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\(\mathbb{A}^1\)-curves on log smooth varieties. (English) Zbl 1430.14004
Dans cet article les auteurs continuent leur étude commencée dans [Q. Chen and Y. Zhu, Algebr. Geom. 1, No. 5, 558–572 (2014; Zbl 1322.14074)] en étudiant la \(\mathbb{A}^{1}\)-connexité du point de vu de la géométrie logarithmique.
Rappelons que les courbes \(\mathbb{A}^{1}\) sur un schéma \(U\) sont des morphismes propres et non constants \(f: \mathbb{A}^{1} \to U\). Se posent alors dans le cas des variétiés non propres les problèmes analogues à ceux de savoir si une variété est réglée, rationnellement connexe, etc. Le résultat principal du présent article (théorème 1.3) est le suivant. Si \(X\) est une log-variété lisse telle que son centre est propre, séparément rationnellement connexe et complètement lisse, alors le lieu ou la log-structure est triviale est séparément \(\mathbb{A}^{1}\)-connexe.
La preuve de ce résultat, d’un intérêt similaire à son énoncé, procède par la géométrie logarithmique. Plus précisément il résulte de l’étude des déformations des log-fonctions stables dégénérées.
Ce résultat donne de nombreux exemples de variétés \(\mathbb{A}^{1}\)-connexes dont nous donnons deux exemples et nous reportons aux théorèmes 1.5 et 1.8 pour des énoncés précis. Les complémentaires de diviseurs amples sont \(\mathbb{A}^{1}\)-connexes. Un groupe algébrique semi-simple \(G\) est séparément \(\mathbb{A}^{1}\)-connexe si la caractéristique du corps de base ne divise pas l’ordre du groupe fondamental de \(G\).
Un autre résultat intéressant est une caractérisation des log-variétés lisses, projectives dont la variété tangente est ample. Elles sont soit isomorphes à \((\mathbb{P}^{n},\emptyset)\) ou à \((\mathbb{P}^{n},\text{un hyperplan})\).
Pour conclure, cet article est dense mais très bien écrit. De plus, les notions clefs sont rappelées de manière succincte, ce rend cet article essentiellement autosuffisant.
MSC:
14A21 Logarithmic algebraic geometry, log schemes
14M22 Rationally connected varieties
14R10 Affine spaces (automorphisms, embeddings, exotic structures, cancellation problem)
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