Cancellation for Surfaces Revisited

Abstract

The celebrated Zariski Cancellation Problem asks as to when the existence of an isomorphism X × A n ≅ X ′ × A n X\times \mathbb {A}^n\cong X’\times \mathbb {A}^n for (affine) algebraic varieties X X and X ′ X’ implies that X ≅ X ′ X\cong X’ . In this paper we provide a criterion for cancellation by the affine line (that is, n = 1 n=1 ) in the case where X X is a normal affine surface admitting an A 1 \mathbb {A}^1 -fibration X → B X\to B with no multiple fiber over a smooth affine curve B B . For two such surfaces X → B X\to B and X ′ → B X’\to B we give a criterion as to when the cylinders X × A 1 X\times \mathbb {A}^1 and X ′ × A 1 X’\times \mathbb {A}^1 are isomorphic over B B . The latter criterion is expressed in terms of linear equivalence of certain divisors on the Danielewski-Fieseler quotient of X X over B B . It occurs that for a smooth A 1 \mathbb {A}^1 -fibered surface X → B X\to B the cancellation by the affine line holds if and only if X → B X\to B is a line bundle, and, for a normal such X X , if and only if X → B X\to B is a cyclic quotient of a line bundle (an orbifold line bundle). If X X does not admit any A 1 \mathbb {A}^1 -fibration over an affine base then the cancellation by the affine line is known to hold for X X by a result of Bandman and Makar-Limanov. If the cancellation does not hold then X X deforms in a non-isotrivial family of A 1 \mathbb {A}^1 -fibered surfaces X λ → B X_\lambda \to B with cylinders X λ × A 1 X_\lambda \times \mathbb {A}^1 isomorphic over B B . We construct such versal deformation families and their coarse moduli spaces provided B B does not admit nonconstant invertible functions. Each of these coarse moduli spaces has infinite number of irreducible components of growing dimensions; each component is an affine variety with quotient singularities. Finally, we analyze from our viewpoint the examples of non-cancellation constructed by Danielewski, tom Dieck, Wilkens, Masuda and Miyanishi, e.a.