Abstract
AbstractWe provide a spectral sequence computing the extension groups of tautological bundles on symmetric products of curves. One main consequence is that, if $$E\ne \mathcal O_X$$
E
≠
O
X
is simple, then the natural map $${{\,\mathrm{\mathsf {Ext}}\,}}^1(E,E)\rightarrow {{\,\mathrm{\mathsf {Ext}}\,}}^1(E^{[n]},E^{[n]})$$
Ext
1
(
E
,
E
)
→
Ext
1
(
E
[
n
]
,
E
[
n
]
)
is injective for every n. Along with previous results, this implies that $$E\mapsto E^{[n]}$$
E
↦
E
[
n
]
defines an embedding of the moduli space of stable bundles of slope $$\mu \notin [-1,n-1]$$
μ
∉
[
-
1
,
n
-
1
]
on the curve X into the moduli space of stable bundles on the symmetric product $$X^{(n)}$$
X
(
n
)
. The image of this embedding is, in most cases, contained in the singular locus. For line bundles on a non-hyperelliptic curve, the embedding identifies the Brill–Noether loci of X with the loci in the moduli space of stable bundles on $$X^{(n)}$$
X
(
n
)
where the dimension of the tangent space jumps. We also prove that $$E^{[n]}$$
E
[
n
]
is simple if E is simple.
Funder
Gottfried Wilhelm Leibniz Universität Hannover
Publisher
Springer Science and Business Media LLC
Subject
Geometry and Topology,Algebra and Number Theory
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