Complements of hypersurfaces in projective spaces

Jérémy Blanc; Pierre-Marie Poloni; Immanuel Van Santen

doi:10.5802/jep.264

Complements of hypersurfaces in projective spaces
[Complémentaires d’hypersurfaces dans les espaces projectifs]

Jérémy Blanc ¹ ; Pierre-Marie Poloni ¹ ; Immanuel Van Santen ¹

¹ Universität Basel, Departement Mathematik und Informatik, Spiegelgasse 1, CH-4051 Basel, Switzerland

Journal de l’École polytechnique — Mathématiques, Tome 11 (2024), pp. 733-768.

Résumé
Abstract

Nous étudions le problème du complémentaire dans les espaces projectifs $ℙ^{n}$ sur tout corps algébriquement clos : Si $H, H^{'} \subseteq ℙ^{n}$ sont des hypersurfaces irréductibles de degré $d$ telles que les complémentaires $ℙ^{n} ∖ H$ , $ℙ^{n} ∖ H^{'}$ sont isomorphes, les hypersurfaces $H$ , $H^{'}$ sont-elles isomorphes ?

Pour $n = 2$ , la réponse est positive si $d \leq 7$ et il y a des contre-exemples lorsque $d = 8$ . En revanche, nous fournissons des contre-exemples pour tous les entiers $n, d \geq 3$ avec $(n, d) \neq (3, 3)$ . De plus, nous montrons que le problème du complémentaire a une réponse affirmative pour $d = 2$ et donnons des résultats partiels dans le cas où $(n, d) = (3, 3)$ . Au cours de l’exposition, nous prouvons que les surfaces projectives normales rationnelles admettant une désingularisation par des arbres de courbes rationnelles lisses sont isomorphes par morceaux si et seulement si elles coïncident dans l’anneau de Grothendieck, répondant ainsi positivement à une question posée par Larsen et Lunts pour de telles surfaces.

We study the complement problem in projective spaces $ℙ^{n}$ over any algebraically closed field: If $H, H^{'} \subseteq ℙ^{n}$ are irreducible hypersurfaces of degree $d$ such that the complements $ℙ^{n} ∖ H$ , $ℙ^{n} ∖ H^{'}$ are isomorphic, are the hypersurfaces $H$ , $H^{'}$ isomorphic?

For $n = 2$ , the answer is positive if $d \leq 7$ and there are counterexamples when $d = 8$ . In contrast, we provide counterexamples for all $n, d \geq 3$ with $(n, d) \neq (3, 3)$ . Moreover, we show that the complement problem has an affirmative answer for $d = 2$ and give partial results in case $(n, d) = (3, 3)$ . In the course of the exposition, we prove that rational normal projective surfaces admitting a desingularization by trees of smooth rational curves are piecewise isomorphic if and only if they coincide in the Grothendieck ring, answering affirmatively a question posed by Larsen and Lunts for such surfaces.

Reçu le : 2023-07-28
Accepté le : 2024-06-17
Publié le : 2024-07-09

DOI : 10.5802/jep.264

Classification : 14J70, 14R05, 14E07, 19A99
Keywords: Complement problem, cylinder over Danielewski surfaces, piecewise isomorphisms
Mot clés : Problème du complémentaire, cylindre sur les surfaces de Danielewski, isomorphismes par morceaux

Affiliations des auteurs :

Jérémy Blanc ¹ ; Pierre-Marie Poloni ¹ ; Immanuel Van Santen ¹

¹ Universität Basel, Departement Mathematik und Informatik, Spiegelgasse 1, CH-4051 Basel, Switzerland

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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Jérémy Blanc; Pierre-Marie Poloni; Immanuel Van Santen. Complements of hypersurfaces in projective spaces. Journal de l’École polytechnique — Mathématiques, Tome 11 (2024), pp. 733-768. doi : 10.5802/jep.264. https://jep.centre-mersenne.org/articles/10.5802/jep.264/

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