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

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KAHW02 - Algebraic K-theory, motivic cohomology and motivic homotopy theory

A conjecture of Morel asserts that the sheaf of $\mathbb A1$-connected components of a space is $\mathbb A1$-invariant.  We will discuss how the sheaves of ``naive” as well as ``genuine” $\mathbb A1$-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 $\mathbb A1$-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.

This talk is part of the Isaac Newton Institute Seminar Series series.

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