Abstract
AbstractAn 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.
Highlights
Recall that a normal projective variety X with log terminal singularities such that the anticanonical divisor âKX is an ample Q-Cartier divisor is said to be Fano
There are 16, 4319, and 473,800,776 isomorphism classes in, respectively, dimension two, three, and four. These classifications are of particular interest: Gorenstein toric Fano varieties are used to construct mirror pairs of Calabi-Yau varieties
A toric variety is a normal variety X that contains an algebraic torus as a dense open subset, together with an action of the torus on X which extends the natural action of the torus on itself
Summary
Gorenstein toric Fano varieties have been classified up to dimension four. There are 16, 4319, and 473,800,776 isomorphism classes in, respectively, dimension two, three, and four (see [KS97, KS98, KS00]) These classifications are of particular interest: Gorenstein toric Fano varieties are used to construct mirror pairs of Calabi-Yau varieties (see [Bat[94], BB96, KS02]). One can attempt to classify those toric Fano varieties with at worst terminal singularities. Every surface of this form is nonsingular, and so the classification reduces to the smooth case above. All the above classifications are subsets of a more general case: toric Fano varieties with at worst canonical singularities.
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