Moduli of symplectic instanton vector bundles of higher rank on projective space ℙ<Superscript>3</Superscript>

Symplectic instanton vector bundles on the projective space ℙ³ constitute a natural generalization of mathematical instantons of rank-2. We study the moduli space I _n;r of rank-2r symplectic instanton vector bundles on ℙ³ with r ≥ 2 and second Chern class n ≥ r, n ≡ r (mod 2). We introduce the notion of tame symplectic instantons by excluding a kind of pathological monads and show that the locus I _n;r ^* of tame symplectic instantons is irreducible and has the expected dimension, equal to 4n(r + 1) −r(2r + 1).     

Strange bundles onP 2 and elementary transformations

Summary Here we begin the study of moduli of vector bundles on a surfaceS with a fixed restriction to a divisorD. Here we stress the caseD≊P ¹. In this way we construct many families of stable rank-2 bundles onP ² with unbalanced general splitting type (in characteristicp>0).

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Riassunto Si comincia qui lo studio dei moduli di fibrati vettoriali su una superficieS con una assegnata restrizione ad un divisoreD (quasi sempre qui conD≊P ¹). In caratteristicap si ottengono così molte famiglie di fibrati stabili suP ² con ?strana? restrizione ad una retta generica.

     

On the restriction of vector bundles to good families of curves

Summary Here we study the restriction to suitable curves of certain vector bundles on projective varieties. We handle in particular: (a) the descent of a vector bundle for the contraction of an extremal ray on a smooth 3-fold (using Mori theory); (b) the stability or semistability (in any characteristic) of certain homogeneus vector bundles on Pⁿ and their restriction to suitable subvarieties.     

Existence of Indecomposable Rank Two Vector Bundles on Higher Dimensional Toric Varieties

In the mid 1970s, Hartshorne conjectured that, for all n > 7, any rank 2 vector bundles on ? ⁿ is a direct sum of line bundles. This conjecture remains still open. In this paper, we construct indecomposable rank two vector bundles on a large class of Fano toric varieties. Unfortunately, this class does not contain ? ⁿ .     

A Vector Bundle of Rank 2 on P1xP3

It is well known that the Horrocks–Mumford bundle F encodesa lot of very interesting geometric information. This is essentiallythe reason for the fact that much work has been done in orderto find other rank-2 bundles on P₄. The only nonsplit vectorbundles of rank 2 on P₄, known up to now, are twists of pullbacksof F by finite coverings f:P₄P₄. So it seems to be a naturalquestion to consider, instead of P₄, other Fano 4-folds. Itis the aim of this note to give an example of a rank-2 vectorbundle on P₁xP₃ and to show that it also admits very interestinggeometric properties.     

On globally generated vector bundles on projective spaces

A classification is given for globally generated vector bundles E of rank k on Pⁿ having first Chern class c₁(E)=2. In particular, we get that they split if k<n unless E is a twisted null-correlation bundle on P³. In view of the well-known correspondence between globally generated vector bundles and maps to Grassmannians, we obtain, as a corollary, a classification of double Veronese embeddings of Pⁿ into a Grassmannian G(k−1,N) of (k−1)-planes in P^N.     

VECTOR BUNDLES ON PROJECTIVE SPACES OVER A DISCRETE VALUATION RING

Let E be a vector bundle on P ⁿ _D, n ≥ 2, with D discrete valuation ring. Here we give a necessary and sufficient condition for the splitting of E as a direct sum of line bundles.     

Holomorphic Hermitian vector bundles over the Riemann sphere

We give a complete classification of isomorphism classes of all SU(2)-equivariant holomorphic Hermitian vector bundles on CP¹. We construct a canonical bijective correspondence between the isomorphism classes of SU(2)-equivariant holomorphic Hermitian vector bundles on CP¹ and the isomorphism classes of pairs ({Hn}_n∈Z,T), where each Hn is a finite dimensional Hilbert space with Hn=0 for all but finitely many n, and T is a linear operator on the direct sum _⊕n∈ZHn such that T(Hn)⊂Hn+2 for all n.     

Moduli of Vector Bundles on Curves in Positive Characteristics

Let X be a projective curve of genus 2 over an algebraically closed field of characteristic 2. The Frobenius map on X induces a rational map on the moduli scheme of rank-2 bundles. We show that up to isomorphism, there is only one (up to tensoring by an order two line bundle) semi-stable vector bundle of rank 2 (with determinant equal to a theta characteristic) whose Frobenius pull-back is not semi-stable. The indeterminacy of the Frobenius map at this point can be resolved by introducing Higgs bundles.     

Linear preservers of matrices of rank-2

Let T be a linear operator on the space of all m×n matrices over any field. we prove that if T maps rank-2 matrices to rank-2 matrices then there exist nonsingular matrices U and V such that either T(X)=UXV for all matrices X, or m=n and T(X)=UX^tV for all matrices X where X^t denotes the transpose of X.     

