Fourier-Mukai transform of vector bundles on surfaces to Hilbert scheme

Let $S$ be an irreducible smooth projective surface defined over an algebraically closed field $k$. For a positive integer $d$, let ${\rm Hilb}^d(S)$ be the Hilbert scheme parametrizing the zero-dimensional subschemes of $S$ of length $d$. For a vector bundle $E$ on $S$, let ${\mathcal H}(E)\, \longrightarrow\, {\rm Hilb}^d(S)$ be its Fourier--Mukai transform constructed using the structure sheaf of the universal subscheme of $S\times {\rm Hilb}^d(S)$ as the kernel. We prove that two vector bundles $E$ and $F$ on $S$ are isomorphic if the vector bundles ${\mathcal H}(E)$ and ${\mathcal H}(F)$ are isomorphic.Comment: To appear in JRM

Reconstructing vector bundles on curves from their direct image on symmetric powers

Let $C$ be an irreducible smooth complex projective curve, and let $E$ be an algebraic vector bundle of rank $r$ on $C$. Associated to $E$, there are vector bundles ${\mathcal F}_n(E)$ of rank $nr$ on $S^n(C)$, where $S^n(C)$ is $$n$-th symmetric power of$C$. We prove the following: Let$E_1$and$E_2$be two semistable vector bundles on$C$, with${\rm genus}(C)\, \geq\, 2$. If${\mathcal F}_n(E_1)\,= \, {\mathcal F}_n(E_2)$for a fixed$n$, then$E_1 \,=\, E_2$

Hyperplane sections of projective bundle associated to the tangent bundle of P2.\mathbb{P}^2.

In this note we give a complete description of all the hyperplane section of the projective bundle associated to the tangent bundle of $\mathbb{P}^2$ under its natural embedding in $\mathbb{P}^7.$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 $\mathcal{H}$ 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 $X$ be the wonderful compactification of a complex symmetric space $G/H$ of minimal rank. For a point $x\,\in\, G$, denote by $Z$ be the closure of $BxH/H$ in $X$, where $B$ is a Borel subgroup of $G$. The universal cover of $G$ is denoted by $\widetilde{G}$. Given a $\widetilde{G}$ equivariant vector bundle $E$ on $X,$ we prove that $E$ is nef (respectively, ample) if and only if its restriction to $Z$ is nef (respectively, ample). Similarly, $E$ is trivial if and only if its restriction to $Z$ is so

Automorphisms of T‾\overline{T}

Let $\overline G$ be the wonderful compactification of a simple affine algebraic group $G$ defined over $\mathbb C$ such that its center is trivial and $G\not= {\rm PSL}(2,\mathbb{C})$. Take a maximal torus $T \subset G$, and denote by $\overline T$ its closure in $\overline G$. We prove that $T$ coincides with the connected component, containing the identity element, of the group of automorphisms of the variety $\overline T$.Comment: Final versio

On equivariant principal bundles over wonderful compactifications

Let $G$ be a simple algebraic group of adjoint type over $\mathbb C$, and let $M$ be the wonderful compactification of a symmetric space $G/H$. Take a $\widetilde G$--equivariant principal $R$--bundle $E$ on $M$, where $R$ is a complex reductive algebraic group and $\widetilde G$ is the universal cover of $G$. If the action of the isotropy group $\widetilde H$ on the fiber of $E$ at the identity coset is irreducible, then we prove that $E$ is polystable with respect to any polarization on $M$. Further, for wonderful compactification of the quotient of $\text{PSL}(n,{\mathbb C})$, $n\,\neq\, 4$ (respectively, $\text{PSL}(2n,{\mathbb C})$, $n \geq 2$) 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 $T$ be a maximal torus of ${\rm PSL}(n, \mathbb C)$. For $n\,\geq\, 4$, we construct a smooth compactification of ${\rm PSL}(n, \mathbb C)/T$ as a geometric invariant theoretic quotient of the wonderful compactification $\overline{{\rm PSL}(n, \mathbb C)}$ for a suitable choice of $T$--linearized ample line bundle on $\overline{{\rm PSL}(n, \mathbb C)}$. We also prove that the connected component, containing the identity element, of the automorphism group of this compactification of ${\rm PSL}(n, \mathbb C)/T$ is ${\rm PSL}(n, \mathbb C)$ 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 $X$ such that the restriction of $E$ to every closed curve $C \,\subset\, X$ is ample, under what conditions $E$ is ample? We first consider the case of an abelian variety $X$. If $E$ is a line bundle on $X$, then we answer the question in the affirmative. When $E$ is of higher rank, we show that the answer is affirmative under some conditions on $E$. We then study the case of $X \,=\, G/P$, where $G$ is a reductive complex affine algebraic group, and $P$ is a parabolic subgroup of $G$. In this case, we show that the answer to our question is affirmative if $E$ is $T$--equivariant, where $T\, \subset\, P$ is a fixed maximal torus. Finally, we compute the Seshadri constant for such vector bundles defined on $G/P$.Comment: Final version; 11 pages; to appear in International Journal of Mathematic