 A remark on relative geometric invariant theory for quasi‐projective varietiesAlexander H. W. Schmitt Mathematische Nachrichten, 2019, vol. 292, issue 2, 428-435 Abstract: Relative geometric invariant theory studies the behavior of semistable points under equivariant morphisms. More precisely, suppose G is a reductive linear algebraic group over an algebraically closed field k, X and Y are quasi‐projective varieties endowed with G‐actions, φ:X→Y is a G‐equivariant projective morphism, the G‐action on Y is linearized in the ample line bundle M, and the G‐action on X is linearized in the φ‐ample line bundle L. For any positive integer n, there is an induced linearization of the G‐action on X in the line bundle L⊗φ★(M⊗n). If Y is projective and n≫0, the set of points in X that are semistable with respect to this linearization is contained in the preimage under φ of the set of points in Y that are semistable with respect to the given linearization in M. The same statement is trivially also true, if Y is affine and M=OY. In this note, we show by means of an example that the statement does not hold for arbitrary quasi‐projective varieties Y. This shows that a claim by Hu of the contrary is not true. Relative geometric invariant theory plays a role in the construction and study of degenerations of moduli spaces. Date: 2019 Downloads: (external link) Related works: Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text Persistent link: https://EconPapers.repec.org/RePEc:bla:mathna:v:292:y:2019:i:2:p:428-435 Ordering information: This journal article can be ordered from Access Statistics for this article Mathematische Nachrichten is currently edited by Robert Denk More articles in Mathematische Nachrichten from Wiley Blackwell |
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