Cracked polytopes and Fano toric complete intersections

Abstract

We introduce the notion of cracked polytope, and -making use of joint work with Coates and Kasprzyk -construct the associated toric variety X as a subvariety of a smooth toric variety Y under certain conditions. Restricting to the case in which this subvariety is a complete intersection, we present a sufficient condition for a smoothing of X to exist inside Y . We exhibit a relative anti-canonical divisor for this smoothing of X, and show that the general member is simple normal crossings.Conventions. Throughout this article N ∼ = Z n will refer to an n-dimensional lattice, and M := hom(N, Z) will refer to the dual lattice. Given a ring R we write N R := N ⊗ Z R and M R := M ⊗ Z R. For brevity we let [k] denote the set {1, . . . , k} for each k ∈ Z ≥1 . We work over an algebraically closed field k of characteristic zero. Cracked polytopesIn this section we introduce the notion of cracked polytope, which will form our main object of study, and, in the Fano setting, characterize its dual polytope. We will assume basic ideas and results from toric geometry -see [9, 10] -throughout.

From cracked polytopes to Fano threefolds

“…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. Theorem 1.1.…”

“…For general choices of S, the variety Y S may be highly singular: for example Y S need not be Q-Gorenstein. In [34] we explore the (restrictive) conditions on S which ensure that Y S is non-singular, and introduce the following notion.…”

“…Taking σ is minimal among such cones, σ corresponds to a non-singular toric stratum Z(σ) of the toric variety TV(Σ). It is shown in [34,Proposition 2.8] that the facet F of P • is a Cayley sum P D 1 · · · P D k , where {D i : 1 ≤ i ≤ k} is a set of nef divisors on Z(σ), and k = dim(σ) + 1. We call a face of P • vertical if it is contained in a factor P D i of some facet F = P D 1 · · · P D k and some i ∈ {1, .…”

“…We explain how to construct an extensible database of Fano manifolds in each dimension. In particular, we develop a combinatorial framework, based on the notion of cracked polytopes introduced in [34]. We show that this framework is flexible enough to obtain every Fano threefold with −K X very ample and b 2 ≥ 2, famously classified by Mori-Mukai [24][25][26][27][28].…”

“…(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.…”