Complements of hypersurfaces in projective spaces

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

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

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

For $n=2$, the answer is positive if $d\le 7$ and there are counterexamples when $d=8$. In contrast, we provide counterexamples for all $n,d\ge 3$ with $(n,d)\ne (3,3)$. 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.

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

Pour $n=2$, la réponse est positive si $d\le 7$ et il y a des contre-exemples lorsque $d=8$. En revanche, nous fournissons des contre-exemples pour tous les entiers $n,d\ge 3$ avec $(n,d)\ne (3,3)$. 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.

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

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