Geometric Models for Algebraic Suspensions

Abstract We analyze the question of which motivic homotopy types admit smooth schemes as representatives. We show that given a pointed smooth affine scheme $X$ and an embedding into affine space, the affine deformation space of the embedding gives a model for the ${\mathbb P}^{1}$ suspension of $X$;...

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Bibliographic Details
Published inInternational mathematics research notices Vol. 2023; no. 20; pp. 17788 - 17821
Main Authors Asok, A, Dubouloz, A, Østvær, P A
Format Journal Article
LanguageEnglish
Published Oxford University Press 15.10.2023
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Summary:Abstract We analyze the question of which motivic homotopy types admit smooth schemes as representatives. We show that given a pointed smooth affine scheme $X$ and an embedding into affine space, the affine deformation space of the embedding gives a model for the ${\mathbb P}^{1}$ suspension of $X$; we also analyze a host of variations on this observation. Our approach yields many examples of ${\mathbb A}^{1}$-$(n-1)$-connected smooth affine $2n$-folds and strictly quasi-affine ${\mathbb A}^{1}$-contractible smooth schemes.
ISSN:1073-7928
1687-0247
DOI:10.1093/imrn/rnad094