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 in:International mathematics research notices 2023-10, Vol.2023 (20), p.17788-17821
Main Authors: Asok, A, Dubouloz, A, Østvær, P A
Format: Article
Language:English
Subjects:
Algebraic Geometry
Mathematics
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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