Cancellation for surfaces revisited. I

The celebrated Zariski Cancellation Problem asks as to when the existence of an isomorphism $X\times\mathbb{A}^n\cong X'\times\mathbb{A}^n$ for (affine) algebraic varieties $X$ and $X'$ implies that $X\cong X'$. 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 $X\to B$ 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 $X\to B$ is a line bundle, and, for a normal such $X$, if and only if $X\to B$ 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 $X_\lambda\to B$, $\lambda\in\Lambda$, of $\mathbb{A}^1$-fibered surfaces with cylinders $X_\lambda\times\mathbb{A}^1$ 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.