Degeneration methods in algebraic geometry (Gavril Farkas, Texas)
Degeneration techniques have been used in algebraic geometry since the
19th century and today they are still ubiquitous especially in
enumerative geometry and the study of different moduli spaces. One of
the main reason people like to work with compact moduli spaces is
precisely to be able to use degeneration methods.
We will focus on degenerations of curves and line (vector) bundles on
them and there are numerous possible topics we could choose from:
-Degenerations of curves and simple examples of stable reduction.
The difference between abstract and embedded degenerations (working out in
examples the geometry of the Hilbert scheme of curves in projective
space or understand examples like e.g. the difference between
degenerations of an abstract curve of genus 3 and embedded degenerations
of a plane quartic).
-Degenerations of coverings of curves and the theory
of admissible coverings. Applications to enumerative problems.
-The study of the Severi varieties of plane curves of degree d and
genus g: Harris's proof (by degeneration) of the irreducibility of the
Severi variety, enumerative questions like finding the degree of the
Severi variety (that is, the number of plane curves of degree d and genus
g passing through 3d+g-1 fixed points).
-Yau-Zaslow formula for the number of (singular) rational curves on a K3
surface.
-degenerations of Jacobian varieties and (maybe) of the moduli space of
vector bundles on a curve. The theory of limit linear series of Eisenbud
and Harris. Applications to divisor class calculations on the moduli space of
stable curves (eg solving problems like "What is the class of the divisor
on the moduli space of genus 3 curves that consists of hyperelliptic
curves").
-Linear series on curves, Gieseker's proof (by degeneration) of the Petri
Theorem for the general curve of genus g.
As a general reference I recommend the book "Moduli of curves" by Harris
and Morrison.