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Abstract
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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 $\dim V^{d-1}_{d}(L)=d-1-\gamma$; then the scheme $V^{\gamma+2}_{\gamma +3}(L)$, if non-empty, 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^{d-1}_{d}(L)$, can not attain its maximum value, i.e. $d-2$. 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 well-known result of Montserrat Teixidor to secant loci', our results answer a question proposed and left unanswered recently by Marian Aprodu and Edoardo Sernesi.
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