A note on the variety of Secant Loci

TL;DR

This paper investigates the dimensions of secant loci on certain algebraic curves, identifying specific conditions under which these dimensions are just below their maximum, and extends existing problems in the field.

Contribution

It characterizes non-hyperelliptic curves of genus at least 9 with particular secant locus properties and generalizes a problem of M. Coppens related to secant loci.

Findings

Identifies conditions for secant loci dimension to be one less than maximum.

Extends the problem of M. Coppens to secant loci.

Provides new insights into the geometry of algebraic curves.

Abstract

We determine non hyper elliptic curves of genus $g(C)\geq 9$, 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.