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CANCELLED $\mathbb A^1$-connected components of ruled surfaces

Presented by: 
Anand Sawant
Thursday 26th March 2020 - 10:10 to 11:10
INI Seminar Room 1
A conjecture of Morel asserts that the sheaf of $\mathbb A^1$-connected components of a space is $\mathbb A^1$-invariant.  We will discuss how the sheaves of ``naive" as well as ``genuine" $\mathbb A^1$-connected components of a smooth projective birationally ruled surface can be determined using purely algebro-geometric methods.  We will discuss a proof of Morel's conjecture for a smooth projective surface birationally ruled over a curve of genus > 0 over an algebraically closed field of characteristic 0.  If time permits, we will indicate why the naive and genuine $\mathbb A^1$-connected components of such a birationally ruled surface do not coincide if the surface is not a minimal model and discuss some open questions and specultions regarding the situation in higher dimensions.  The talk is based on joint work with Chetan Balwe.
University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons