Cancellation for surfaces revisited. II

Abstract: Let $X$ and $X'$ be affine algebraic varieties over a field $\mathbb{k}$. The celebrated Zariski Cancellation Problem asks as to when the existence of an isomorphism $X\times\mathbb{A}^n\cong X'\times\mathbb{A}^n$ implies $X\cong X'$. In Part I of this paper (arXiv:1610.01805) we provided a criterion for cancellation 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. In the present Part II we classify all pairs $(X,X')$ of smooth affine surfaces $\mathbb{A}^1$-fibered over $B$ with only reduced fibers whose cylinders $X\times\mathbb{A}^1$, $X'\times\mathbb{A}^1$ are isomorphic over $B$. Our criterion of isomorphism of cylinders over $B$ is expressed in terms of linear equivalence of certain divisors on the Danielewski-Fieseler quotient of $X$ over $B$. Under a mild restriction we construct a coarse moduli of such surfaces.