# A relative Hilbert–Mumford criterion

@article{Gulbrandsen2015ARH, title={A relative Hilbert–Mumford criterion}, author={Martin G. Gulbrandsen and Lars Halvard Halle and Klaus Hulek}, journal={Manuscripta Mathematica}, year={2015}, volume={148}, pages={283-301} }

We generalize the classical Hilbert–Mumford criteria for GIT (semi-)stability in terms of one parameter subgroups of a linearly reductive group G over a field k, to the relative situation of an equivariant, projective morphism $${X \rightarrow \,{\rm Spec}\,\, A}$$X→SpecA to a noetherian k-algebra A. We also extend the classical projectivity result for GIT quotients: the induced morphism $${X^{ss} /G \rightarrow \,{\rm Spec}\,\, A^G}$$Xss/G→SpecAG is projective. As an example of applications to… Expand

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