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Given a vector bundle   V   of rank   r   over a curve   X  , we define and study a surjective rational mapHilb  d    (  P  V  )   ⤍   Quot   0  ,  d(   V  ∗   )generalising the natural mapSym  d   X  →   Quot   0  ,  d(   O  X   )    . We then give a generalisation of the geometric Riemann–Roch theorem to vector bundles of higher rank over   X  . We use this to give a geometric description of the tangent cone to the higher rank Brill–Noether locus    B   r  ,  d   k    at a suitable bundle   E  . This gives a generalisation of the Riemann–Kempf singularity theorem. As a corollary, if    k  =  r    andh  0    (  X  ,  E  )   =  r  +  n   , we show that the   n  th secant variety of the rank one locus of    P  End    E    is contained in the tangent cone to    B   r  ,  d   r   .

A Riemann–Kempf singularity theorem for higher rank Brill–Noether loci