Applications of exponential maps to epimorphism and cancellation problems/ Parnashree Ghosh

Thesis (Ph.D.)- Indian statistical Institute, 2023

Includes bibliography

Preliminaries -- Triviality of a family of linear hyperplanes -- An infinite family of higher dimensional counterexamples to ZCP -- Generalised Danielewski varieties and invariants of generalised Asanuma varieties

Guided by Prof. Neena Gupta

Throughout this thesis k will always denote a field. The main aims of this
thesis are the following:

(a) To study “Generalised Asanuma varieties” and deduce Epimorphism re-
sults for a certain family of linear hyperplanes over fields of arbitrary charac-
teristic.

(b) To determine isomorphism classes and automorphisms of Generalised
Asanuma varieties and use the classification to demonstrate an infinite family
of pairwise non-isomorphic varieties which are counter examples to the Zariski

Cancellation Problem (ZCP) in higher dimensions (≥ 3) and in positive char-
acteristic.

(c) To study Generalised Danielewski varieties in higher dimensions and in
arbitrary characteristic and use the results to determine some invariants
(Derksen and Makar-Limanov invariants) of some subfamilies of Generalised
Asanuma varieties and to demonstrate a new infinite family of counterexam-
ples to the General Cancellation Problem in arbitrary characteristic. In Chapter 3 we discuss (a) under the heading “Triviality of a family
of linear hyperplanes” and in Chapter 4 we discuss (b) under the title “An
infinite family of higher dimensional counterexamples to ZCP”. In Chapter
5, entitled “Generalised Danielewski varieties and invariants of generalised
Asanuma varieties”, we will study Danielewski varieties in a more general
set up and thereby provide a new family of counterexamples to the General
Cancellation Problem.