Programme

12:00

Herbert
Lange (Erlangen)

Clifford
indices for vector bundles of rank 2 on curves

This
is a report on recent joint work with Peter Newstead on
generalized Clifford indices. After recalling the classical
Clifford index Cliff(C)
of a smooth projective curve C,
the Clifford index Cliff_{2}(C)
for semistable vector bundles of rank 2 will be defined. We then
work out some analogous properties of Cliff_{2}(C)
which are well known for Cliff(C).

15:00

Angela
Ortega (Berlin)

Rank
two BrillNoether theory and the maximal rank conjecture

Rank
two BrillNoether Theory deals with linear series of rank 2 on a
curve C, more precisely with the cycles BN_{C}(d,k)
in the moduli space of semistable rank 2 vector bundles on C
of degree d, defined by the condition of admitting
at least k
sections. Unlike classical
BrillNoether theory, the dimension of BN_{C}(d,k)
on a general curve is not governed by the Brill Nother number.


Related
to the nonemptiness problem of BN_{C}(d,k),
Mercat's conjecture gives a uniform bound for the number of
independent sections on a rank 2 vector bundle. In this talk, I
will explain the link between rank 2 BrillNoether theory and the
Koszul geometry of the curve C
and
show how the maximal rank conjecture implies, in some cases,
Mercat's conjecture. Using these ideas, we are able to prove
Mercat's conjecture for a bounded genus. We also show that, for
k=4,
there exist BrillNoether general curves in any genus > 11 for
which Mercat's conjecture fails. This is joint work with G.
Farkas.

16:30

Peter
Newstead (Liverpool)

BrillNoether
theory for
vector bundles with fixed determinant

In
the BrillNoether theory of stable vector bundles, there is a
natural definition of expected dimension, although this cannot
always be attained when the rank is greater than 1. For fixed
determinant, there is an obvious lower bound on the dimension of
components given by subtracting g
from
the bound for variable determinant. However, for rank 2 and
canonical determinant, the bound cannot be attained and there is
an alternative natural definition, which is attained in most
(probably all) cases. Very recently Osserman has obtained results
for other determinants. The main purpose of the talk is to show
that in some cases, Osserman's bounds can be attained. The
natural context is that of coherent systems. This is joint work
with Grzegorczyk.



An
activity of SFB 647

"RaumZeitMaterie"

www.raumzeitmaterie.de

Organized
by
A.
Schmitt

Poster

Impressum

Venue

KonradZuseZentrum
für Informationstechnik Berlin Takustrasse 7 D14195
BerlinDahlem Germany

www.zib.de
