"Introduction to explicit methods in algebraic geometry"

 UK/Poland Summer algebraic geometry school

 Topic:  Explicit construction of moduli and parameter spaces.
 Lecturers: Alastair Craw and Diane Maclagan

 Overview:

 These lectures are intended as an introduction to some of the ways that 
moduli problems arise in algebraic geometry, with strong emphasis on 
explicit descriptions of the resulting spaces.

 One place where such problems arise is when one is interested in a family
of geometric objects, such as subschemes of a projective variety.  By
restricting to objects satisfying a numerical condition, like Hilbert
polynomial of a subscheme, one may obtain a moduli space of such objects.
One can then expect to improve one's understanding of the objects by
studying geometric or topological properties of the moduli space itself,
including dimension, connectedness and the structure of its irreducible
components.

 For a second situation, imagine that one is interested in geometric or
topological properties of a given scheme.  In this case, it may be
possible to introduce a moduli problem for which the underlying moduli
space is the scheme in question and, moreover, where the tautological
family on the moduli space gives insight into properties of the space. For
instance,  if the tautological family defines a collection of vector
bundles on the scheme, do the classes of these bundles freely generate the
Grothendieck group of vector bundles, say, or even the bounded derived
category of coherent sheaves?

Background: These lectures should be accessible to a beginning
graduate student in algebraic geometry.

Outline (subject to change:)

 Maclagan - Hilbert schemes
 Lecture 1: Introduction/Hilbert scheme of subschemes of P^n;
 Lecture 2: Details of constructions/connectedness of the Hilbert scheme;
 Lecture 3: Hilbert schemes of points on surfaces;
 Lecture 4: Multigraded Hilbert schemes;
 Lecture 5: Examples of multigraded Hilbert schemes (GHilb for abelian
            groups, Hilbert schemes of toric varieties, toric Hilbert
 	    schemes, etc), open questions.

 Craw - Quiver representations in toric geometry
 Lecture 1: Introduction/GIT for torus actions;
 Lecture 2: Toric varieties following Cox, toric quiver varieties;
 Lecture 3: Quivers of sections and multilinear series;
 Lecture 4: Bound quivers, and toric varieties as fine moduli of algebras;
 Lecture 5: Bound McKay quiver and the coherent component, open questions.