Remarks on the Geometry of Secant Loci

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 $\dim V^{d-1}_{d}(L)=d-1-\gamma$, 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, $4\le d\le h^0(L)-2$, it will be proved that the dimension of the scheme $V^{d-1}_{d}(L)$ cannot 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.