Geometric models for algebraic suspensions

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.