Applications of Exponential Maps to Epimorphism and Cancellation Problems

Abstract

Throughout this talk k will denote a field. The talk is primarily divided into two parts. In the first part we will discuss one of the formidable open problems in the area of Affine Algebraic Geometry, called the Epimorphism Problem. Question 1. If k[X1,...,Xn] (H) = k[n−1], then is k[X1, . . . , Xn] = k[H][n−1]? For n = 2, the answer to above question is affirmative when k is a field of characteristic zero. This result is known as the Epimorphism Theorem proved by Abhyankar-Moh and independently by Suzuki. However, in positive characteristic there are counter examples due to Segre-Nagata. The famous Abhyankar-Sathaye conjecture asserts affirmative answer to Question 1 for n ⩾ 3 over fields of char- acteristic zero. So far we only have partial answers to this conjecture. The first affirmative result for n = 3 is due to Sathaye for linear planes over fields of char- acteristic zero. Later, Russell extended this result over fields of arbitrary character- istic. In this talk we consider the following varieties. Let m a positive integer, V an affine subvariety of Am+3 defined by a linear relation of the form xr1 1 · · · xrm m y = F (x1, . . . , xm, z, t), A the coordinate ring of V and G = Xr1 1 · · · Xrm m Y −F (X1, . . . , Xm, Z, T ). We name these varieties as “Generalised Asanuma varieties”. Earlier, Gupta had studied the case m = 1, and had obtained several necessary and sufficient con- ditions for V to be isomorphic to the affine 3-space and G to be a coordinate in k[X1, Y, Z, T ]. We study the general higher-dimensional variety V for each m ⩾ 1 and obtain analogous conditions for V to be isomorphic to Am+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 do- mains of the type A under certain conditions. These results show that for each d ⩾ 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. We further give complete description of two important invariants called Makar- Limanov and Derksen invariants of a certain subfamily of Generalised Asanuma varieties. In the second part of this talk we discuss about another major problem called the Cancellation Problem which investigates the following: Question 2. Let D and E be two affine domains over a field k such that D[1] =k E[1]. Does this imply D ∼=k E? The answer to Question 2 is affirmative for one dimensional affine domains. This result is due to Abhyankar, Eakin and Heinzer. However, there are coun- terexamples in dimensions greater than or equal to two. Danielewski constructed 1 a family of two dimensional pairwise non-isomorphic smooth complex varieties which are counterexamples to the Cancellation Problem. A. J. Crachiola further extended Danielewski’s examples over arbitrary characteristic. Dubouloz con- structed higher dimensional (⩾ 2) analogues of the Danielewski varieties over the field of complex numbers, which are counterexamples to this problem. Over fields of arbitrary characteristic, we establish an infinite family of a higher dimensional varieties which are pairwise non-isomorphic and are counter examples to the Can- cellation Problem. Moreover, this family accommodates the counter examples due to Dubouloz over C.