Canonical models of toric hypersurfaces

Abstract

Let Z ⊂ T d be a non-degenerate hypersurface in d-dimensional torus T d ∼ = (C * ) d defined by a Laurent polynomial f with a given d-dimensional Newton polytope P . It follows from a theorem of Ishii that Z is birational to a smooth projective variety X of Kodaira dimension κ ≥ 0 if and only if the Fine interior F (P ) of P is nonempty. We define a unique projective model Z of Z having at worst canonical singularities which allows us to obtain minimal models Z of Z by crepant morphisms Z → Z. Moreover, we show that κ = min{d − 1, dim F (P )} and that general fibers in the Iitaka fibration of the canonical model Z are nondegenerate (d − 1 − κ)-dimensional toric hypersurfaces of Kodaira dimension 0. Using the rational polytope F (P ), we compute the stringy E-function of minimal models Z and obtain a combinatorial formula for their stringy Euler numbers.

Kanev and Todorov surfaces in toric 3-folds

“…In this article we study some surfaces of general type that arise as hypersurfaces in toric 3-folds. This article illustrates results from the article ( [Bat20]) via concrete examples. In the latter article it was shown quite generally how to construct minimal and canonical models for nondegenerate hypersurfaces in toric varieties.…”

“…In this section we summarize some results from ( [Bat20]). The results of this section stay true in any dimension.…”

“…But Z ∆ might have bad singularities. In ( [Bat20]) it was shown, that via natural operations on the polytope ∆ we can partially resolve the singularities of Z ∆ and make the canonical divisor of the hypersurface nef. In particular the Fine interior F (∆) of ∆ is important within this context: If ∆ is defined by the equations…”

The Kodaira-Spencer map for nondegenerate toric hypersurfaces

“…In this article we study natural families of minimal surfaces in toric 3-folds: We vary f over all nondegenerate Laurent polynomials U reg (∆) with fixed 3-dimensional Newton polytope ∆. By the results from ( [Bat20]) there is a projective toric variety P Σ to a fan Σ, such that the compactification Z Σ,f of Z f := {f = 0} ⊂ (C * ) 3 in P Σ is smooth and has nef canonical divisor K Z Σ,f , independently of f ∈ U reg (∆). We say that Z Σ,f is a minimal model of Z f .…”

The plurigenera of nondegenerate minimal toric hypersurfaces

“…In the article ( [Bat20]) it was shown how to construct minimal models and other birational models with mild singularities of nondegenerate toric hypersurfaces. That is we start with a nondegenerate Laurent polynomial f with an n-dimensional Newton polytope ∆ ⊂ M R , consider the zero set Z f := {f = 0} ⊂ (C * ) n and ask for a projective variety Z Σ birational to Z f with at most terminal singularities and with K Z Σ nef, which we also call a minimal model of Z f .…”