Rational Parametrisations via Lie Algebras I

              Michael Harrison
            University of Sydney

 This will be the first of two talks on a computational
method based on Lie Algebra techniques to determine
whether twisted forms of certain algebraic varieties
are trivial and, if so, to find an explicit isomorphism
to standard non-twisted models.
 The method has been applied to rational Del Pezzo
surfaces that are twists of the projective plane, the
plane blown up at one point and the product of two
projective lines.
 This talk will concentrate on giving an overview of the
method with specific reference to the plane case
(Severi-Brauer surfaces). The first key point about such
surfaces is that their automorphism group determines the
twist. The second key point is that their anticanonical
embedding is given by quadratic equations which allows the
Lie algebra of the automorphism group to be determined
explicitly by linear algebra.