Cet ouvrage fait partie de la bibliothèque YouScribe
Obtenez un accès à la bibliothèque pour le lire en ligne
En savoir plus

ADDITIVE GROUP ACTIONS ON DANIELEWSKI VARIETIES AND THE CANCELLATION PROBLEM

15 pages
Niveau: Supérieur, Doctorat, Bac+8
ADDITIVE GROUP ACTIONS ON DANIELEWSKI VARIETIES AND THE CANCELLATION PROBLEM ADRIEN DUBOULOZ Prepublication de l'Institut Fourier no 680 (2005) www-fourier.ujf-grenoble.fr/prepublications.html Abstract. The cancellation problem asks if two complex algebraic varieties X and Y of the same dimension such that X ? C and Y ? C are isomorphic are isomorphic. Iitaka and Fujita [15] established that the answer is positive for a large class of varieties of any dimension. In 1989, Danielewski [4] constructed a famous counter-example using smooth affine surfaces with additive group actions. His construction was further generalized by Fieseler [10] and Wilkens [22] to describe a larger class of affine surfaces. Here we construct higher dimensional analogues of these surfaces. We study algebraic actions of the additive group C+ on certain of these varieties, and we obtain counter-examples to the cancellation problem in every dimension n ≥ 2. Keywords: Danielewski varieties, Cancellation Problem, additive group actions, Makar- Limanov invariant. Resume. Le probleme dit de simplification demande si deux varietes algebriques complexes X et Y telles X ?C et Y ?C soient isomorphes sont isomorphes. Iitaka et Fujita ont montre a la fin des annees 70 que la reponse est affirmative pour une large classe de varietes. Les varietes affines-reglees ne font pas partie de cette classe, et, en 1989, Danielewski a construit un contre-exemple a partir de deux surfaces affines de ce type.

  • additive group

  • danielewski varieties

  • makar limanov invariant

  • bundle over

  • actions de groupes additifs

  • then x?1i


Voir plus Voir moins
ADDITIVE GROUP ACTIONS ON DANIELEWSKI VARIETIES AND THE CANCELLATION PROBLEM ADRIEN DUBOULOZ
PrepublicationdelInstitutFouriern o 680 (2005) www-fourier.ujf-grenoble.fr/prepublications.html Abstract. The cancellation problem asks if two complex algebraic varieties X and Y of the same dimension such that X  C and Y  C are isomorphic are isomorphic. Iitaka and Fujita [15] established that the answer is positive for a large class of varieties of any dimension. In 1989, Danielewski [4] constructed a famous counter-example using smooth ane surfaces with additive group actions. His construction was further generalized by Fieseler [10] and Wilkens [22] to describe a larger class of ane surfaces. Here we construct higher dimensional analogues of these surfaces. We study algebraic actions of the additive group C + on certain of these varieties, and we obtain counter-examples to the cancellation problem in every dimension n  2. Keywords : Danielewski varieties, Cancellation Problem, additive group actions, Makar-Limanov invariant.
Resume. Leproblemeditdesimpli cationdemandesideuxvarietesalgebriquescomplexes X et Y telles X  C et Y  C soientisomorphessontisomorphes.IitakaetFujitaontmontre ala ndesannees70quelareponseestarmativepourunelargeclassedevarietes.Les varietesanes-regleesnefontpaspartiedecetteclasse,et,en1989,Danielewskiaconstruit uncontre-exempleapartirdedeuxsurfacesanesdecetype.Danscetarticle,ongeneralise laconstructiondeDanielewskipourobtenirdesvarietesanesquisontlesespacestotauxde br es principaux sous le groupe additif, de base un schema non separe, en l’occurrence, un espaceanedontleshyperplansdecoordonnesonetemultiplies.Graˆceaunetechniquede deformationequivariantedeveloppeeparKalimanetMakar-Limanov,ondetermineensuite touteslesactionsdegroupesadditifssurcertainesdecesvarietes.Celaconduit nalementa desgeneralisationsnaturellesducontre-exampledeDanielewski,valablesentoutedimension n  2. Mots clefs :varietesdeDanielewski,ProblemedeSimpli cation,groupesadditifs,invariant de Makar-Limanov. Mathematics Subject Classi cation (2000) : 14R10,14R20.
1
Un pour Un
Permettre à tous d'accéder à la lecture
Pour chaque accès à la bibliothèque, YouScribe donne un accès à une personne dans le besoin