A note on the variety of Secant Loci

We determine non hyper elliptic curves of genus $g(C)\ge 9$, such that for some very ample line bundle on them and for some integers d and r with some prescribed assumptions, the dimension of secant loci, attains one less than its maximum value. Then we proceed to generalize and extend a problem of M. Coppens to Secant Loci.