2,318 research outputs found
Fourier-Mukai transform of vector bundles on surfaces to Hilbert scheme
Let be an irreducible smooth projective surface defined over an
algebraically closed field . For a positive integer , let be the Hilbert scheme parametrizing the zero-dimensional subschemes
of of length . For a vector bundle on , let be its Fourier--Mukai transform constructed
using the structure sheaf of the universal subscheme of as the kernel. We prove that two vector bundles and on
are isomorphic if the vector bundles and
are isomorphic.Comment: To appear in JRM
Reconstructing vector bundles on curves from their direct image on symmetric powers
Let be an irreducible smooth complex projective curve, and let be an
algebraic vector bundle of rank on . Associated to , there are vector
bundles of rank on , where is nCE_1E_2C{\rm genus}(C)\, \geq\, 2{\mathcal F}_n(E_1)\,= \, {\mathcal F}_n(E_2)nE_1
\,=\, E_2$
Hyperplane sections of projective bundle associated to the tangent bundle of
In this note we give a complete description of all the hyperplane section of
the projective bundle associated to the tangent bundle of under
its natural embedding in Comment: comments are welcome, revised version, some minor mistakes in the
previous version is correcte
Tangent bundle of \PP^2 and morphism from \PP^2 to \text{Gr}(2, \CC^{4})
In this note we study the image of \PP^2 in \text{Gr}(2, \CC^{4}) given
by tangent bundle of \PP^2. We show that there is component of
the Hibert scheme of surfaces in \text{Gr}(2, \CC^{4}) with no point of it
corresponds to a smooth surface.Comment: To appear in " proceedings of the march conference in Hyderabad.
Equivariant vector bundles on complete symmetric varieties of minimal rank
Let be the wonderful compactification of a complex symmetric space
of minimal rank. For a point , denote by be the closure of
in , where is a Borel subgroup of . The universal cover of
is denoted by . Given a equivariant vector
bundle on we prove that is nef (respectively, ample) if and only
if its restriction to is nef (respectively, ample). Similarly, is
trivial if and only if its restriction to is so
Automorphisms of
Let be the wonderful compactification of a simple affine
algebraic group defined over such that its center is trivial
and . Take a maximal torus , and
denote by its closure in . We prove that
coincides with the connected component, containing the identity element, of the
group of automorphisms of the variety .Comment: Final versio
On equivariant principal bundles over wonderful compactifications
Let be a simple algebraic group of adjoint type over , and let
be the wonderful compactification of a symmetric space . Take a
--equivariant principal --bundle on , where is a
complex reductive algebraic group and is the universal cover of
. If the action of the isotropy group on the fiber of at
the identity coset is irreducible, then we prove that is polystable with
respect to any polarization on . Further, for wonderful compactification of
the quotient of , (respectively,
, ) by the normalizer of the projective
orthogonal group (respectively, the projective symplectic group), we prove that
the tangent bundle is stable with respect to any polarization on the wonderful
compactification
On a smooth compactification of PSL(n, C)/T
Let be a maximal torus of . For ,
we construct a smooth compactification of as a
geometric invariant theoretic quotient of the wonderful compactification
for a suitable choice of --linearized
ample line bundle on . We also prove that
the connected component, containing the identity element, of the automorphism
group of this compactification of is itself
How not to share a set of secrets
This note analyses one of the existing space efficient secret sharing schemes
and suggests vulnerabilities in its design. We observe that the said algorithm
fails for certain choices of the set of secrets and there is no reason for
preferring this particular scheme over alternative schemes. The paper also
elaborates the adoption of a scheme proposed by Hugo Krawczyk as an extension
of Shamir's scheme, for a set of secrets. Such an implementation is space
optimal and works for all choices of secrets. We also propose two new methods
of attack which are valid under certain assumptions and observe that it is the
elimination of random values that facilitates these kinds of attacks.Comment: Added a new section demonstrating two new kinds of attack; 10 page
Positivity of vector bundles on homogeneous varieties
We study the following question: Given a vector bundle on a projective
variety such that the restriction of to every closed curve is ample, under what conditions is ample? We first consider
the case of an abelian variety . If is a line bundle on , then we
answer the question in the affirmative. When is of higher rank, we show
that the answer is affirmative under some conditions on . We then study the
case of , where is a reductive complex affine algebraic group,
and is a parabolic subgroup of . In this case, we show that the answer
to our question is affirmative if is --equivariant, where is a fixed maximal torus. Finally, we compute the Seshadri constant for such
vector bundles defined on .Comment: Final version; 11 pages; to appear in International Journal of
Mathematic
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