Bounded birationality and isomorphism problems are computable

Abstract

Let X, Y be two irreducible subvarieties of the projective space Pn, and d≥1 an integer number. The main result of this paper is an algorithm to construct explicitly, in terms of d and the ideals defining X and Y, a quasi-affine algebraic variety parametrising the set of all birational maps f from X onto Y which can be extended to a self-rational map of Pn of algebraic degree ≤d. We also prove similar results for the case f is a dominant rational map, regular morphism, isomorphism or regular embedding. Similar results are valid for varieties over an arbitrary algebraically closed field, and also for maps on non-projective varieties.