A correspondence of good G-sets under partial geometric quotients

  • Johannes Schmitt
Original Paper


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.


Geometric invariant theory Good quotients Toric varieties Variation of GIT 

Mathematics Subject Classification

14L24 14L30 



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.


  1. Arzhantsev, I.V., Hausen, J.: Geometric invariant theory via Cox rings. J. Pure Appl. Algebra 213(1), 154–172 (2009). doi: 10.1016/j.jpaa.2008.06.005 (ISSN 0022-4049)
  2. Berchtold, F., Hausen, J.: GIT equivalence beyond the ample cone. Mich. Math. J. 54(3), 483–515 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  3. Białynicki-Birula, A.: Quotients by actions of groups. In: Algebraic Quotients. Torus Actions and Cohomology. The Adjoint Representation and the Adjoint Action, volume 131 of Encyclopaedia Math. Sci. Springer, Berlin, pp. 1–82 (2002)Google Scholar
  4. Białynicki-Birula, A., Święcicka, J.: On exotic orbit spaces of tori acting on projective varieties. In: Group Actions and Invariant Theory (Montreal, PQ, 1988), volume 10 of CMS Conf. Proc. American Mathematical Society, Providence, pp. 25–30 (1989)Google Scholar
  5. Cox, D. A., Little, J. B., Schenck, H. K.: Toric Varieties, volume 124 of Graduate Studies in Mathematics. American Mathematical Society, Providence (2011) (ISBN 978-0-8218-4819-7)Google Scholar
  6. Dolgachev, I.V., Hu, Y.: Variation of geometric invariant theory quotients. Inst. Hautes Études Sci. Publ. Math. 87, 5–56 (1998) (With an appendix by Nicolas Ressayre]Google Scholar
  7. Gelfand, I.M., MacPherson, R.D.: Geometry in Grassmannians and a generalization of the dilogarithm. Adv. Math. 44(3), 279–312 (1982). doi: 10.1016/0001-8708(82)90040-8 (ISSN 0001-8708)
  8. Grothendieck, A.: Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. III. Inst. Hautes Études Sci. Publ. Math. 28, 255 (1966)Google Scholar
  9. Hassett, B.: Moduli spaces of weighted pointed stable curves. Adv. Math. 173(2), 316–352 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  10. Hausen, J.: Geometric invariant theory based on Weil divisors. Compos. Math. 140(6), 1518–1536 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  11. Herzog, J., Hibi, T.: Monomial Ideals, volume 260 of Graduate Texts in Mathematics. Springer, London (2011) (ISBN 978-0-85729-105-9]Google Scholar
  12. Hu, Y., Keel, S.: Mori dream spaces and GIT. Mich. Math. J. 48, 331–348 (2000). doi: 10.1307/mmj/1030132722 (ISSN 0026-2285; Dedicated to William Fulton on the occasion of his 60th birthday)
  13. Kapranov, M.M.: Chow quotients of Grassmannians. I. In: I. M. Gelfand Seminar, volume 16 of Adv. Soviet Math. American Mathematical Society, Providence, pp. 29–110 (1993)Google Scholar
  14. Keicher, S.: Computing the GIT-fan. Int. J. Algebra Comput. 22(7), 1250064 (2012). doi: 10.1142/S0218196712500646 (ISSN 0218-1967)
  15. Kirwan, F.C., Lee, R., Weintraub, S.H.: Quotients of the complex ball by discrete groups. Pac. J. Math. 130(1), 115–141 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  16. Mumford, D., Fogarty, J., Kirwan, F.: Geometric Invariant Theory, volume 34 of Ergebnisse der Mathematik und ihrer Grenzgebiete (2), 3rd edn. Springer, Berlin (1994) (ISBN 3-540-56963-4)Google Scholar
  17. Polito, M.: \(\text{ SL }(2,\mathbf{C})\)-quotients de \((\mathbf{P}^1)^n\). C. R. Acad. Sci. Paris Sér. I Math. 321(12), 1577–1582 (1995)MathSciNetzbMATHGoogle Scholar
  18. Popov, V.L.: Picard groups of homogeneous spaces of linear algebraic groups and one-dimensional homogeneous vector fiberings. Izv. Akad. Nauk SSSR Ser. Mat. 38, 294–322 (1974)MathSciNetGoogle Scholar
  19. Ramanathan, A.: Moduli for principal bundles over algebraic curves. II. Proc. Indian Acad. Sci. Math. Sci. 106(4), 421–449 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  20. Ressayre, N.: Geometric invariant theory and the generalized eigenvalue problem. Invent. Math. 180(2), 389–441 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  21. Romagny, M.: Group actions on stacks and applications. Mich. Math. J. 53(1), 209–236 (2005)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© The Managing Editors 2017

Authors and Affiliations

  1. 1.Departement MathematikETH ZürichZurichSwitzerland

Personalised recommendations