Naive ${\mathbb{A}}^1$-homotopies on 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 $\le 1$ 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.

15 pages, final version before page proofs, accepted for publication in IMRN. arXiv admin note: text overlap with arXiv:1911.05549

Cited by in corpus (1)

• ${\mathbb{A}}^1$-connected components of blow-up of threefolds fibered over a surface