Explicit methods in algebraic geometry (XPL)
Organised by Reid, Brown, Corti, Farkas

Explicit geometric examples are the alpha and omega of algebraic
geometry: they provide points of entry to the field for graduate
students and points of contact with neighbouring areas; explicit
constructions act as motivation and sanity checks for abstract
arguments, and more often than not suggest the final goal, as well as
generalisations, abstract statements and essential ingredients in
rigorous proofs. The introduction to [CR] explained this point at length
in one context; that book opened up the new field of explicit birational
geometry to put Mori fibre spaces under the microscope, a subject that
continues to develop rapidly.

What this theme lacks in structure and organisation, it more than makes
up for in substance: there is a huge amount of great stuff here, and
problems for 50 PhD projects. Newer methods to complement traditional
ideas such as pluricanonical linear systems and embeddings include toric
geometry, the use of ``key varieties'' such as weighted quasihomogeneous
spaces, Mukai's vector bundles method, orbifold techniques, commutative
and computer algebra. Explicit methods have recently led to several
prominent advances: these include the construction of curves and their
moduli, the explicit classification of the steps of the MMP, and of the
birational links between Mori fibre spaces. Several long-standing
problems in this area seem to be nearing their goal, including the
existence of simply connected Campedelli surfaces (Lee and Park [LP])
and the irreducibility of the moduli of simply connected Godeaux
surfaces (separate but related work of Reid and Schreyer).

This topic will happen throughout the year, but more specifically during
the winter term Jan--Apr 2008, with short workshops on several of the
main topics and techniques.

Participants. Confirmed: Brown, Corti, Farkas, LEE Yongnam, Pignatelli,
Reid, Szendroi, Zucconi

Targetted: Altmann, Bauer, Brion, Buckley, Catanese, Craw, Chelstov,
Etienne Mann, CHEN Meng (Fudan), Tom Coates, Craw, Kawakita, Kerber,
Mella, Mendes Lopes, Mukai, Pardini, PARK Jongil, Ranestad, Ryder,
Takagi, Tziolas, Verrill, Zucconi

[LEE Yongnam and PARK Jongil] A simply connected surface of general type
with pg=0 and K2=2, math.AG/0609072, 24 pp.

[Frank-Olaf Schreyer] An experimental approach to numerical Godeaux
surfaces, Oberwolfach report 7/2005, pp.~434--436


========================

Problem areas: curves and their moduli, construction of surfaces with
small invariants, such as Godeaux and Campedelli surfaces, fibred
surfaces and relative algebras, related questions for 3-folds of general
type
  toric varieties, diptych varieties, homogeneous spaces and other
`key varieties',
  Fano database, T&J, Calabi-Yaus, orbifold RR,
  Hilbert functions for orbifolds, Fano 4-folds and canonical 4-folds
  orbifolds and stacks, quantum cohomology, GHilb and McKay corr
  Sarkisov program for Fanos, explicit contractions and flips
  surfaces of general type such as           
  3-folds, explicit birational geometry,         
  toric methods and quasihomogeneous spaces, computations                       

one or more weekend conferences during term time