Résume | Algebraic varieties (zeroes of polynomial equations) often present
singularities: points around which the variety fails to be a manifold,
and where the usual techniques of calculus encounter difficulties. The
problem of understanding singularities can be traced to the very
beginning of algebraic geometry, and we now have at our disposal many
tools for their study. Among these, one of the most successful is what
is known as resolution of singularities, a process that transforms
(often in an algorithmic way) any variety into a smooth one, using a
sequence of simple modifications.
In the 60's Nash proposed a novel approach to the study of
singularities: the arc space. These spaces are natural higher-order
analogs of tangent spaces; they parametrize germs of curves mapping
into the variety. Just as for tangent spaces, arc spaces are easy to
understand in the smooth case, but Nash pointed out that their
geometric structure becomes very rich in the presence of
singularities.
Roughly speaking, the Nash problem explores the connection between the
topology of the arc space and the process of resolution of
singularities. The mere existence of such a connection has sparked in
recent years a high volume of activity in singularity theory, with
connections to many other areas, most notably birational geometry and
the minimal model program.
The objective of this two-part lecture is to give an overview of the
recent developments on the Nash problem. In the first talk I will
introduce the arc space, explore its connection with valuation theory,
and give a precise description of the Nash problem. In the second part
I will discuss our contribution to the Nash problem (this is joint
work with T. de Fernex). I will give an almost complete proof of our
main theorem, which states that terminal valuations are in the image
of the Nash map. In dimension two, this provides a new proof of the
Nash conjecture (originally proven by Fernandez de Bobadilla and Pe
Pereira). |