Cancellation for surfaces revisited. II

Abstract

Let X and X ′ be affine algebraic varieties over a field k. The celebrated Zariski Cancellation Problem asks as to when the existence of an isomorphism X×A n ≅ X ′ × A n implies X ≅ 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 A 1 -fibration X → B over a smooth affine curve B. If X does not admit such an 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 A 1 -fibered over B with only reduced fibers whose cylinders X × A 1 , X ′ × 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. We construct a course moduli space of such surfaces with a given cylinder. This moduli space has an infinite number of irreducible components whose dimension grows. Each irreducible component of this moduli space is an affine variety with quotient singularities. Contents 18 6.2. Special isomorphisms and regularization 19 7. On moduli spaces of GDF surfaces 22 7.1. Coarse moduli spaces of GDF surfaces 23 7.2. The automorphism group of a GDF surface 24 7.3. Configuration spaces and configuration invariants 25 7.4. Versal deformation families of trivializing sequences 27 7.5. Proof of Theorem 7.3 31 References 33