مشخصات پژوهش

‎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.