The variety of bad zero-schemes

The locus of reduced bad zero-schemes, \frak{B}_0 \subset X^{[b_0]}, for a linear system |V| on a nonsingular, n-dimensional, algebraic variety X is defined. The pairs (X,V) for which \frak{B}_0 has the maximal dimension, nb_0-1, are characterized. For n=2, the non reduced bad zero-schemes are also discussed.