Description

Date and Time: 10 November 2020, 5:30pm IST/ 12:00GMT/07:00am EDT (joining

time 5:15pm - 5:30pm IST)

Speaker: Amartya Kumar Datta, ISI Kolkata

Google meet link: https://meet.google.com/jom-etrz-bdd

Title: G_a-actions on Affine Varieties: Some Applications - Part 1

Abstract: One of the hardest problems that come up in affine algebraic geometry is to decide whether a certain d-dimensional factorial affine domain is ``trivial'', i.e., isomorphic to the polynomial ring in d variables. There are instances when the ring of invariants of a suitably chosen G_a-action has been able to distinguish between two rings (i.e., to prove they are non-isomorphic), when all other

known invariants failed to make the distinction. It was using one such

invariant that Makar-Limanov proved the non-triviality of the

Russell-Koras threefold, leading to the solution of the Linearization

Problem; and again, it was using an invariant of G_a-actions that Neena

Gupta proved the nontriviality of a large class of Asanuma threefolds

leading to her solution of the Zariski Cancellation Problem in positive

characteristic.

G_a actions are also involved in the algebraic characterisation of the

affine plane by M. Miyanishi and the algebraic characterisation of the

affine 3-space.by Nikhilesh Dasgupta and Neena Gupta. Miyanishi's

characterisation had led to the solution of Zariski's Cancellation Problem

for the affine plane. Using G_a-actions, a simple algebraic proof for

this cancellation theorem was obtained three decades later by

Makar-Limanov.

In this talk (in two parts), we will discuss the concept of G_a-actions

along with the above applications, and the closely related theme of

Invariant Theory. The concept of G_a-action can be reformulated in the

convenient ring-theoretic language of ``locally nilpotent derivation'' (in

characteristic zero) and ``exponential map'' (in arbitrary

characteristic). The ring of invariants of a G_a- action corresponds to

the kernel of the corresponding locally nilpotent derivation (in

characteristic zero) and the ring of invariants of an exponential map. We

will recall these concepts. We will also mention a theorem on G_a actions

on affine spaces (or polynomial rings) due to C.S. Seshadri.

We will also discuss the close alignment of the kernel of a locally

nilpotent derivation on a polynomial ring over a field of characteristic

zero with Hilbert's fourteenth problem. While Hilbert Basis Theorem had

its genesis in a problem on Invariant Theory, Hilbert's fourteenth

problem seeks a further generalisation: Zariski generalises it still

further. The connection with locally nilpotent derivations has helped

construct some low-dimensional counterexamples to Hilbert's problem. We

will also mention an open problem about the kernel of a locally nilpotent

derivation on the polynomial ring in four variables; and some partial

results on it due to Daigle-Freudenburg, Bhatwadekar-Daigle,

Bhatwadekar-Gupta-Lokhande and Dasgupta-Gupta. Finally, we will state a

few technical results on the ring of invariants of a G_a action on the

polynomial ring over a Noetherian normal domain, obtained by

Bhatwadekar-Dutta and Chakrabarty-Dasgupta-Dutta-Gupta.

time 5:15pm - 5:30pm IST)

Speaker: Amartya Kumar Datta, ISI Kolkata

Google meet link: https://meet.google.com/jom-etrz-bdd

Title: G_a-actions on Affine Varieties: Some Applications - Part 1

Abstract: One of the hardest problems that come up in affine algebraic geometry is to decide whether a certain d-dimensional factorial affine domain is ``trivial'', i.e., isomorphic to the polynomial ring in d variables. There are instances when the ring of invariants of a suitably chosen G_a-action has been able to distinguish between two rings (i.e., to prove they are non-isomorphic), when all other

known invariants failed to make the distinction. It was using one such

invariant that Makar-Limanov proved the non-triviality of the

Russell-Koras threefold, leading to the solution of the Linearization

Problem; and again, it was using an invariant of G_a-actions that Neena

Gupta proved the nontriviality of a large class of Asanuma threefolds

leading to her solution of the Zariski Cancellation Problem in positive

characteristic.

G_a actions are also involved in the algebraic characterisation of the

affine plane by M. Miyanishi and the algebraic characterisation of the

affine 3-space.by Nikhilesh Dasgupta and Neena Gupta. Miyanishi's

characterisation had led to the solution of Zariski's Cancellation Problem

for the affine plane. Using G_a-actions, a simple algebraic proof for

this cancellation theorem was obtained three decades later by

Makar-Limanov.

In this talk (in two parts), we will discuss the concept of G_a-actions

along with the above applications, and the closely related theme of

Invariant Theory. The concept of G_a-action can be reformulated in the

convenient ring-theoretic language of ``locally nilpotent derivation'' (in

characteristic zero) and ``exponential map'' (in arbitrary

characteristic). The ring of invariants of a G_a- action corresponds to

the kernel of the corresponding locally nilpotent derivation (in

characteristic zero) and the ring of invariants of an exponential map. We

will recall these concepts. We will also mention a theorem on G_a actions

on affine spaces (or polynomial rings) due to C.S. Seshadri.

We will also discuss the close alignment of the kernel of a locally

nilpotent derivation on a polynomial ring over a field of characteristic

zero with Hilbert's fourteenth problem. While Hilbert Basis Theorem had

its genesis in a problem on Invariant Theory, Hilbert's fourteenth

problem seeks a further generalisation: Zariski generalises it still

further. The connection with locally nilpotent derivations has helped

construct some low-dimensional counterexamples to Hilbert's problem. We

will also mention an open problem about the kernel of a locally nilpotent

derivation on the polynomial ring in four variables; and some partial

results on it due to Daigle-Freudenburg, Bhatwadekar-Daigle,

Bhatwadekar-Gupta-Lokhande and Dasgupta-Gupta. Finally, we will state a

few technical results on the ring of invariants of a G_a action on the

polynomial ring over a Noetherian normal domain, obtained by

Bhatwadekar-Dutta and Chakrabarty-Dasgupta-Dutta-Gupta.

Date

Tue, November 10, 2020

Start Time

5:30pm IST

Priority

5-Medium

Access

Public

Created by

DEFAULT ADMINISTRATOR

Updated

Tue, November 10, 2020 3:39pm IST