2004/09/26 by Tommaso de Fernex, de Fernex, Tommaso, Antonio Lanteri +1
Computer Science · Mathematics · #14C20 (primary) #14J25 (secondary) #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Combinatorics #FOS: Mathematics #Line bundle #Locus (genetics) #Mathematics #Morphism #Polynomial and algebraic computation #Pure mathematics #Vertex (graph theory) #math.AG #msc:14C20 #msc:14J25
paper · pdf · doi:10.48550/arxiv.math/0409499
13 pages; to appear in Adv. Geom
arxiv created 2004/09/26 · openalex publication_date 2004/09/26 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
The bad locus of a base-point free linear system L on a normal complex projective variety X is defined as the subset B(L) of 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 the dimension of X is at least 2, and that 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.