Ample and Spanned Rank-2 Vector Bundles with k-Jet Ample Determinant on Algebraic Surfaces

We investigate ample and spanned rank-2 vector bundles E with k-jet ample determinant on algebraic surfaces S. In paticular, we classify such polarized pairs (S,E) with small second Chern class. The case of c_2(E)=k+1 is a generalization of the result of Ballico and Lanteri.     

A note on 1-cycles on the moduli space of rank-2 bundles over a curve

Over a smooth complex projective curve of genus ≥3, we study 1-cycles on the moduli space of rank-2 stable vector bundles with fixed determinant of degree 1. We show the first Chow group of the moduli space is isomorphic to the zeroth Chow group of the curve.     

Geometric and cohomological properties of rank 2 vector bundles on curves

Here we study the postulation of curves embedded in a smooth quadric hypersurface of P ⁴ and P ⁵ and relate this subject to the study of cohomological properties of rank 2 spanned vector bundles on smooth projective curves. Received: May 25, 2000; in final form: January 3, 2001?Published online: May 29, 2002     

On globally generated vector bundles on projective spaces II

Extending the main result of Sierra and Ugaglia (2009)  [12], we classify globally generated vector bundles on Pⁿ${\mathbb{P}}^n$ with first Chern class equal to 3.     

Qregularity and an extension of the Evans-Griffiths criterion to vector bundles on quadrics

Here we define the concept of Qregularity for coherent sheaves on a smooth quadric hypersurface Q_n⊂Pⁿ⁺¹. In this setting we prove analogs of some classical properties. We compare the Qregularity of coherent sheaves on Q_n with the Castelnuovo-Mumford regularity of their extension by zero in Pⁿ⁺¹. We also classify the coherent sheaves with Qregularity −∞. We use our notion of Qregularity in order to prove an extension of the Evans-Griffiths criterion to vector bundles on quadrics. In particular, we get a new and simple proof of Knörrer’s characterization of ACM bundles.     

n-Dimensional Fano varieties of wild representation type

The aim of this work is to contribute to the classification of projective varieties according to their representation type, providing examples of n  -dimensional varieties of wild representation type, for arbitrary n?2$n?2$. More precisely, we prove that all Fano blow-ups of Pⁿ${\mathbb{P}}^n$ at a finite number of points are of wild representation type exhibiting families of dimension of order r²$r^2$ of simple (hence, indecomposable) ACM rank r   vector bundles for any r?n$r?n$. In the two dimensional case, the vector bundles that we construct are moreover Ulrich bundles and μ-stable with respect to certain ample divisor.     

Remarks on Ramanujam-Kawamata-Viehweg Vanishing Theorem

In this article we prove a general result on a nef vector bundle E on a projective manifold X of dimension n depending on the vector space H^n,n(X,E): It is also shown that H^n,n(X,E) = 0 for an indecomposable nef rank 2 vector bundles E on some specific type of n dimensional projective manifold X. The same vanishing shown to hold for indecomposable nef and big rank 2 vector bundles on any variety with trivial canonical bundle.     

A Note on vector bundles on Hirzebruch surfaces

In literature, two basic construction methods have been used to study vector bundles on a Hirzebruch surface. On the one hand, we have Serre?s method and elementary modifications, describing rank-2 bundles as extensions in a canonical way (Brînz?nescu and Stoia, 1984 [4], [5], Brînz?nescu, 1996 [6], Brosius, 1983 [7], Friedman, 1998 [9]), and on the other hand, we have a Beilinson-type spectral sequence (Buchdahl, 1987 [8]). Morally, the Beilinson spectral sequence indicates how to recover a bundle from the cohomology of its twists and from some sheaf morphisms (the differentials of the sequence). The aim of this Note is to show that the canonical extension of a rank-2 bundle can be deduced from the Beilinson spectral sequence of a suitable twist, called the normalization. In the final part we give a cohomological criterion for a topologically trivial vector bundle on a Hirzebruch surface to be trivial. To emphasize the relations and the differences between these two construction methods mentioned above, two different proofs are given.     

On Buchsbaum bundles on quadric hypersurfaces

Let E be an indecomposable rank two vector bundle on the projective space ℙ ⁿ , n ≥ 3, over an algebraically closed field of characteristic zero. It is well known that E is arithmetically Buchsbaum if and only if n = 3 and E is a null-correlation bundle. In the present paper we establish an analogous result for rank two indecomposable arithmetically Buchsbaum vector bundles on the smooth quadric hypersurface Q _n ⊂ ℙ ⁿ⁺¹, n ≥ 3. We give in fact a full classification and prove that n must be at most 5. As to k-Buchsbaum rank two vector bundles on Q ₃, k ≥ 2, we prove two boundedness results.     

M.P. Appell's Function and Vector Bundles of Rank 2 on Elliptic Curves

This paper is devoted to the function introduced by M.P. Appell in connection with decomposition of elliptic functions of the third kind into simple elements. We show that this function is related to global sections of rank-2 vector bundles on elliptic curves. We derive analogues of theta-identities for this function and prove the divisibility property for the action of the modular group, that should be considered as a replacement of the functional equation.