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

Chetan Balwe; Anand Sawant

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arXiv · 2002.08761

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

Chetan Balwe·Anand Sawant

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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.

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BibTeX RIS

Chetan Balwe, Anand Sawant. 2020-02-19. Naive $\mathbb A^1$-homotopies on ruled surfaces. https://arxiv.org/abs/2002.08761

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