Cancellation for surfaces revisited. I

DOI10.1090/MEMO/1371zbMATH Open1527.14002arXiv1610.01805MaRDI QIDQ6278357

Hubert Flenner, Shulim Kaliman, Mikhail Zaidenberg

Publication date: 6 October 2016

Abstract: The celebrated Zariski Cancellation Problem asks as to when the existence of an isomorphism $Ximesmathbb{A}^{n}cong{X}^{\prime }imesmathbb{A}^n$ for (affine) algebraic varieties $X$ and $X^{\prime }$ implies that $Xcong{X}^{\prime }$. In this paper we provide a criterion for cancellation by the affine line (that is, $n=1$) in the case where $X$ is a normal affine surface admitting an $mathbb{A}^1$-fibration $XoB$ over a smooth affine curve $B$. If $X$ does not admit such an $mathbb{A}^1$-fibration then the cancellation by the affine line is known to hold for $X$ by a result of Bandman and Makar-Limanov. It occurs that for a smooth $mathbb{A}^1$-fibered affine surface $X$ over $B$ the cancellation by an affine line holds if and only if $XoB$ is a line bundle, and, for a normal such $X$, if and only if $XoB$ is a cyclic quotient of a line bundle (an orbifold line bundle). When the cancellation does not hold for $X$ we include $X$ in a non-isotrivial deformation family $X_{l}ambdaoB$, $lambdainLambda$, of $mathbb{A}^1$-fibered surfaces with cylinders $X_{l}ambdaimesmathbb{A}^1$ isomorphic over $B$. This gives large families of examples of non-cancellation for surfaces which extend the known examples constructed by Danielewski, tom Dieck, Wilkens, Masuda and Miyanishi, e.a.

Mathematics Subject Classification ID

Research exposition (monographs, survey articles) pertaining to algebraic geometry (14-02) Affine spaces (automorphisms, embeddings, exotic structures, cancellation problem) (14R10) Fine and coarse moduli spaces (14D22)

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