Laurent Inversion
Abstract
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.
Keywords
Cite
@article{arxiv.1707.05842,
title = {Laurent Inversion},
author = {Tom Coates and Alexander Kasprzyk and Thomas Prince},
journal= {arXiv preprint arXiv:1707.05842},
year = {2021}
}
Comments
29 pages, 16 figures. This supersedes our earlier preprint with the same name (arXiv:1505.01855 [math.AG]). The new version is much more systematic, and works beyond the toric complete intersection case; it also draws connections to the work of Doran--Harder on amenable collections and Batyrev--Borisov on nef partitions