# Wednesday, October 11 ## $\S 7.1$: Polyhedra and Toric Varieties :::{.remark} Last time: quasiprojective varieties and projective morphisms. Note projective $\implies$ quasiprojective and affine $\implies$ quasiprojective, but neither of projective/affine imply each other. The implications are strict: $\AA^1\times \PP^1$ is quasiprojective but neither affine nor projective. Goal: quasiprojective toric varieties, which are neither affine nor projective. These are described by polyhedra (intersections of a finite number of half-spaces). Note that this may be an unbounded region. Polytopes are special cases of polyhedra where the region is bounded and is the convex hull of finitely many points. ::: :::{.remark} Recall constructing affine toric varieties from cones: $X_C = \spec \CC[C \intersect M]$ for $C \subseteq M_\RR$ a strongly convex cone. Regular functions on $X_C$ correspond to integral points in $C$ by construction. For a polytope $P \subseteq M_\RR$, embed $P$ in $M_\RR \times \ZZ$ at height 1 and let $C_P$ be the cones induced by coning $P$ to the origin. Then $X_{C_P} \da \spec \CC[M_{C_P}]$ is the affine cone of a projective variety. Consider $\cstar_\actson X_{C_P}$ by $(t, f) \mapsto t^n\cdot f$; the only fixed point is $f=0$. So one can consider the projective toric variety $\qty{ \spec \CC[M_{C_P} ] \smz }/\cstar = \Proj \CC[M_{C_P} ]$. ::: :::{.remark} For affine varieties, cones biject with dual cones. However, for projective varieties, the polytope includes more data than then fan -- namely, the data of an ample line bundle. Many polytopes correspond to the same fan $\Sigma$, however this choice can be made unique after fixing a strongly convex PL function on $\Sigma$. ::: :::{.definition title="Recession cone"} The recession cone associated to a polyhedron $P$ given by the intersection of closed affine half-spaces is the cone obtained by moving every hyperplane to the origin and intersecting the half-spaces there. ::: :::{.remark} E.g. if $P$ is $\HH \intersect A \intersect B$ where $A = \ts{\Re(z) \geq -1}$ and $B = \ts{\Re(z) \leq 1}$ then the recession cone is $\ts{\Re(z) = 0 , \Im(z) > 0 }$. How to think about this: contracting bounded edges, i.e. viewing $P$ from far away. Note that if $P$ is already a cone, the recession cone of $P$ is $P$ itself. ::: :::{.definition title="Lattice polyhedron"} A lattice polyhedron $P$ is a **lattice polyhedron** ineffective 1. The vertices of $P$ are in $M$, and 2. The recession cone of $P$ is a strongly convex rational polyhedral cone. :::