Canonical models of toric hypersurfaces

2020/08/13 by Batyrev, Victor V.
#Algebraic Geometry (math.AG) #Combinatorics (math.CO) #FOS: Mathematics #FOS: Physical sciences #High Energy Physics - Theory (hep-th) #Symplectic Geometry (math.SG)

paper · doi:10.48550/arxiv.2008.05814

Abstract

Let Z be a nondegenerate hypersurface in d-dimensional torus (ℂ^*)^d defined by a Laurent polynomial f with a d-dimensional Newton polytope P. The subset F(P) ⊂ P consisting of all points in P having integral distance at least 1 to all integral supporting hyperplanes of P is called the Fine interior of P. If F(P) ≠ ∅ we construct a unique projective model \widetildeZ of Z having at worst canonical singularities and obtain minimal models Z of Z by crepant morphisms Z→ \widetildeZ. We show that the Kodaira dimension κ=κ(\widetildeZ) equals min \ d-1, dim F(P) \ and the general fibers in the Iitaka fibration of the canonical model \widetildeZ are non\-degenerate (d-1-κ)-dimensional toric hypersurfaces of Kodaira dimension 0. Using F(P), we obtain a simple combinatorial formula for the intersection number (K_\widetildeZ)^d-1.

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