Bounded birationality and isomorphism problems are computable

TL;DR

This paper presents an algorithm to explicitly parametrize birational maps between subvarieties of projective space, advancing the understanding of rationality problems and providing solutions for specific cases.

Contribution

It introduces a constructive algorithm for parametrizing birational maps of bounded degree, and applies it to address the rationality problem for certain varieties.

Findings

Algorithm constructs parametrizing variety for birational maps

Solves rationality problem for some simple cases

Extends results to various types of rational maps and fields

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.