Workshop in Algebraic Geometry, Ljubljana – November 2003

ABSTRACTS:

Janez Bernik (University of Ljubljana): Reduction theorems for groups of matrices

Let G be a linear algebraic group over an algebraically closed field k. If char(k)=0, assume that the unipotent radical of G is trivial. Then G is solvable, nilpotent or abelian if and only if every finite subgroup is solvable, nilpotent or abelian respectively.

This is a recent result obtained with R. Guralnick and M. Mastnak. Time permitting, we shall discuss some possible applications.

Anita Buckley (University of Ljubljana): Smooth Calabi-Yau 3-folds in codimension 4

Riemann-Roch formula and Hilbert series for smooth 3-folds can be used to predict possible resolutions of Calabi-Yaus. Resolutions of Gorenstein rings up to codimension 3 are well known. We also get some interesting examples in codimension 4 by using the method of unprojections (joint work with Balazs Szendroi and Stavros Papadakis).

Barbara Fantechi (SISSA, Triest): Quantum cohomology of orbifolds.

Emilia Mezzetti (University of Triest): On the Gauss map of projective varieties

I will consider projective varieties with degenerate Gauss image, whose focal hypersurfaces are non-reduced schemes. Examples of this situation are provided by the secant varieties of Severi and Scorza varieties. The Severi varieties are moreover characterized by a uniqueness propety.

Miles Reid (University of Warwick): Deformations of toric varieties and graded rings over Mori flips

We describe several infinite families of 6-dimensional varieties, determined by combinatorics of continued fractions, that have two different toric 4-fold sections; they serve as key varieties for the Â* cover of Mori flips of type A (joint work with Gavin Brown).

Balazs Szendroi (Utrecht University): Examples of equivalences of derived categories motivated by motivic

integration

Motivic integration has been very successful in the study of birational geometry and the McKay correspondence. However, it typically proves an equality of numbers (such as Euler characteristics) without constructing a natural map between spaces (such as K-theory). Natural maps between spaces can arise from functors between categories, such as the category of sheaves. I will briefly survey some results in this area, and give examples of natural equivalences between categories suggested by the motivic integration approach.