On the varieties of special divisors

Based on a relation between the varieties W_d^r(C) of special divisors on a curve C and subloci of effective divisors on C imposing a suitable number of conditions on a certain linear series we develop a tool for the construction of irreducible components of W_d^r(C). Using this we discover new irreducible components of W_d^r(C), for a general k-gonal curve C of genus g, and in some cases we can identify the duals of these components in K_C − W_d^r(C) = W_d′^r′(C)(d′ = 2g − 2 − d, r′ = g − 1 − d + r).