Problems on Minkowski sums of convex lattice polytopes

This paper was submitted to the Oberwolfach Conference “Combinatorial Convexity and Algebraic Geometry”, October 1997. Let $M={\mathbb{Z}}^r$. For convex lattice polytopes $P,P^{\mathrm{\prime }}$ in ${\mathbb R}^r$,whenis$(M \cap P)+ (M \cap P’) = M \cap (P + P’)$? 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 $P^{\mathrm{\prime }}$ corresponds to an ample (or, more generally, a nef) divisor $D^{\mathrm{\prime }}$ on the same $X$. Then the question correspons to the surjectivity of the canonical map $$H^0(X,{\mathcal{O}}_X(D))\otimes H^0(X,{\mathcal{O}}_X(D^{\mathrm{\prime }}))\to H^0(X,{\mathcal{O}}_X(D+D^{\mathrm{\prime }}))\mathrm{.}$$ 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.