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.