Laurent Inversion

We describe a practical and effective method for reconstructing the deformation class of a Fano manifold X from a Laurent polynomial f that corresponds to X under Mirror Symmetry. We explore connections to nef partitions, the smoothing of singular toric varieties, and the construction of embeddings of one (possibly-singular) toric variety in another. In particular, we construct degenerations from Fano manifolds to singular toric varieties; in the toric complete intersection case, these degenerations were constructed previously by Doran–Harder. We use our method to find models of orbifold del Pezzo surfaces as complete intersections and degeneracy loci, and to construct a new four-dimensional Fano manifold.