2023/07/30 by Batyrev, Victor V.
#14E30 #14J45 (Primary) #14J70 (Secondary) #14M25 #Algebraic Geometry (math.AG) #Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.2307.16306
The Fine interior F(P) of a d-dimensional lattice polytope P ⊂ \Bbb Rd is the set of all points y ∈ P having integral distance at least 1 to any integral supporting hyperplane of P. We call a lattice polytope F-hollow if its Fine interior is empty. The main theorem claims that up to unimodular equivalence in each dimension d there exist only finitely many d-dimensional F-hollow lattice polytopes P, so called \em sporadic, which do not admit a lattice projection onto a k-dimensional F-hollow lattice polytope P' for some 1 ≤ k ≤ d-1. The proof is purely combinatorial, but it is inspired by \Bbb Q-Fano fibrations in the Minimal Model Program, since we show that non-degenerate toric hypersurfaces Z ⊂ (\Bbb C^*)d defined by zeros of Laurent polynomials with a given Newton polytope P have negative Kodaira dimension if and only if P is F-hollow. The finiteness theorem for d-dimensional sporadic F-hollow Newton polytopes P gives rise to finitely many families \mathcal F(P) of (d-1)-dimensional \Bbb Q-Fano hypersurfaces with at worst canonical singularities.