  The bad locus and the rude locus of an ample and base point free linear
system on a smooth complex projective variety are introduced and studied. The
bad locus is defined as the set of points that force divisors through them to
be reducible. The rude locus is defined as the set of points such that divisors
that are singular at them are forced to be reducible. The existence of a
nonmempty bad locus is shown to be exclusively a two dimensional phenomenon.
Polarized surfaces of small degree, or whose degree is the square of a prime,
with nonempty bad loci are completely classified. Several explicit examples are
offered to describe the variety of behaviors of the two loci.
