Gröbner Techniques for Low-Degree Hilbert Stability

Abstract

We give a method for verifying, by a symbolic calculation, the stability or semistability with respect to a linearization of fixed, possibly small, degree m, of the Hilbert point of a scheme having a suitably large automorphism group. We also implement our method and apply it to analyze the stability of bicanonical models of certain curves. Our examples are very special, but they arise naturally in the log minimal model program for . In some examples, this connection provides a check of our computations; in others, the computations confirm predictions about conjectural stages of the program.

2000 AMS Subject Classification:

• Primary 14L24
• 14H10; Secondary 14D22
• 13P10

Keywords:

• Hilbert stability
• state polytope

Acknowledgments

It is a great pleasure to thank Brendan Hassett, David Hyeon, and Yongnam Lee, whose work got us interested in these GIT questions, and Dave Bayer, Johan de Jong, Bill Graham, Anders Jensen, Julius Ross, Greg Smith, David Smyth, Mike Stillman, and Bernd Sturmfels for helpful discussions. The first author wishes to acknowledge support from a Fordham University Faculty Fellowship during the early stages of this project. The second thanks Sonja Mapes for her expert instruction on programming in Macaulay2, and Dan Grayson for suggesting many improvements to the StatePolytope package. We are grateful to the University of Georgia's Research Computing Cluster for allowing us to use their resources, and to the National Center for Supercomputing Applications for an allocation of computing time under grant TG-DMS090027.

Notes

¹This is the state polytope of [Bayer and Morrison 88, Section 2] defined entirely in the fixed degree m, as opposed to that of [Sturmfels 96, Theorem 2.5], which is the Minkowski sum of the former for all degrees up to m.

²His last appeared 59 years later!

³Both the source code files used and the resulting output can be found at http://www.math.uga.edu/~davids/gs/gs.html.

⁴See http://www.math.uga.edu/~davids/gs/wiman3/kkt.pdf .