Bad loci of free linear systems

Abstract

Abstract The bad locus of a free linear system ℒ on a normal complex projective variety X is defined as the set B( ℒ ) ⊆ 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.