Proof. First look at the minimal generators along the edge of . Then we can always choose on the generators
to be e2 without a problem by a simple rotation, and the other is then of the form (m, y) for m a positive
integer, y some variable which we don’t know yet. However we can apply the following automorphism to the
lattice
✓
◆ ✓
◆ ✓
◆
1 0
m 0
m
0
·
=
c 1
y 1
cm + y 1
Note that this automorphism preserves the e2 basis vector we designated earlier. However, we also see that
this automorphism allows us to change y, as long as it is preserved modulo m. Therefore, we can take y = k
for some 0  k  m, because why not at this point. However, if y = 0modm, we are in the boring standard
case, so I have no clue why we care. The fact that k and m are relatively prime comes from the fact that we
take a minimal generator, and non-coprime pairs are not minimal.
With this , it’s clear that we can generate the corresponding semigroup by the elements ie⇤1 + je⇤2 where
jm
C · X i · Y j where the sum is over the allowed pairs (i, j).
k i, and thus we have that A =
Now we do a similar procedure as before, reusing the same notation. Let N 0 be the lattice generated
1 ⇤
by me1 ke2 and e2 . Then M 0 is clearly generated by m
e1 and e⇤2 , which again corresponds to U and Y
1
k
⇤
(see the parallels?). Then S 0 is generated by m e1 and m e⇤1 + e⇤2 and thus we have
A
0
= C[U, U k Y ] = C[U, V ]
2
where V = U k Y
and thus we have that U
0
= C2 again.
Now consider the same group G as before, but with the action on C2 given by
⇠ · (u, v) = (⇠u, ⇠ k v)
Then we have that
U = U 0 /G = C2 /G
again clear by looking at the polynomials that comprise A and noting that A = AG0 . To be more explicit,
we note that G acts on the ring C[M 0 ] = C[U, U 1 , V, V 1 by the action that takes U ! ⇠U and Y ! Y .
(the second equality for C[M 0 ] is a direct isomorphism related to how V is defined). Thus, the ring of
invariants is given by C[X, X 1 , Y, Y 1 ] = C[M ]. This gives the string of equalities
A =A
0
\ C[M ] = A
0
\ (C[M 0 ])G = (A 0 )G
Now we take this construction further. 2 dimensions is for the weak. Let’s go for the kill and attack n
dimensions. Let N 0 ⇢ N be a sub lattice of finite index. Then we have that M ⇢ M 0 , and that there is a
canonical dual pairing
M 0 /M ⇥ N/N 0 ! Q/Z ,! C
The first map is given but the dual pairing, and the second is given by the map q ! e2⇡iq . This gives a
natural action of G = N/N 0 on C[M 0 ] by
v·
u
= e2⇡ihu,vi ·
u
and its clear that under his action we have that M is the fixed portion, and thus one suspects that under
this action, we have
C[M 0 ]G = C[M ]
(1)
In particular, we have that G acts on the torus TN 0 and that the quotient by the action gives us TN . We
now prove equation 1.
proof of equation 1. Take {ei } to be a basis for N such that {mi ei } is a basis for N 0 , where mi are suitable
positive integers. Then we have that C[M 0 ] is the a laurent polynomial ring with generators we can call Xi ,
such that (Ui )mi = Xi where C[M ] is the laurent polynomial ring with generators Ui . Then we have that
N/N 0 =
Z/mi Z
1
n
and an
P arbitrary element of this group (a1 , . . . an ) acts on the monomial U1 · · · · · Un by multiplication by
2⇡i( ai / mi )
e
for each respective coordinate. Note this action gives us the action we saw earlier in the n=2
case.
Now consider the setting where is an n-simplex and n0 is the sub lattice generated by the minimal
elements in \ N along the n edges. Then letting G = N/N 0 and constructing the cone 0 as before, we
have that
U = U 0 /G = C[M ]
by a similar processes as all the the other times.
Now lets do something cool. If is any simplex, then U is given by the product of the quotient above and
a torus. This can be generalized to a simplicial fan, i.e. the case where
only consists of simplicial cones,
and thus we get that X( ) is an orbifold.
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Orbifolds (or V-manifold, as Satake coined them) are super cool objects that are basically manifolds that are
locally quotiented out by some finite group action. However, it may be simpler for the reader to just think
of them as the di↵erential geometric analogue of a smooth Deligne-Mumford stack, a trivially easy object to
understand.
Example 1.2 (Weighted Projective Space). An example of a quotiented toric variety (and also of an orbifold)
is given by weighted complex projective space, denoted by Pn (d0 , . . . , dn ). To construct this, we recall that
the n th projective space is is constructed by the fan whose cones are generated by proper subsets of
{v0 , . . . vn }, where any n of the vectors is linearly independent and the sum is zero. However, take the lattice
N to be generated by d1i · vi . Then we have that
Pn (d0 . . . dn ) = Cn+1 {0}/C⇤
where the C⇤ action is given by ⇠ · (x0 , . . . , xn ) = (⇠ d0 x0 , . . . , ⇠ dn xn ).
As an orbifold, one notes that the only nontrivial isotropy groups are located at the coordinate points
of the projective space, and that this is indeed not a global quotient. As a toric variety, the gluing is the
same as projective space, with the origin of each affine patch having nontrivial isotropy.
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