Bad loci of free linear systems

The bad locus of a free linear system L on a normal complex projective variety X is defined as the set B(L) \subset X of points that are not contained in any irreducible and reduced member of L. In this paper we provide a geometric description of such locus in terms of the morphism defined by L. In particular, assume that dim X = 2 and L is the complete linear system associated to an ample and spanned line bundle. It is known that in this case B(L) is empty unless X is a surface. Then we prove that, when the latter occurs, B(L) is not empty if and only if L defines a morphism onto a two dimensional cone, in which case B(L) is the inverse image of the vertex of the cone.