# Toric Geometry

# Toric Varieties

A polyhedron in \QQ^{d} is a subset PP defined by finitely many weak affine inequalities,

P={xQd:j=1daijxjbj,aij,bjQ}.P = \{ x \in \mathbb{Q}^{d} : \sum_{j=1}^{d} a_{ij}x_{j} \geq b_{j},\, a_{ij}, b_{j} \in \mathbb{Q} \}.

# Quotients of affine toric varieties

Let RR be a finitely generated integral algebra over C\CC, so that Spec(R)=X\Spec(R) = X is an affine variety over C\CC. An action of G=CG = \CC^{\ast} on Spec(R)\Spec(R) is equivalent to a Z\ZZ-grading of RR over C\CC, say R=iZRiR = \oplus_{i \in \ZZ} R_{i}. To study the GIT quotients X//GX // G linearised on the trivial bundle, choose any nZn \in \ZZ and define a Z\ZZ-grading on R[z]R[z] by RiR[z]iR_{i} \subset R[z]_{i}, zR[z]nz \in R[z]_{-n}. As X=Spec(R)=Proj(R[z])X = \Spec(R) = \Proj(R[z]), this Z\ZZ-grading is equivalent to a linearisation on the trivial bundle O\mc{O} of the GG-action of XX. The quotient is then Proj(R[z]G)=Proj(R[z]0)=ProjiNRnizi\Proj(R[z]^{G}) = \Proj(R[z]_{0}) = \Proj \oplus_{i\in \NN} R_{ni}z^{i}.

For n=0n = 0, this is just the usual affine quotient Proj(R0[z])=Spec(R0)\Proj(R_{0}[z]) = \Spec(R_{0}). For n>0n > 0,

ProjiNRni=ProjiNRi, \Proj\oplus_{i \in \NN} R_{ni} = \Proj \oplus_{i \in \NN}R_{i},

and similarly for n<0n < 0,

ProjiNRni=ProjiNRi. \Proj\oplus_{i \in \NN} R_{ni} = \Proj \oplus_{i \in \NN}R_{-i}.

So we only need to concern ourselves with the quotients when n=0,+1n = 0, +1, and 1-1, which we shall refer to as X//0X // 0, X//+X // +, and X//X // -, respectively. Observe that both X//±X // \pm are projective over X//0X // 0.


Let R=C[x1,,xn;y]R =\CC[x_{1},\ldots, x_{n};y] where the xix_{i}'s each have degree 0, and yy has degree 11, so that Proj(R)=Cn\Proj(R) = \CC^{n}. To lift the action of G=CG = \CC^{\ast} on Cn\CC^{n} to an action on RR, we must choose a linearisation of the GG-action on Cn\CC^{n}. Such a linearisation must have the form

txi=tαixi,andty=tλy, t \cdot x_{i} = t^{-\alpha_{i}}x_{i},\qquad \text{and} \qquad t \cdot y = t^{\lambda}y,

for αi,λZ\alpha_{i}, \lambda \in \ZZ.

The maximal compact subgroup K=S1GK = S^{1} \subset G then acts on Cn\CC^{n} with the corresponding moment map

μ(z1,,xn)=12α1z12++12αnzn2λ. \mu(z_{1},\ldots, x_{n}) = \frac{1}{2}\alpha_{1}|z_{1}|^{2} + \ldots + \frac{1}{2}\alpha_{n}|z_{n}|^{2} - \lambda.