Naive $\mathbb A^1$-homotopies on ruled surfaces

TL;DR

This paper explicitly classifies naive $A^1$-homotopy classes of morphisms into certain ruled surfaces and explores the differences between naive and genuine $A^1$-connected components, especially in non-minimal models.

Contribution

It provides an explicit description of $A^1$-chain homotopy classes for ruled surfaces and analyzes the discrepancy between naive and genuine $A^1$-connected components.

Findings

Naive and genuine $A^1$-connected components differ for non-minimal ruled surfaces.

Sections over schemes of dimension ≤ 1 agree for both types of components.

The Morel-Voevodsky singular construction is not $A^1$-local on certain ruled surfaces.

Abstract

We explicitly describe the $\mathbb A^1$-chain homotopy classes of morphisms from a smooth henselian local scheme into a smooth projective surface, which is birationally ruled over a curve of genus $> 0$. We consequently determine the sheaf of naive $\mathbb A^1$-connected components of such a surface and show that it does not agree with the sheaf of its genuine $\mathbb A^1$-connected components when the surface is not a minimal model. However, the sections of the sheaves of both naive and genuine $\mathbb A^1$-connected components over schemes of dimension $\leq 1$ agree. As a consequence, we show that the Morel-Voevodsky singular construction on a smooth projective surface, which is birationally ruled over a curve of genus $> 0$, is not $\mathbb A^1$-local if the surface is not a minimal model.