Projecting lattice polytopes according to the Minimal Model Program

Abstract

The Fine interior $F(P)$ of a $d$-dimensional lattice polytope $P \subset {\Bbb R}^d$ 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 {\em 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 ${\Bbb Q}$-Fano fibrations in the Minimal Model Program, since we show that non-degenerate toric hypersurfaces $Z \subset ({\Bbb C}^*)^d$ 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.