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Register a new user$mathbb{A}^1$-connected components of classifying spaces and purity for torsors
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In this paper, we study the Nisnevich sheafification $mathcal{H}^1_{acute{e}t}(G)$ of the presheaf associating to a smooth scheme the set of isomorphism classes of $G$-torsors, for a reductive group $G$. We show that if $G$-torsors on affine lines are extended, then $mathcal{H}^1_{acute{e}t}(G)$ is homotopy invariant and show that the sheaf is unramified if and only if Nisnevich-local purity holds for $G$-torsors. We also identify the sheaf $mathcal{H}^1_{acute{e}t}(G)$ with the sheaf of $mathbb{A}^1$-connected components of the classifying space ${rm B}_{acute{e}t}G$. This establishes the homotopy invariance of the sheaves of components as conjectured by Morel. It moreover provides a computation of the sheaf of $mathbb{A}^1$-connected components in terms of unramified $G$-torsors over function fields whenever Nisnevich-local purity holds for $G$-torsors.
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In this essay we study various notions of projective space (and other schemes) over $mathbb{F}_{1^ell}$, with $mathbb{F}_1$ denoting the field with one element. Our leading motivation is the Hiden Points Principle, which shows a huge deviation between the set of rational points as closed points defined over $mathbb{F}_{1^ell}$, and the set of rational points defined as morphisms $texttt{Spec}(mathbb{F}_{1^ell}) mapsto mathcal{X}$. We also introduce, in the same vein as Kurokawa [13], schemes of $mathbb{F}_{1^ell}$-type, and consider their zeta functions.
We give a general structure theorem for affine A 1-fibrations on smooth quasi-projective surfaces. As an application, we show that every smooth A 1-fibered affine surface non-isomorphic to the total space of a line bundle over a smooth affine curve fails the Zariski Cancellation Problem. The present note is an expanded version of a talk given at the Kinosaki Algebraic Geometry Symposium in October 2019.
Let S be a smooth del Pezzo surface that is defined over a field K and splits over a Galois extension L. Let G be either the split reductive group given by the root system of $S_L$ in Pic $S_L$, or a form of it containing the Neron-Severi torus. Let $mathcal{G}$ be the G-torsor over $S_L$ obtained by extension of structure group from a universal torsor $mathcal{T}$ over $S_L$. We prove that $mathcal{G}$ does not descend to S unless $mathcal{T}$ does. This is in contrast to a result of Friedman and Morgan that such $mathcal{G}$ always descend to singular del Pezzo surfaces over $mathbb{C}$ from their desingularizations.
We give a criterion for a flat fibration with affine plane fibers over a smooth scheme defined over a field of characteristic zero to be a Zariski locally trivial $mathbb{A}^2$-bundle. An application is a positive answer to a version of the Dolgachev-Weisfeiler Conjecture for such fibrations: a flat fibration $mathbb{A}^m$ $rightarrow$ $mathbb{A}^n$ with all fibers isomorphic to $mathbb{A}^2$ is the trivial $mathbb{A}^2$-bundle.
This paper considers the moduli spaces (stacks) of parabolic bundles (parabolic logarithmic flat bundles with given spectrum, parabolic regular Higgs bundles) with rank 2 and degree 1 over $mathbb{P}^1$ with five marked points. The stratification structures on these moduli spaces (stacks) are investigated. We confirm Simpsons foliation conjecture of moduli space of parabolic logarithmic flat bundles for our case.
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