Problems on Minkowski sums of convex lattice polytopes

TL;DR

This paper investigates the conditions under which Minkowski sums of convex lattice polytopes correspond to algebraic geometric properties of toric varieties, focusing on the surjectivity of certain canonical maps.

Contribution

It explores various geometric conditions affecting Minkowski sums of convex lattice polytopes within the framework of toric geometry, highlighting open questions about surjectivity.

Findings

Identifies conditions where Minkowski sum equalities hold or fail.

Connects geometric properties of polytopes to algebraic properties of toric varieties.

Highlights open problems in the surjectivity of canonical maps for nonsingular toric varieties.

Abstract

This paper was submitted to the Oberwolfach Conference "Combinatorial Convexity and Algebraic Geometry", October 1997. Let $M={\mathbb Z}^r$. For convex lattice polytopes $P,P'$ in ${\mathbb R}^r$, when is $(M \cap P)+ (M \cap P') = M \cap (P + P')$? Without any additional condition, the equality obviously does not hold. When the pair $(M,P)$ corresponds to a complex projective toric variety $X$ and an ample divisor $D$ on $X$, it is reasonable to assume that $P'$ corresponds to an ample (or, more generally, a nef) divisor $D'$ on the same $X$. Then the question correspons to the surjectivity of the canonical map \[ H^0(X,{\mathcal O}_X(D))\otimes H^0(X,{\mathcal O}_X(D'))\to H^0(X,{\mathcal O}_X(D+D')).\] When $X$ is nonsingular, the map is hoped to be surjective, but this remains to be an open question after more than ten years. The paper explores various variations on the question in terms of toric geometry.