134 research outputs found
Discriminant loci of ample and spanned line bundles
Let be a triplet where is an irreducible smooth complex
projective variety, is an ample and spanned line bundle on and
spans . The discriminant locus \Cal D(X,V) \subset
|V| is the algebraic subset of singular elements of . We study the
components of \Cal D(X,V) in connection with the jumping sets of ,
generalizing the classical biduality theorem. We also deal with the degree of
the discriminant (codegree of ) giving some bounds on it and
classifying curves and surfaces of codegree 2 and 3. We exclude the possibility
for the codegree to be 1. Significant examples are provided
Higher Order Bad Loci
Zero-schemes on smooth complex projective varieties, forcing all elements of
ample and free linear systems to be reducible are studied. Relationships among
the minimal length of such zero-schemes, the positivity of the line bundle
associated with the linear system, and the dimension of the variety are
established. A generalization to higher dimension subschemes is studied in the
last section.Comment: 23 pages. Refereed version, to appear in JPA
Ample subvarieties and rationally connected fibrations
Under some positivity assumptions, extension properties of rationally
connected fibrations from a submanifold to its ambient variety are studied.
Given a family of rational curves on a complex projective manifold X inducing a
covering family on a submanifold Y with ample normal bundle in X, the main
results relate, under suitable conditions, the associated rational connected
fiber structures on X and on Y. Applications of these results include an
extension theorem for Mori contractions of fiber type and a classification
theorem in the case Y has a structure of projective bundle or quadric
fibration.Comment: 27 pages; v2: minor changes and corrections following the referee's
comments. v3: few typos corrected. To appear in Math. An
Peculiar Loci of Ample and Spanned Line Bundles
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.Comment: to appear in Manuscripta Mathematic
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