A STATE POLYTOPE DECOMPOSITION FORMULA

作者：Hyeon Donghoon^*; Kim Jaekwang

来源：Proceedings of the Edinburgh Mathematical Society, 2016, 59(3): 759-776.

DOI：10.1017/S0013091515000401

• geometric invariant theory
• state polytope
• initial ideal
• Grobner basis

We give a decomposition formula for computing the state polytope of a reducible variety in terms of the state polytopes of its components: if a polarized projective variety X is a chain of subvarieties X-i satisfying some further conditions, then the state polytope of X is the Minkowski sum of the state polytopes of X-i translated by a vector tau, which can be readily computed from the ideal of X-i. The decomposition is in the strongest sense in that the vertices of the state polytope of X are precisely the sum of vertices of the state polytopes of Xi translated by t. We also give a similar decomposition formula for the Hilbert-Mumford index of the Hilbert points of X. We give a few examples of the state polytope and the Hilbert-Mumford index computation of reducible curves, which are interesting in the context of the log minimal model program for the moduli space of stable curves.

• 出版日期2016-8