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            <title xml:lang="en">$\mathbb{A}^1$-cylinders over smooth affine surfaces of negative Kodaira dimension</title>
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                <forename type="first">Adrien</forename>
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            <idno type="halRefHtml">&lt;i&gt;Kinosaki Algebraic Geometry Symposium 2019&lt;/i&gt;, Sep 2019, Toyo-oka, Japan</idno>
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                  <date type="start">2019-09-21</date>
                  <date type="end">2019-09-25</date>
                  <settlement>Toyo-oka</settlement>
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              <p>The Zariski Cancellation problem for smooth affine surfaces asks whether two suchsurfaces whose products with the affine line are isomorphic are isomorphic themselves. Byresults of Iitaka-Fujita, the answer is positive for surfaces of non-negative Kodaira dimen-sion. By a characterization due to Miyanishi, surfaces of negative Kodaira dimension arefibered by the affine line, and by a celebrated result of Miyanishi-Sugie, the answer to theproblem is positive if one of the surfaces is the affine plane. On the other hand, exam-ples of non-isomorphicA1-fibered affine surfaces with isomorphicA1-cylinders were firstconstructed by Danielewski in 1989, and then by many other authors. All these counter-examples are essentially constructed by variants of the method employed by Danielewski,nowadays known as the ”Danielewski fiber product trick”. In this talk, I will explain thatthis method is actually more than a trick: re-interpreted in a suitable way, it providesa necessary and sufficient criterion for twoA1-fibered surfaces over a same affine basecurveCto have isomorphic relativeA1-cylinders overC.  The characterization can beroughly stated as follows: two such surfaces have isomorphic relativeA1-cylinders overCif and only if their respectively relative (log)-canonical classes are equal when viewed ascertain naturally definedQ-divisors on a non-separated orbifold curve ̆CdominatingC,fully determined by the structure of the fibers of theA1-fibrations at hand. (Joint workin progress with S. Kaliman and M. Zaidenberg).</p>
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