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.
Bad loci of free linear systems / T. De Fernex, A. Lanteri. - In: ADVANCES IN GEOMETRY. - ISSN 1615-715X. - 6:1(2006), pp. 93-107. [10.1515/ADVGEOM.2006.007]
Bad loci of free linear systems
A. Lanteri
2006
Abstract
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.Pubblicazioni consigliate
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