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摘要翻译：
这篇论文被提交给Oberwolfach会议“组合凸性和代数几何”，1997年10月。设$m={\mathbb Z}^r$。对于${\mathbb R}^R$中的凸点阵多边形$P,P'$,$(M\cap P)+(M\cap P')=M\cap(P+P')$是什么时候？如果没有任何附加条件，这种平等显然是不成立的。当对$(M，P)$对应于一个复杂的射影多环变量$x$和$x$上的一个充分的除数$d$时，可以合理地假定$P'$对应于同一个$x$上的一个充分的（或者更一般地，一个nef）除数$d'$。这个问题涉及到规范映射[H^0(X,{\mathcal O}_X(D))\times H^0(X,{\mathcal O}_X(D'))\\到H^0(X,{\mathcal O}_X(D+D'))\\]的满射性。当$X$为非奇异时，人们希望该映射是满射的，但这在十多年后仍是一个悬而未决的问题。本文从曲面几何的角度探讨了这个问题的各种变化。
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英文标题：
《Problems on Minkowski sums of convex lattice polytopes》
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作者：
Tadao Oda
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最新提交年份：
2008
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分类信息：
一级分类：Mathematics 数学
二级分类：Algebraic Geometry 代数几何
分类描述：Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology
代数簇，叠，束，格式，模空间，复几何，量子上同调
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一级分类：Mathematics 数学
二级分类：Combinatorics 组合学
分类描述：Discrete mathematics, graph theory, enumeration, combinatorial optimization, Ramsey theory, combinatorial game theory
离散数学，图论，计数，组合优化，拉姆齐理论，组合对策论
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英文摘要：
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.
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PDF链接：
https://arxiv.org/pdf/0812.1418
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