On the triviality of the affine threefold $x^my = F(x, z, t)$ - Part 2

Abstract: In this talk we will discuss a theory for affine threefolds of the form $x^my = F(x, z, t)$ which will yield several necessary and sufficient conditions for the coordinate ring of such a threefold to be a polynomial ring. For instance, we will see that this problem of four variables reduces to the equivalent but simpler two-variable question as to whether F(0, z, t) defines an embedded line in the affine plane. As one immediate consequence, one readily sees the non-triviality of the famous Russell-Koras threefold $x^2y+x+z^2+t^3=0$ (which was an exciting open problem till the mid 1990s) from the obvious fact that $z^2+t^3$ is not a coordinate. The theory on the above threefolds connects several central problems on Affine Algebraic Geometry. It links the study of these threefolds with the famous Abhyankar-Moh “Epimorphism Theorem” in characteristic zero and the Segre-Nagata lines in positive characteristic. We will also see a simplified proof of the triviality of most of the Asanuma threefolds (to be defined in the talk) and an affirmative solution to a special case of the Abhyankar-Sathaye Conjecture. Using the theory, we will also give a recipe for constructing infinitely many counterexample to the Zariski Cancellation Problem (ZCP) in positive characteristic. This will give a simplified proof of the speaker's earlier result on the negative solution for the ZCP.

Mathematics

Audience: advanced learners

IIT Bombay Virtual Commutative Algebra Seminar

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