Week 1
Mon 3rd-Fri 7th Sep
"Explicit construction of moduli and parameter spaces"
Lecturers: Alastair Craw (Glasgow) and Diane Maclagan (Warwick)
The school is organised in collaboration with the long-running Polish
September school, with Buczynski, Langer and Wisniewski as coorganisers.
The plan is to have 2 lecture courses accessible to beginning graduate
students in algebraic geometry, together with problem sessions and ample
time for discussion. There will be room for a number of invited lectures
or contributions from participants.
please register online at https://www.warwick.ac.uk/mrc/events.php
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?
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: Projective toric 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.
Mon 3rd-Fri 7th Sep
"Explicit construction of moduli and parameter spaces"
Lecturers: Alastair Craw (Glasgow) and Diane Maclagan (Warwick)
Titles: Craw - Quiver representations in toric geometry 1-5
Maclagan - Hilbert schemes 1-5
Program (1st draft)
Mon 3rd Sep 11-12 Craw 1
Mon 3rd Sep 2-3 Maclagan 1
Mon 3rd Sep 3:30-4:30 and 5-6 problem sessions
Tue 4th Sep 9:30-10:30 Maclagan 2
Tue 4th Sep 11-12 Craw 2
Tue 4th Sep 1:30-3 and 3:30-4:30 problem sessions
Tue 4th Sep 5-6 invited lecture Prof Adrian Langer (Warsaw)
On D-affinity of quadrics
Wed 5th Sep 9-10 Craw 3
Wed 5th Sep 10:30-11:30 Maclagan 3
Wed 5th Sep 12-1 problem session
Free afternoon
Thu 6th Sep 9:30-10:30 Maclagan 4
Thu 6th Sep 11-12 Craw 4
Thu 6th Sep 1:30-3 and 3:30-5 problem sessions
Fri 7th Sep 9:30-10:30 Craw 5
Fri 7th Sep 11-12 Maclagan 5
Fri 7th Sep 1:30-3 problem session
Fri 7th Sep 4- Colloquium lecture Prof Jaroslaw Wisniewski (Warsaw)
Phylogenetic trees and algebraic geometry
Confirmed senior participants:
Brown, Buczynski, Craw, Langer, Reid, Siksek, Wisniewski