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A correspondence of good G-sets under partial geometric quotients

  • Johannes Schmitt
Original Paper
  • 73 Downloads

Abstract

For a complex variety \(\widehat{X}\) with an action of a reductive group \({\widehat{G}}\) and a geometric quotient \(\pi : \widehat{X} \rightarrow X\) by a closed normal subgroup \(H \subset {\widehat{G}}\), we show that open sets of X admitting good quotients by \(G={\widehat{G}} / H\) correspond bijectively to open sets in \(\widehat{X}\) with good \({\widehat{G}}\)-quotients. We use this to compute GIT-chambers and their associated quotients for the diagonal action of \(\text {PGL}_2\) on \((\mathbb {P}^1)^n\) in certain subcones of the \(\text {PGL}_2\)-effective cone via a torus action on affine space. This allows us to represent these quotients as toric varieties with fans determined by convex geometry.

Keywords

Geometric invariant theory Good quotients Toric varieties Variation of GIT 

Mathematics Subject Classification

14L24 14L30 

Notes

Acknowledgements

I want to thank Gergely Bérczi, Brent Doran and Frances Kirwan for their advice during the preparation of this paper. I am also grateful to the anonymous referee for pointing out several important references and a connection to the Gelfand–MacPherson correspondence. I am supported by the Grant SNF-200020162928.

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Authors and Affiliations

  1. 1.Departement MathematikETH ZürichZurichSwitzerland

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