Abstract
This paper classifies all toric Fano 3-folds with terminal singularities. This is achieved by solving the equivalent combinatoric problem; that of finding, up to the action of GL(3,Z), all convex polytopes in Z^3 which contain the origin as the only non-vertex lattice point.
Highlights
Background and IntroductionA toric variety of dimension n over an algebraically closed field k is a normal variety X that contains a torus T ∼= (k∗)n as a dense open subset, together with an action T × X → X of T on X that extends the natural action of T on itself.Let M ∼= Zn be the lattice of characters of the torus T, with dual lattice N = Hom(M, Z)
There is a well-known description of what it means for a toric variety X to be a Fano variety in terms of its fan ∆: Let {ρi}i∈I be the set of rays of ∆
Since the origin lies in the strict interior of P there must exist distinct vertices y1, . . . , y4 of P not equal to x1 or x2 such that conv{x1, x2, y1, y2} is a Fano square and conv{x1, x2, y1, y2, y3, y4} is a Fano octahedron
Summary
There exist three vertices of P which lie in a plane containing the origin, and the origin lies strictly in the interior of the triangle defined by these three points This possibility will be discussed in further detail below. Since the origin lies in the strict interior of P there must exist vertices y1 and y2 lying on either side of the plane containing our Fano triangle. Either x = x, which gives a Fano polytope equivalent to the one previously found, or x lies outside of P If this is the case we have that x lies on the opposite side to the origin of the plane defined by {e2, −e1 − e2, x}. Given any Fano polytope P with vertices {x1, . . . , xk} we make the following definition (cf. Definition 3.2):
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