  Given a vector bundle $V$ over a curve $X$, we define and study a surjective
rational map $\mathrm{Hilb}^d (\mathbb{P} V ) - \mathrm{Quot}^{0, d} ( V^* )$
generalising the natural map $\mathrm{Sym}^d X \to \mathrm{Quot}^{0, d}
({\mathcal O}_X)$. 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 Brill--Noether locus $B^r_{r,
d}$ at a suitable bundle $E$ with $h^0 (E) = r+n$. This gives a generalisation
of the Riemann--Kempf singularity theorem. As a corollary, we show that the
$n$th secant variety of the rank one locus of $\mathbb{P} \mathrm{End} E$ is
contained in the tangent cone.
