An inductive approach to classifying all toric Fano varieties is given. As an application of this technique, we present a classification of the toric Fano threefolds with at worst canonical singularities. Up to isomorphism, there are 674,688 such varieties.

We discuss n4 configurations of n points and n planes in three-dimensional projective space. These have four points on each plane, and four planes through each point. When the last of the 4n incidences between points and planes happens as a consequence of the preceding 4n−1 the configuration is called a ‘theorem’. Using a graph-theoretic search algorithm we find that there are two 84 and one 94 ‘theorems’. One of these 84 ‘theorems’ was already found by Möbius in 1828, while the 94 ‘theorem’ is related to Desargues’ ten-point configuration. We prove these ‘theorems’ by various methods, and connect them with other questions, such as forbidden minors in graph theory, and sets of electrons that are energy minimal.

The theorem referred to in the title is a technical result that is needed for the classification of elliptic and K3 fibrations birational to Fano 3-fold hypersurfaces in weighted projective space. We present a complete proof of the curve exclusion theorem, which first appeared in the author's PhD thesis and has since been relied upon in solutions to many cases of the fibration classification problem. We give examples of these solutions and discuss them briefly.