Abstract

For a very ample line bundle $L$ on a smooth projective algebraic curve $C$, we prove that under some circumstances on positive integers $\gamma, d$,
if one had $\dim V^{d1}_{d}(L)=d1\gamma$; then the scheme $V^{\gamma+2}_{\gamma +3}(L)$, if nonempty, would be $2$dimensional.
Furthermore; for a $2$very ample line bundle $L$ on $C$ and an integer $d$, $4\leq d\leq h^0(L)2$, it would be proved that the dimension of
the scheme $ V^{d1}_{d}(L)$, can not attain its maximum value, i.e. $d2$. This will be used to prove irreduciblity of the
highest secant loci of $L$, namely $V^{h^0(L)2}_{h^0(L)1}(L)$. Then we discuss on the existence of very ample line bundles having reducible highest secant loci on $k$gonal curves. Extending a wellknown result of Montserrat Teixidor to secant loci', our results answer a question proposed and left unanswered recently by Marian Aprodu and Edoardo Sernesi.
