Abstract
The first part of this paper is a refinement of Winkelmann’s work on invariant rings and quotients of algebraic group actions on affine varieties, where we take a more geometric point of view. We show that the (algebraic) quotient \(X/\!/ G\) given by the possibly not finitely generated ring of invariants is “almost” an algebraic variety, and that the quotient morphism \(\pi :X \rightarrow X/\!/G\) has a number of nice properties. One of the main difficulties comes from the fact that the quotient morphism is not necessarily surjective. These general results are then refined for actions of the additive group \({{\mathbb {G}}_{a}}\), where we can say much more. We get a rather explicit description of the so-called plinth variety and of the separating variety, which measures how much orbits are separated by invariants. The most complete results are obtained for representations. We also give a complete and detailed analysis of Roberts’ famous example of a an action of \({{\mathbb {G}}_{a}}\) on 7-dimensional affine space with a non-finitely generated ring of invariants.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bourbaki, N.: Commutative algebra. Chapters 1–7, Elements of Mathematics (Berlin), Translated from the French, Reprint of the 1989 English translation. Springer, Berlin (1998)
Decker, W., Greuel, G.-M., Pfister, G., Schönemann, H.: Singular 3-1-6—a computer algebra system for polynomial computations. http://www.singular.uni-kl.de
Derksen, H., Kemper, G.: Computational Invariant Theory, Invariant Theory and Algebraic Transformation Groups, I, Encyclopaedia of Mathematical Sciences, vol. 130. Springer, Berlin (2002)
Derksen, H., Kemper, G.: Computing invariants of algebraic groups in arbitrary characteristic. Adv. Math. 217(5), 2089–2129 (2008)
Derksen, H., Kemper, G.: Finite separating sets and quasi-affine quotients. J. Pure Appl. Algebra 217(2), 247–253 (2013)
Dufresne, E., Elmer, J., Sezer, M.: Separating invariants for arbitrary linear actions of the additive group. Manuscr. Math. 143, 207–219 (2014)
Dufresne, E., Kohls, M.: The separating variety for the basic representations of the additive group. J. Algebra 377, 269–280 (2013)
Elmer, J., Kohls, M.: Separating invariants for the basic \({\mathbb{G}}_{a}\)-actions. Proc. Am. Math. Soc. 140(1), 135–146 (2012)
Hartshorne, R.: Algebraic Geometry, Graduate Texts in Mathematics, vol. 52. Springer, New York (1977)
Igusa, Jun-ichi: Geometry of absolutely admissible representations. In: Number Theory, Algebraic Geometry and Commutative Algebra, in Honor of Yasuo Akizuki, Tokyo, Kinokuniya, pp. 373–452 (1973)
Kemper, G.: Computing invariants of reductive groups in positive characteristic. Transform. Groups 8(2), 159–176 (2003)
Kraft, H.: Geometrische Methoden in der Invariantentheorie, Aspects of Mathematics. D. Friedr. Vieweg & Sohn, Braunschweig (1984)
Krull, W.: Jacobsonsche Ringe, Hilbertscher Nullstellensatz, Dimensionstheorie. Math. Z. 54, 354–387 (1951)
Kuroda, S.: A generalization of Roberts’ counterexample to the fourteenth problem of Hilbert. Tohoku Math. J. (2) 56(4), 501–522 (2004)
Luna, D.: Adhérences d’orbite et invariants. Invent. Math. 29(3), 231–238 (1975)
Matsumura, H.: Commutative Ring Theory, Cambridge Studies in Advanced Mathematics, vol. 8, 2nd edn. Translated from the Japanese by M. Reid Cambridge University Press, Cambridge (1989)
Mumford, D.: Abelian Varieties, Tata Institute of Fundamental Research Studies in Mathematics, vol. 5. Published for the Tata Institute of Fundamental Research, Bombay, 2008, With appendices by C. P. Ramanujam and Yuri Manin, Corrected reprint of the second edition (1974)
Mumford, D.: The Red Book of Varieties and Schemes, Expanded ed., Lecture Notes in Mathematics, vol. 1358. Includes the Michigan lectures (1974) on curves and their Jacobians, With contributions by Enrico Arbarello. Springer, Berlin (1999)
Mumford, D., Fogarty, J., Kirwan, F.: Geometric invariant theory, 3rd edn. Ergebnisse der Mathematik und ihrer Grenzgebiete (2) [Results in Mathematics and Related Areas (2)], vol. 34. Springer, Berlin (1994)
Nagata, M.: Lectures on the Fourteenth Problem of Hilbert. Tata Institute of Fundamental Research, Bombay (1965)
Robbiano, L., Sweedler, M.: Subalgebra Bases. In: Commutative Algebra (Salvador, 1988), Lecture Notes in Mathematics, vol. 1430, pp. 61–87. Springer, Berlin (1990)
Roberts, P.: An infinitely generated symbolic blow-up in a power series ring and a new counterexample to Hilbert’s fourteenth problem. J. Algebra 132(2), 461–473 (1990)
Schur, I.: Vorlesungen über Invariantentheorie, Bearbeitet und herausgegeben von Helmut Grunsky. Die Grundlehren der mathematischen Wissenschaften, Band, vol. 143. Springer, Berlin (1968)
Slodowy, P.: Der Scheibensatz für algebraische Transformationsgruppen, Algebraische Transformationsgruppen und Invariantentheorie, DMV Sem., vol. 13, pp. 89–113. Birkhäuser, Basel (1989)
Winkelmann, J.: Invariant rings and quasiaffine quotients. Math. Z. 244(1), 163–174 (2003)
Zariski, O., Samuel, P.: Commutative Algebra, vol. II. The University Series in Higher Mathematics, D. Van Nostrand Co., Inc., Princeton, N. J.-Toronto-London-New York (1960)
Author information
Authors and Affiliations
Corresponding author
Additional information
The authors were partially supported by the SNF (Schweizerischer Nationalfonds).
Rights and permissions
About this article
Cite this article
Dufresne, E., Kraft, H. Invariants and separating morphisms for algebraic group actions. Math. Z. 280, 231–255 (2015). https://doi.org/10.1007/s00209-015-1420-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00209-015-1420-0