744 research outputs found
Ample Vector Bundles and Branched Coverings, II
In continuation of our work in Comm. in Algebra, vol. 28 (2000), we study
ramified coverings of projective manifolds, in particular over Fano manifolds
and investigate positivity properties of the associated vector bundle. Moreover
we study the topology of low degree coverings and the structure of the
ramification divisor.Comment: LaTeX, 21 page
Numerical homotopies to compute generic points on positive dimensional algebraic sets
Many applications modeled by polynomial systems have positive dimensional
solution components (e.g., the path synthesis problems for four-bar mechanisms)
that are challenging to compute numerically by homotopy continuation methods. A
procedure of A. Sommese and C. Wampler consists in slicing the components with
linear subspaces in general position to obtain generic points of the components
as the isolated solutions of an auxiliary system. Since this requires the
solution of a number of larger overdetermined systems, the procedure is
computationally expensive and also wasteful because many solution paths
diverge. In this article an embedding of the original polynomial system is
presented, which leads to a sequence of homotopies, with solution paths leading
to generic points of all components as the isolated solutions of an auxiliary
system. The new procedure significantly reduces the number of paths to
solutions that need to be followed. This approach has been implemented and
applied to various polynomial systems, such as the cyclic n-roots problem
Line bundles for which a projectivized jet bundle is a product
We characterize the triples (X,L,H), consisting of holomorphic line bundles L
and H on a complex projective manifold X, such that for some positive integer
k, the k-th holomorphic jet bundle of L, J_k(L), is isomorphic to a direct sum
H+...+H. Given the geometrical constrains imposed by a projectivized line
bundle being a product of the base and a projective space it is natural to
expect that this would happen only under very rare circumstances. It is shown,
in fact, that X is either an Abelian variety or projective space. In the former
case L\cong H is any line bundle of Chern class zero. In the later case for k a
positive integer, L=O_{P^n}(q) with J_k(L)=H+...+H if and only if
H=O_{P^n}(q-k) and either q\ge k or q\le -1.Comment: Latex file, 5 page
Kodaira Dimension of Subvarieties
In this article we study how the birational geometry of a normal projective
variety is influenced by a normal subvariety One of the most
basic examples in this context is provided by the following situation. Let
be a surjective holomorphic map with connected fibers between
compact connected complex manifolds. It is well known that given a general
fiber of we have This article grew
out of the realization that this result should be true with replaced
by the codimension \cod_X A for a pair consisting of a normal
subvariety of a compact normal variety under weak semipositivity
conditions on the normal sheaf of and the weak singularity condition
\cod_A (A\cap\sing X)\ge 2. We shall now state our main results in the
special case of a submanifold in a projective manifold and we also
simplify the semipositivity notion
An intrinsic homotopy for intersecting algebraic varieties
Recently we developed a diagonal homotopy method to compute a numerical
representation of all positive dimensional components in the intersection of
two irreducible algebraic sets. In this paper, we rewrite this diagonal
homotopy in intrinsic coordinates, which reduces the number of variables,
typically in half. This has the potential to save a significant amount of
computation, especially in the iterative solving portion of the homotopy path
tracker. There numerical experiments all show a speedup of about a factor two
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