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Register a new user$mathbb{A}^1$-cylinders over smooth $mathbb{A}^1$-fibered surfaces
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Authors
Adrien Dubouloz
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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.
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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.
An algebraic variety is called $mathbb{A}^{1}$-cylindrical if it contains an $mathbb{A}^{1}$-cylinder, i.e. a Zariski open subset of the form $Ztimesmathbb{A}^{1}$ for some algebraic variety Z. We show that the generic fiber of a family $f:Xrightarrow S$ of normal $mathbb{A}^{1}$-cylindrical varieties becomes $mathbb{A}^{1}$-cylindrical after a finite extension of the base. Our second result is a criterion for existence of an $mathbb{A}^{1}$-cylinder in X which we derive from a careful inspection of a relative Minimal Model Program ran from a suitable smooth relative projective model of X over S.
For every prime $p$, Mohan Kumar constructed examples of stably free modules of rank $p$ on suitable $(p+1)$-dimensional smooth affine varieties. This note discusses how to detect the corresponding unimodular rows in motivic cohomology. Using the recent developments in the $mathbb{A}^1$-obstruction classification of vector bundles, this provides an alternative proof of non-triviality of Mohan Kumars stably free modules. The reinterpretation of Mohan Kumars examples also allows to produce interesting examples of stably trivial torsors for other algebraic groups.
Every deformed Koras-Russell threefold of the first kind $Y = left{ x^{n}z=y^{m}-t^{r} + xh(x,y,t)right}$ in $mathbb{A}^{4}$ is the algebraic quotient of proper Zariski locally trivial $mathbb{G}_a$-action on $mathrm{SL}_2 times mathbb{A}^1$.
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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