Let
C
be a smooth irreducible complex projective curve of genus
g
≥
2
and
M
the moduli space of stable vector bundles on
C
of rank
n
and degree
d
with
gcd
(
n
,
d
)
=
1
. A generalised Picard sheaf ...is the direct image on
M
of the tensor product of a universal bundle on
M
×
C
by the pullback of a vector bundle
E
0
on
C
. In this paper, we investigate the stability of generalised Picard sheaves and, in the case where these are locally free, their deformations. When
g
≥
3
,
n
≥
2
(with some additional restrictions for
g
=
3
,
4
) and the rank and degree of
E
0
are coprime, this leads to the construction of a fine moduli space for deformations of Picard bundles.
In this paper, we construct some examples of rank-2 Brill–Noether loci with “unexpected” properties on general curves. The key examples are in genus 6, but we also have interesting examples in genus ...5 and in higher genus. We relate some of our results to the recent proof of Mercat’s conjecture in rank 2 by Bakker and Farkas.
Higher rank Brill-Noether theory is completely known for curves of genus ≤ 3. In this paper, we investigate the theory for curves of genus 4. Some of our results apply to curves of arbitrary genus.
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