Fri, July 31, 2020
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5:00pm [5:30pm] Neena Gupta, ISI Kolkata
Date and Time: Friday 31 July 2020, 5:30 pm IST / 12:00 GMT / 08:00am EDT (joining time : 5:15 pm IST - 5:30 pm IST) Google Meet link: Speaker: Neena Gupta, ISI Kolkata. Title: 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.