Geometric models for algebraic suspensions

TL;DR

This paper explores geometric models for algebraic suspensions in motivic homotopy theory, demonstrating how affine deformation spaces can represent ${f P}^1$ suspensions and providing examples of highly connected smooth schemes.

Contribution

It introduces a geometric approach using affine deformation spaces to model algebraic suspensions and constructs new examples of ${f A}^1$-contractible and connected smooth schemes.

Findings

Affine deformation spaces model ${f P}^1$ suspensions.

Constructs ${f A}^1$-$(n-1)$-connected smooth affine $2n$-folds.

Provides examples of strictly quasi-affine ${f A}^1$-contractible schemes.

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.