SL 2 -equivariant polynomial automorphisms of the binary forms

Alexandre Kurth

Annales de l'institut Fourier (1997)

  • Volume: 47, Issue: 2, page 585-597
  • ISSN: 0373-0956

Abstract

top
We consider the space of binary forms of degree n ≥ 1 denoted by R n : = ℂ [ x , y ] n . We will show that every polynomial automorphism of R n which commutes with the linear SL 2 ( ℂ ) -action and which maps the variety of forms with pairwise distinct zeroes into itself, is a multiple of the identity on R n .

How to cite

top

Kurth, Alexandre. "${\rm SL}_2$-equivariant polynomial automorphisms of the binary forms." Annales de l'institut Fourier 47.2 (1997): 585-597. <http://eudml.org/doc/75238>.

@article{Kurth1997,
abstract = {We consider the space of binary forms of degree $n\ge 1$ denoted by $R_n :=\{\Bbb C\}[x,y]_n$. We will show that every polynomial automorphism of $R_n$ which commutes with the linear $\{\rm SL\}_2 (\{\Bbb C\})$-action and which maps the variety of forms with pairwise distinct zeroes into itself, is a multiple of the identity on $R_n$.},
author = {Kurth, Alexandre},
journal = {Annales de l'institut Fourier},
keywords = {algebraic transformation groups; equivariant automorphism; binary forms; line bundle; lifting automorphisms},
language = {eng},
number = {2},
pages = {585-597},
publisher = {Association des Annales de l'Institut Fourier},
title = {$\{\rm SL\}_2$-equivariant polynomial automorphisms of the binary forms},
url = {http://eudml.org/doc/75238},
volume = {47},
year = {1997},
}

TY - JOUR
AU - Kurth, Alexandre
TI - ${\rm SL}_2$-equivariant polynomial automorphisms of the binary forms
JO - Annales de l'institut Fourier
PY - 1997
PB - Association des Annales de l'Institut Fourier
VL - 47
IS - 2
SP - 585
EP - 597
AB - We consider the space of binary forms of degree $n\ge 1$ denoted by $R_n :={\Bbb C}[x,y]_n$. We will show that every polynomial automorphism of $R_n$ which commutes with the linear ${\rm SL}_2 ({\Bbb C})$-action and which maps the variety of forms with pairwise distinct zeroes into itself, is a multiple of the identity on $R_n$.
LA - eng
KW - algebraic transformation groups; equivariant automorphism; binary forms; line bundle; lifting automorphisms
UR - http://eudml.org/doc/75238
ER -

References

top
  1. [1] E. ARTIN, Braids and Permutations, Ann. Math., 48 (1947), 643-649. Zbl0030.17802MR9,6c
  2. [2] R. HARTSHORNE, Algebraic Geometry, GTM, 52, Springer-Verlag, Berlin-New York (1977). Zbl0367.14001MR57 #3116
  3. [3] F. KNOP, H. KRAFT, D. LUNA, T. VUST, Local Properties of Algebraic Group Actions, In : H. Kraft, P. Slodowy, T.A. Springer : Algebraische Transformations-gruppen und Invariantentheorie, Algebraic Transformation Groups and Invariant Theory, DMV Seminar Band, 13, Birkhäuser (1989), 63-75. Zbl0722.14032MR1044585
  4. [4] F. KNOP, H. KRAFT, T. VUST, The Picard Group of a G-Variety, In : H. Kraft, P. Slodowy, T.A. Springer : Algebraische Transformationsgruppen und Invarianten-theorie, Algebraic Transformation Groups and Invariant Theory, DMV Seminar Band, 13, Birkhäuser (1989), 77-87. Zbl0705.14005MR1044586
  5. [5] H. KRAFT, Geometrische Methoden in der Invariantentheorie, Aspekte der Mathematik, D1, Vieweg (1985). Zbl0669.14003
  6. [6] H. KRAFT, Klassische Invariantentheorie : Eine Einführung, In : H. Kraft, P. Slodowy, T.A. Springer : Algebraische Transformationsgruppen und Invarianten-theorie, Algebraic Transformation Groups and Invariant Theory, DMV Seminar Band, 13, Birkhäuser (1989), 41-62. Zbl0726.13003MR1044584
  7. [7] H. KRAFT, Algebraic Automorphisms of Affine Space, In : “Topological Methods in Algebraic Transformation Groups” ; Progress in Mathematics, vol. 80, Birkhäuser Verlag Boston Basel Berlin (1989), 81-105. Zbl0719.14030MR91g:14044
  8. [8] H. KRAFT, G.W. SCHWARZ, Reductive Group Actions with one-dimensional Quotient, Publ. Math. IHES, 76 (1992). Zbl0783.14026MR94e:14065
  9. [9] A. KURTH, Equivariant Polynomial Automorphisms, Ph.D. Thesis Basel (1996). 
  10. [10] V. LIN, Around the 13th Hilbert Problem for Algebraic Functions, Israel Mathematical Conference Proceedings, vol. 9 (1996), 307-327. Zbl0846.32014MR96h:32027
  11. [11] D. LUNA, Slices étales, Bull. Soc. Math. France, Mémoire, 33 (1973), 81-105. Zbl0286.14014MR49 #7269
  12. [12] J.-P. SERRE, Cohomologie Galoisienne, Lecture Notes in Math., 5, Springer-Verlag, Berlin New York (1964). Zbl0128.26303
  13. [13] J.-P. SERRE, Local Fields, GTM, 67, Springer-Verlag, Berlin New York (1979). Zbl0423.12016
  14. [14] E.H. SPANIER, Algebraic Topology, Springer-Verlag, Berlin New York (1966). Zbl0145.43303MR35 #1007
  15. [15] H. WEYL, The Classical Groups, Their Invariants and Representations, 2nd., ed., Princeton Univ. Press, Princeton, N.J., 1946. Zbl1024.20502

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.