2014/08/10 by Aiichi Yamasaki, Yamasaki, Aiichi
Mathematics · #12G05 #14E05 #14E08 #Advanced Algebra and Geometry #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #math.AG #msc:12G05 #msc:14E05 #msc:14E08
paper · pdf · doi:10.48550/arxiv.1408.2233
incorporates all of the content of arXiv:1308.0909
openalex publication_date 2014/08/10 · arxiv created 2015/09/20 · arxiv updated 2015/09/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let k be a field with char k \not= 2, X be an affine surface defined by the equation z2=P(x)y2+Q(x) where P(x), Q(x) ∈ k[x] are separable polynomials. We will investigate the rationality problem of X in terms of the polynomials P(x) and Q(x). The necessary and sufficient condition is s ≤ 3 with minor exceptions, where s=s1+s2+s3+s4, s1 (resp. s2, resp. s3) being the number of c ∈ k such that P(c)=0 and Q(c) \not∈ k(c)2 (resp. Q(c)=0 and P(c) \not∈ k(c)2, resp. P(c)=Q(c)=0 and -(Q)/(P)(c) \not∈ k(c)2). s4=0 or 1 according to the behavior at x=∞. X is a conic bundle over ℙk1, whose rationality was studied by Iskovskikh. Iskovskikh formulated his results in geometric language. This paper aims to give an algebraic counterpart.