Projecting Lattice Polytopes According to the Minimal Model Program

The Fine interior \(F(P)\) of a d-dimensional lattice polytope \(P \subset \mathbb {R}^d\) is the set of all points \(y \in P\) 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 sporadic, which do not admit a lattice projection onto a k-dimensional F-hollow lattice polytope \(P'\) for some \(1 \leq k \leq d-1\) . The proof is purely combinatorial, but it is inspired by \(\mathbb {Q}\) -Fano fibrations in the Minimal Model Program, since we show that non-degenerate toric hypersurfaces \(Z \subset (\mathbb {C}^*)^d\) 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 \(\mathbb {Q}\) -Fano hypersurfaces with at worst canonical singularities.