凸格多面体的Minkowski和问题
Problems on Minkowski sums of convex lattice polytopes
摘要：
本文提交给Oberwolfach会议“组合凸性和代数几何”，1997年10月。令$M={\mathbb Z}^r$。对于${\mathbb R}^r$中的凸格多面体$P,P'$，当是$(M \cap P)+ (M \cap P') = M \cap (P + P')$？如果没有任何附加条件，等式显然不成立。当对$(M,P)$对应于复射影复曲面簇$X$和$X$上的充分除数$D$时，可以合理地假设$P'$对应于同一$X$上的充分（或更一般地，nef）除数$D'$。当\[ H^0(X,{\mathcal O}_X(D))\otimes H^0(X,{\mathcal O}_X(D'))\to H^0(X,{\mathcal O}_X(D+D')).\]是非奇异的时，人们希望它是满射的，但十多年过去了，这仍然是一个悬而未决的问题。本文探讨了各种变化的问题方面的复曲面几何。
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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