Abstract
A classical tool, used by the algebraic geometers of the so-called
''Italian school'' (in particular by Guido Castelnuovo and Federigo Enriques)
was the study of the so-called ''limiting cases''.
The idea is to consider geometric objects varying in a family and
''degenerating'', in the limit, to very particular configurations. These can
be on one side very complicated, on the other may be simpler than the objects
one started with. The point is to deduce properties of the general object of
the family from those, hopefuly simpler, of the ''limit''. This technique,
classically proved to be very fruitful, but was also very much criticized from
the viewpoint of mathematical rigour. However in recent times it was revived
and put on a firm basis, and it is today an essential tool for the resolution
of difficult problems. In this talk I intend to start from very simple and
concrete examples, at the level of secondary school, and, as such, already
considered by Emma Castelnuovo (who inherited them from her father and uncle).
These examples will illustrate the usefulness of these methods, but also the
conceptual difficulties which they present. I will then try to illustrate some
open research problems which can be attacked with these techniques and the
interesting results which is possible to prove using them.
