2020/02/19 by Chetan Balwe, Balwe, Chetan, Anand Sawant +1
Mathematics · #14F42 #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #Commutative Algebra (math.AC) #Dimension (graph theory) #FOS: Mathematics #Genus #Geometry #Homotopy #Homotopy and Cohomology in Algebraic Topology #Homotopy category #K-Theory and Homology (math.KT) #Mathematical analysis #Mathematics #Morphism #Pure mathematics #Ruled surface #Scheme (mathematics) #Sheaf #Surface (topology) #math.AC #math.AG #math.KT #msc:14F42
paper · pdf · doi:10.48550/arxiv.2002.08761
15 pages, final version before page proofs, accepted for publication in IMRN. arXiv admin note: text overlap with arXiv:1911.05549
openalex publication_date 2020/02/19 · arxiv created 2021/07/21 · arxiv updated 2021/07/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We explicitly describe the \mathbb A1-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 A1-connected components of such a surface and show that it does not agree with the sheaf of its genuine \mathbb A1-connected components when the surface is not a minimal model. However, the sections of the sheaves of both naive and genuine \mathbb A1-connected components over schemes of dimension ≤ 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 A1-local if the surface is not a minimal model.