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