Fri, July 31, 2020
Public Access Category: All |
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- Time:
- 5:30pm
- Description:
- 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: https://meet.google.com/xxm-cidr-yqa
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.