Algebraic Geometry Lecture 24 – Toric Varieties
Joe Grant1
Last week we went
Toric variety −→ (Toric) fan.
This week we go the other way.
Let our lattice be N∼
=Z2. Consider the two-dimensional cone σgenerated by (0,1) and (2,−1).
Find the dual cone σ∨⊂M=N∨= Hom(N, Z). Mhas a basis of functionals h1, h2that we’ll
write as h1= (1,0) and h2= (0,1). The dual cone is
{m∈M:m(s)>0 for every s∈σ}.
These are the “vectors in Mat most orthogonal to everything in N”. To see σ∨it helps to tensor
with R.
Over R,σ∨⊗Ris generated by (1,2) and (1,0). But σ∨itself (which is really M∩“the dual of
σ”) has three generators, (1,0),(1,2), and (1,1). This gives the semigroup associated to σ∨, called
Sσ. (If it comes from a fan then it’s finitely generated by Gordon’s theorem.)
Write this group multiplicatively, so (a, b)·(c, d) = (a+c, b +d). Then we can form the
semigroup algebra CSσwhose elements are linear combinations of elements of Sσwith the induced
multiplication.
pRecall group algebras: Let Gbe a finite group, then elements of CGlook like λ1g1+. . . +λngn,
and we multiply them by, for example,
(5g1+ 2g2)(√−3g3)=5√−3g1g3+ 2√−3g2g3.
Semigroup algebras work the same way.
y
If we write X= (1,0), Y = (0,1), we see Sσhas generators
X, X Y, XY 2.
So CSσ=C[X, X Y, XY 2]. We define a ring homomorphism
C[u, v, w]→CSσ
by
u7→ X v 7→ XY w 7→ XY 2.
This is clearly surjective and has kernel huw −v2i, so by the first isomorphism theorem,
CSσ∼
=C[u, v, w]/(uw −v2).
Finally to get our toric variety we let
Uσ= Spec CSσ,
then we can see that Uσ={(u, v, w)∈C3:uw −v2= 0}.
1Typed by Lee Butler
1
