Problems in Affine Algebraic Geometry : on Triviality and Embedding of Linear Hyperplanes and Rigidity of Pham-Brieskorn surfaces

Abstract

My thesis consists of two topics from Affine Algebraic Geometry: one of the topics is on Linear varieties (varieties defined by polynomials which are linear in one variable) and the second topic explores when Pham-Brieskorn surfaces do not admit non-trivial Ga-actions. Linear varieties over a field k have been playing a central role in the study of some of the challenging problems on affine spaces like the Zariski Cancellation Problem and the Lineariza- tion Problem. Breakthroughs on such problems have occurred by examining two questions on linear polynomials of the form H := α(X1, . . . , Xm)Y − F (X1, . . . , Xm, Z, T ) ∈ D := k[X1, . . . , Xm, Y, Z, T ] : (i) Whether H defines a hyperplane i.e., the affine variety V ∈ Am+3 k defined by H is isomorphic to the affine space Am+2 k . (ii) If V is isomorphic to an affine space, then whether H is a coordinate in D. Question (i) connects to the Characterization Problem of identifying affine spaces among affine varieties; Question (ii) is a special case of the formidable Embedding Problem for affine spaces. In Chapter 3 of the thesis, using K-theory and Ga-actions, we address these questions under certain conditions on α and F . For instance, we show that when the characteristic of k is zero, F ∈ k[Z, T ] and H defines a hyperplane, then H is a coordinate in D along with X1, X2, . . . , Xm. Our results yield certain family of higher-dimensional hyperplanes satisfying the Ab- hyankar–Sathaye Conjecture on the Epimorphism Problem and an infinite family of higher dimensional non-isomorphic varieties which are counterexamples to the Zariski Cancellation Problem in positive characteristic and A2-fibration Problem in positive characteristic. We have also discussed the above two questions by replacing the field k with a Noetherian integral domain R. In Chapter 4 of the thesis, we have discussed the rigidity of Pham-Brieskorn rings. Over any field k, for n ∈ Z>3 and a1, . . . , an ∈ Z>1, Pham-Brieskorn rings are denoted by B(a1,...,an) and defined by B(a1,...,an) := k[X1, . . . , Xn]/(Xa1 1 + · · · + Xan n ). We showed that every non-domain Pham-Brieskorn ring, for n ∈ Z>3 is non-rigid. For any three integers a, b, c > 1, we give some sufficient conditions on (a, b, c) for which Pham- Brieskorn domain B(a,b,c) is rigid. This gives an alternative approach to show that over a field k of characteristic p > 0, there does not exist any non-trivial exponential map on k[X, Y, Z, T ]/(XmY + T pr q + Zpe ) = k[x, y, z, t], for m, q > 1, p - mq and e > r > 1, which fixes y, a crucial result used in “On the cancellation problem for the affine space A3 in characteristic p, Invent. Math. 195” by Neena Gupta to show that the Zariski Cancellation Problem does not hold for the affine 3-space. We also provide a sufficient condition for B(a,b,c) to be stably rigid.