  For a smooth projective curve, the cycles of e-secant k-planes are among the
most studied objects in classical enumerative geometry and there are well-known
formulas due to Castelnuovo, Cayley and MacDonald concerning them. Despite
various attempts, surprisingly little is known about the enumerative validity
of such formulas. The aim of this paper is to completely clarify this problem
in the case of the generic curve C of given genus. Using degeneration
techniques and a few facts about the birational geometry of moduli spaces of
stable pointed curves we determine precisely under which conditions the cycle
of e-secant k-planes in non-empty and we compute its dimension. We also
precisely determine the dimension of the variety of linear series on C carrying
e-secant k-planes. In a different direction, in the last part of the paper we
study the distribution of ramification points of the powers of a line bundle on
C having prescribed ramification at a given point.
