Varieties of secants on the generic curve

From CRCG-Wiki

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. We completely clarify this problem in the case of the generic curve C of given genus and determine precisely under which conditions the cycle of e-secant k-planes is non-empty and we compute its dimension.