2022/06/30 by Parnashree Ghosh, Ghosh, Parnashree, Neena Gupta +1 · 1 citation
Computer Science · Mathematics · #13A02 #13A50 #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Coding theory and cryptography #Commutative Algebra (math.AC) #FOS: Mathematics #Finite Group Theory Research #Primary: 14R10 #Secondary: 13B25
paper · pdf · doi:10.48550/arxiv.2206.15210
openalex publication_date 2022/06/30 · openalex created_date 2022/07/03 · openalex updated_date 2026/07/28
Let k be a field, m a positive integer, \mathbbV an affine subvariety of \mathbbAm+3 defined by a linear relation of the form x1^r1⋯ xm^rmy=F(x1, … , xm,z,t), A the coordinate ring of \mathbbV and G= X1r1⋯ XmrmY-F(X1, …, Xm,Z,T). In \citecom, the second author had studied the case m=1 and had obtained several necessary and sufficient conditions for \mathbbV to be isomorphic to the affine 3-space and G to be a coordinate in k[X1, Y,Z,T]. In this paper, we study the general higher-dimensional variety \mathbbV for each m \geqslant 1 and obtain analogous conditions for \mathbbV to be isomorphic to \mathbbAm+2 and G to be a coordinate in k[X1, …, Xm, Y,Z,T], under a certain hypothesis on F. Our main theorem immediately yields a family of higher-dimensional linear hyperplanes for which the Abhyankar-Sathaye Conjecture holds. We also describe the isomorphism classes and automorphisms of integral domains of the type A under certain conditions. These results show that for each d \geqslant 3, there is a family of infinitely many pairwise non-isomorphic rings which are counterexamples to the Zariski Cancellation Problem for dimension d in positive characteristic.