9,936 research outputs found
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
The effect of points fattening on Hirzebruch surfaces
The purpose of this note is to study initial sequences of zero-dimensional
subschemes of Hirzebruch surfaces and classify subschemes whose initial
sequence has the minimal possible growth.Comment: 9 page
The bottleneck degree of algebraic varieties
A bottleneck of a smooth algebraic variety is a pair
of distinct points such that the Euclidean normal spaces at
and contain the line spanned by and . The narrowness of bottlenecks
is a fundamental complexity measure in the algebraic geometry of data. In this
paper we study the number of bottlenecks of affine and projective varieties,
which we call the bottleneck degree. The bottleneck degree is a measure of the
complexity of computing all bottlenecks of an algebraic variety, using for
example numerical homotopy methods. We show that the bottleneck degree is a
function of classical invariants such as Chern classes and polar classes. We
give the formula explicitly in low dimension and provide an algorithm to
compute it in the general case.Comment: Major revision. New introduction. Added some new illustrative lemmas
and figures. Added pseudocode for the algorithm to compute bottleneck degree.
Fixed some typo
Classifying smooth lattice polytopes via toric fibrations
We define Q-normal lattice polytopes. Natural examples of such polytopes are
Cayley sums of strictly combinatorially equivalent lattice polytopes, which
correspond to particularly nice toric fibrations, namely toric projective
bundles. In a recent paper Batyrev and Nill have suggested that there should be
a bound, N(d), such that every lattice polytope of degree d and dimension at
least N(d) decomposes as a Cayley sum. We give a sharp answer to this question
for smooth Q-normal polytopes. We show that any smooth Q-normal lattice
polytope P of dimension n and degree d is a Cayley sum of strictly
combinatorially equivalent polytopes if n is greater than or equal to 2d+1. The
proof relies on the study of the nef value morphism associated to the
corresponding toric embedding.Comment: Revised version, minor changes. To appear in Advances in Mat
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