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.

(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.

Date

Fri, July 31, 2020

Start Time

5:30pm IST

Priority

5-Medium

Access

Public

Created by

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Updated

Fri, July 31, 2020 8:13am IST