Volume 26, issue 1 (2022)

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$\mathbb A^{1}$–connected components of ruled surfaces

Chetan Balwe and Anand Sawant

Geometry & Topology 26 (2022) 321–376
DOI: 10.2140/gt.2022.26.321
Abstract

A conjecture of Morel asserts that the sheaf of ${\mathbb{𝔸}}^{\phantom{\rule{-0.17em}{0ex}}1}$–connected components of a space is ${\mathbb{𝔸}}^{\phantom{\rule{-0.17em}{0ex}}1}$–invariant. Using purely algebrogeometric methods, we determine the sheaf of ${\mathbb{𝔸}}^{\phantom{\rule{-0.17em}{0ex}}1}$–connected components of a smooth projective surface, which is birationally ruled over a curve of genus $>0$. As a consequence, we show that Morel’s conjecture holds for all smooth projective surfaces over an algebraically closed field of characteristic $0$.

Keywords
$\mathbb A^1$–connected components, ghost homotopies, ruled surfaces
Mathematical Subject Classification
Primary: 14F42, 55Q05