Reconstructing vector bundles on curves from their direct image on symmetric powers
Abstract
Let
C
be an irreducible smooth complex projective curve, and let
E
be an algebraic vector bundle of rank
r
on
C
. Associated to
E
, there are vector bundles
F
n
(E)
of rank
nr
on
S
n
(C)
, where
S
n
(C)
is
n
−thsymmetricpowerof
C
.Weprovethefollowing:Let
E_1
and
E_2
betwosemistablevectorbundleson
C
,with
{\rm genus}(C)\, \geq\, 2
.If
{\mathcal F}_n(E_1)\,= \, {\mathcal F}_n(E_2)
forafixed
n
,then
E_1 \,=\, E_2$.