# 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 \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}.