Bounded birationality and isomorphism problems are computable

Abstract

Let $X,Y$ be two irreducible subvarieties of the projective space $\mathbb{P}^n$, and $d\geq 1$ an integer number. The main result of this paper is an algorithm to construct {\bf 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 $\mathbb{P}^n$ of degree $\leq d$. Based on this result, we propose an approach towards the rationality problem (see Section 3 below), solve it for some simple cases (varieties of general type or curves), and state a rough strategy for reducing it to some simpler cases via Iitaka's fibrations. 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.