Objectif In contrast with the 2-dimensional case, the study of automorphisms of C³ is still in its early beginnings. The most striking recent result is probably the proof by Shestakov and Umirbaev that a large class of automorphisms of C³ is non tame. Unfortunately their proof is based on tricky calculations and lacks conceptuality. On the other hand, Mori theory has emerged since the 1980s as the right framework to study birational maps in higher dimension, of which automorphisms of C³ are special cases. However up to now the known factorization algorithm ("Sarkisov program") has been essentially applied to varieties with a small group of birational selfmaps (typically a smooth 3-dimensional quartic, where one thus obtains an irrationality criterion). Very recently, Hacon and McKernan had made an announcement that the Sarkisov program in any dimension follows as a corollary from their major breakthrough; however their result is still very theoretical. We believe, and this is the first main objective of our proposal, that in the case of polynomial automorphisms one can modify the available proofs to obtain a canonical factorization by mean of steps in the logarithmic Minimal Model Program, and that this factorization could be computable in practice. Based partially on the elementary case of surfaces, and partially on explicit computations for automorphisms of low degree, our conviction is that the right point of view is to look for links not between Mori fiber spaces but between compactifications of C³ with a minimal number of irreducible components on the boundary. Among expected consequences we should mention a birational proof of the relations in the tame group in dimension 3 (which available proof depends on the "black box" of Shestakov and Umirbaev), and obtain some natural candidates to generate the automorphism group of C³. Champ scientifique engineering and technologymaterials engineeringfibersnatural sciencesmathematicspure mathematicsgeometry Mots‑clés Algebraic geometry Birational geometry Generators and relations Group theory Mori theory Polynomial automorphisms Programme(s) FP7-PEOPLE - Specific programme "People" implementing the Seventh Framework Programme of the European Community for research, technological development and demonstration activities (2007 to 2013) Thème(s) FP7-PEOPLE-IEF-2008 - Marie Curie Action: "Intra-European Fellowships for Career Development" Appel à propositions FP7-PEOPLE-IEF-2008 Voir d’autres projets de cet appel Régime de financement MC-IEF - Intra-European Fellowships (IEF) Coordinateur THE UNIVERSITY OF WARWICK Contribution de l’UE € 237 684,37 Adresse Kirby Corner Road - University House CV4 8UW Coventry Royaume-Uni Voir sur la carte Région West Midlands (England) West Midlands Coventry Type d’activité Higher or Secondary Education Establishments Contact administratif Peter Hedges (Dr.) Liens Contacter l’organisation Opens in new window Site web Opens in new window Coût total Aucune donnée
