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. arXiv:1707.05842v1 [math.AG]

From cracked polytopes to Fano threefolds

“…Scaffolding a polytope P determines an embedding of X P into an ambient space Y S . This is the main result of [13]; see also the treatment given in [34, §3]. where e i denotes the standard basis of Div TM Z ∼ = Z .…”

“…Fixing a complete fan -which we refer to as the shape -we say a polytope is cracked along Σ if its intersection with each maximal cone of Σ is unimodular, see Definition 2.1. In [13] we show that embeddings of X P into toric varieties, compactifying the embedding of affine varieties described above, are described by scaffoldings. Moreover, in [34] we show that embeddings of X P into non-singular toric ambient spaces Y correspond to the combinatorial condition that the scaffolding is full, see Definition 2.7 and Theorem 2.8.…”

“…The method Laurent inversion -introduced in [13] -was developed to construct models of Fano manifolds embedded in toric varieties. To describe this method we first fix a splitting N =N ⊕ N U of N .…”

“…For each i ∈ [c] the degree of the line bundle L i cutting out X P in Y S is given by the sum of the columns in C i . In particular, there is a distinguished binomial z m 1 − z m 2 in L i , where m 1 is the sum of standard basis vectors in (Z n+r ) corresponding to the columns of C i , and m 2 is the unique lift of L i ∈ L to (Z r ) : the subspace of (Z n+r ) corresponding to the first r columns of R. It is shown in [13], see also [34, §3], that X P is the vanishing locus of these c binomials.…”

“…(i) Intersecting the fan Σ with a lattice polytope P , we describe how the embedding of (C ) n determined by Σ may be compactified to an embedding of the toric variety X P in a non-singular toric variety Y . This is based on [34] and joint work [13] with Coates and Kasprzyk. (ii) The embedding of affine spaces determined by Σ admits various possible deformations, and we explicitly construct embedded deformations in Y by homogenizing the coordinate rings of such families.…”