2018/01/03 by Tuyen Trung Truong, Truong, Tuyen Trung
Computer Science · Mathematics · #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Commutative Algebra and Its Applications #FOS: Mathematics #Polynomial and algebraic computation
paper · pdf · doi:10.48550/arxiv.1801.00901
openalex publication_date 2018/01/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 \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 ℙn of degree ≤ 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.