# Stability and equivariant maps

@article{Reichstein1989StabilityAE, title={Stability and equivariant maps}, author={Zinovy Reichstein}, journal={Inventiones mathematicae}, year={1989}, volume={96}, pages={349-383} }

SummaryConsider a linearized action of a reductive algebraic group on a projective algebraic varietyX over an algebraically closed field. In this situation Mumford [1] defined the concept of stability for points ofX. Given an equivariant morphismY→X we introduce a suitable linearization of the action onY and relate stability inY to stability inX. In particular, we prove a relative Hilbert-Mumford theorem which says that stability inX andY can be tested simultaneously by 1-parameter subgroups…

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## References

SHOWING 1-4 OF 4 REFERENCES

Partial desingularisations of quotients of nonsingular varieties and their Betti numbers

- Mathematics
- 1985

When a reductive group G acts linearly on a nonsingular complex projective variety X one can define a projective "quotient" variety X//G using Mumford's geometric invariant theory. If the condition…

Orbits on Linear Algebraic Groups

- Mathematics
- 1971

Let G be a linear algebraic group and let p: G GL(V) be a rational representation of G. When G is linearly reductive, D. Mumford has shown that if a point x C V has 0 in the Zariski-closure cl (Gs x)…

Geometric Invariant Theory

- Mathematics
- 1965

“Geometric Invariant Theory” by Mumford/Fogarty (the first edition was published in 1965, a second, enlarged edition appeared in 1982) is the standard reference on applications of invariant theory to…