A tropical approach to secant dimensions
A tropical approach to secant dimensions
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DOI：
10.1016/j.jpaa.2007.05.022
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发表时间：
2008-02-01
影响因子：
0.8
通讯作者：
Draisma, Jan
中科院分区：
文献类型：
--
作者：
Draisma, Jan
就某些组合多面体优化问题而言，热带几何在割线变化的维度上产生了很好的下界。这种方法对于像塞格雷-维罗内嵌件这样的环形品种尤其成功。特别地，它给出了Hirschowitz定理的一个引人注目的图形证明，即除了二次和四次投影平面外，所有投影平面的Veronese嵌入都是无缺陷的；事实上，在热带下界不能给出正确维度的地方，没有已知的塞格雷-维罗内塞嵌入。简短的自包含介绍割线品种和所需的热带几何包括在内。(c) 2007 Elsevier B.V.版权所有
Tropical geometry yields good lower bounds, in terms of certain combinatorial-polyhedral optimisation problems, on the dimensions of secant varieties. The approach is especially successful for toric varieties such as Segre-Veronese embeddings. In particular, it gives an attractive pictorial proof of the theorem of Hirschowitz that all Veronese embeddings of the projective plane except for the quadratic one and the quartic one are non-defective; and indeed, no Segre-Veronese embeddings are known where the tropical lower bound does not give the correct dimension. Short self-contained introductions to secant varieties and the required tropical geometry are included. (c) 2007 Elsevier B.V. All rights reserved.