Cocharacter-Closure and the Rational Hilbert-Mumford Theorem

View/ Open
Date
2014-12-20MFO Scientific Program
Research in Pairs 2012Series
Oberwolfach Preprints;2014,16Author
Bate, Michael
Herpel, Sebastian
Martin, Benjamin
Röhrle, Gerhard
Metadata
Show full item recordOWP-2014-16
Abstract
For a field $k$, let $G$ be a reductive $k$-group and $V$ an affine $k$-variety on which $G$ acts. Using the notion of cocharacter-closed $G(k)$-orbits in $V$ , we prove a rational version of the celebrated Hilbert-Mumford Theorem from geometric invariant theory. We initiate a study of applications stemming from this rationality tool. A number of examples are discussed to illustrate the concept of cocharacter-closure and to highlight how it differs from the usual Zariski-closure.