𝔸1–connected components of ruled surfaces

Balwe, Chetan; Sawant, Anand

doi:10.2140/gt.2022.26.321

Download this article
	For screen
	For printing

Recent Issues

The Journal

Submission Guidelines

Submission Page

Policies for Authors

Ethics Statement

ISSN 1364-0380 (online)

ISSN 1465-3060 (print)

Author Index

To Appear

Other MSP Journals

$\mathbb A^{1}$ –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 $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mrow><mi mathvariant="double-struck">𝔸</mi></mrow><mrow><mspace width="-0.17em" class="negthinspace"/><mn>1</mn></mrow></msup></math>$ –connected components of a space is $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mrow><mi mathvariant="double-struck">𝔸</mi></mrow><mrow><mspace width="-0.17em" class="negthinspace"/><mn>1</mn></mrow></msup></math>$ –invariant. Using purely algebrogeometric methods, we determine the sheaf of $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mrow><mi mathvariant="double-struck">𝔸</mi></mrow><mrow><mspace width="-0.17em" class="negthinspace"/><mn>1</mn></mrow></msup></math>$ –connected components of a smooth projective surface, which is birationally ruled over a curve of genus $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"> <mo class="MathClass-rel">&gt;</mo> <mn>0</mn></math>$ . As a consequence, we show that Morel’s conjecture holds for all smooth projective surfaces over an algebraically closed field of characteristic $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mn>0</mn></math>$ .

Keywords

$\mathbb A^1$ –connected components, ghost homotopies, ruled surfaces

Mathematical Subject Classification

Primary: 14F42, 55Q05

References

Publication

Received: 20 July 2020

Revised: 14 January 2021

Accepted: 17 February 2021

Published: 5 April 2022

Proposed: Mark Gross

Seconded: Haynes R Miller, Mark Behrens

Authors

Chetan Balwe
Department of Mathematical Sciences Indian Institute of Science Education and Research Mohali India

Anand Sawant
School of Mathematics Tata Institute of Fundamental Research Mumbai India

Volume 26, issue 1 (2022)