The Wadge hierarchy on Zariski topologies

Abstract

We study the relation of continuous reducibility, or Wadge reducibility, between subsets of an affine variety. We show that on any curve the relation of continuous reducibility is a bqo, though it may have large finite antichains. We determine the Wadge hierarchy on irreducible curves and on countable irreducible affine varieties of any dimension. Under a technical assumption of adequateness, we prove that on the class of finite differences of closed subsets of an irreducible affine variety the structure of the Wadge hierarchy depends only on the dimension of the affine variety and coincides with the difference hierarchy; we also show that this assumption of adequateness is satisfied by a large class of affine varieties. In contrast, we show that for large cardinalities the behaviour of the Wadge hierarchy outside the class of finite differences of closed sets can be much wilder, for example there may exist antichains of size the continuum.