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Laurent inversion

Coates, Tom; Kasprzyk, Alexander; Prince, Thomas


Tom Coates

Thomas Prince


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.


Coates, T., Kasprzyk, A., & Prince, T. (2019). Laurent inversion. Pure and Applied Mathematics Quarterly, 15(4), 1135–1179.

Journal Article Type Article
Acceptance Date May 19, 2019
Online Publication Date Mar 20, 2020
Publication Date 2019
Deposit Date Jul 9, 2019
Publicly Available Date Jul 10, 2019
Journal Pure and Applied Mathematics Quarterly
Print ISSN 1558-8599
Electronic ISSN 1558-8602
Publisher International Press
Peer Reviewed Peer Reviewed
Volume 15
Issue 4
Pages 1135–1179
Keywords Mirror symmetry, Fano manifolds, Toric degenerations
Public URL
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