Consider a linearized action of a reductive algebraic group on a projective algebraic varietyX over an algebraically closed field. In this situation Mumford  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. It is hoped that this result will have applications to moduli problems.
In the caseY→X is a blowing up of a sheaf of ideals inX we give a simple explicit description of the properly stable and semi-stable loci inY. Our relative Hilbert-Mumford theorem is used here in an essential way.
We also consider the following resolution problem introduced by Kirwan . The equivariant mapY→X is called a stable resolution if it is an isomorphism over the properly stable locus inX and every point ofY is either unstable or properly stable. Kirwan  gave a canonical procedure for constructing a stable resolution over the complex numbers. We show that a stable resolution can be obtained by resolving the singularities of a naturally defined subvariety ofX. This gives an alternative (non-canonical) procedure for constructing stable resolutions.
KeywordsComplex Number Algebraic Group Stable Locus Explicit Description Resolution Problem
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