Hostname: page-component-586b7cd67f-t7fkt Total loading time: 0 Render date: 2024-11-22T21:21:52.764Z Has data issue: false hasContentIssue false
  • English
  • Français

A State Polytope Decomposition Formula

Part of: Algebraic groups

Published online by Cambridge University Press:  14 December 2015

Donghoon Hyeon  and
Jaekwang Kim
Donghoon Hyeon
Affiliation:
Department of Mathematical Sciences, Seoul National University, Seoul 151-747, Republic of Korea ([email protected]) Center for Geometry and Its Applications, POSTECH, Pohang, Gyungbuk 790-784, Republic of Korea
Jaekwang Kim
Affiliation:
Department of Mathematics, POSTECH, Pohang, Gyungbuk 790-784, Republic of Korea ([email protected])
Get access Rights & Permissions [Opens in a new window]

Abstract

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 Xi satisfying some further conditions, then the state polytope of X is the Minkowski sum of the state polytopes of Xi translated by a vector τ, which can be readily computed from the ideal of Xi . 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 τ. 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.

Keywords

geometric invariant theorystate polytopeinitial idealGrübner basis

MSC classification

Secondary: 14L24: Geometric invariant theory
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 59 , Issue 3 , August 2016 , pp. 759 - 776
Copyright
Copyright © Edinburgh Mathematical Society 2015 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1. Alper, J., Fedorchuk, M. and Smyth, D. I., Singularities with Gm -action and the log minimal model program for , J. Reine Angew. Math. DOI:10.1515/crelle-2014-0063.Google Scholar
2. Bayer, D. and Morrison, I., Standard bases and geometric invariant theory, I, Initial ideals and state polytopes, J. Symb. Computat. 6(2–3) (1988), 209–217.CrossRefGoogle Scholar
3. Grayson, D. R. and Stillman, M. E., Macaulay 2, a software system for research in algebraic geometry (available at http://www.math.uiuc.edu/Macaulay2/).Google Scholar
4. Hassett, B. and Hyeon, D., Log minimal model program for the moduli space of curves: the first flip, Annals Math. (2) 177(3) (2013), 911–968.CrossRefGoogle Scholar
5. Hassett, B., Hyeon, D. and Lee, Y., Stability computation via Gröbner basis, J. Korean Math. Soc. 47(1) (2010), 41–62.Google Scholar
6. Hyeon, D. and Morrison, I., Stability of tails and 4-canonical models, Math. Res. Lett. 17(4) (2010), 721–729.CrossRefGoogle Scholar
7. Kempf, G. R., Instability in invariant theory, Annals Math. (2) 108(2) (1978), 299–316.CrossRefGoogle Scholar
8. Morrison, I. and Swinarski, D., Gröbner techniques for low-degree Hilbert stability, Exp. Math. 20(1) (2011), 34–56.CrossRefGoogle Scholar
9. Mumford, D., Geometric invariant theory, 1st edn, Ergebnisse der Mathematik und ihrer Grenzgebiete (Springer, 1965).CrossRefGoogle Scholar
10. Schubert, D., A new compactification of the moduli space of curves, Compositio Math. 78(3) (1991), 297–313.Google Scholar
11. Sturmfels, B., Gröbner bases and convex polytopes, University Lecture Series, Volume 8 (American Mathematical Society, Providence, RI, 1996).Google Scholar