The Osculating cone to special Brill-Noether Loci

In this paper, we describe the osculating cone to Brill-Noether loci $W_d^0(C)$ at smooth isolated points of $W_d^1(C)$ for a smooth canonically embedded curve $C$ of even genus $g=2(d-1)$. In particular, we show that the canonical curve $C$ is a component of the osculating cone. The proof is based on techniques introduced by George Kempf in 1986.