Refined de Jonqui\`eres divisors and secant varieties on algebraic curves

The aim of this paper is to provide another perspective on secant varieties on algebraic curves by reformulating the problem in terms of refined de Jonqui`eres divisors, that is divisors on the curve with prescribed multiplicities and dimensions of their spaces of global sections. We are able to both recover some already known results and to obtain some new statements concerning the dimension theory of secant varieties. We do this via the study of the dimension theory of refined de Jonqui`eres divisors in some relevant cases and degeneration arguments.