Interpolation Problems for Vector Bundles on Algebraic Curves

In this paper we study meromorphic maps between vector bundles on a Riemann surface. We are mainly interested in stable vector bundles. For a huge number of numerical data we prove the existence of a meromorphic map between two vector bundles with a prescribed number of zeroes and a prescribed number of poles.     

Remarks on Normal Generation of Special Line Bundles on Algebraic Curves

Let L be a very ample line bundle with h ¹(L) ≥2 on a curve of genus g. We prove that L is normally generated if deg(L) ≥2g ? 1 ? 4h ¹(L) for large enough genus g.     

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.     

On Semistable Mori Conic Bundles

We study Fano-Mori contractions from threefolds to surfaces satisfying the semistability assumption. A number of examples are constructed. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 14, Algebra, 2004.     

Generalized Adjunction of k-Very Ample Vector Bundles on Algebraic Surfaces

We study the k-very ampleness of the adjoint bundle K _S+det E associated to a k-very ample vector bundle E on an algebraic surface S. We extend the results of Beltrametti and Sommese to the vector bundle case and give the classification of the pairs (S, E) such that the preservation of k-very ampleness fails.     

Invariant Curves by Vector Fields on Algebraic Varieties

If C is a reduced curve which is invariant by a one-dimensionalfoliation F of degree d_F on the projective space then it isshown that d_F–1+a is a bound for the quotient of the twocoefficients of the Hilbert–Samuel polynomial for C, wherea is an integer obtained from a concrete problem of imposingsingularities to projective hypersurfaces, and so a bound isobtained for the degree of C when it is a complete intersection.Concrete values of a can be derived for several interestingapplications. The results are presented in the form of intersection-theoreticalinequalities for one-dimensional foliations on arbitrary smoothalgebraic varieties.     

Construction of Rank 2 Semistable Torsion Free Sheaves Which Are Not Limit of Vector Bundles

We show the following theorem of compensated compactness type: Ifu _n ⇁u weakly in the spaceH ^1,p (Ω, ℝ ^k ) and if also in the sense of distributions then ∂_α(∣∇u∣ ^p-2∂_α u)=0. This result has applications in the partial regularity theory ofp-stationary mappings Ω→S ^k −1.     

Vector Bundles Near Negative Curves: Moduli and Local Euler Characteristic

We study moduli of vector bundles on a two-dimensional neighbourhood Z _k of an irreducible curve ? ? ?¹ with ?² = ?k and give an explicit construction of their moduli stacks. For the case of instanton bundles, we stratify the stacks and construct moduli spaces. We give sharp bounds for the local holomorphic Euler characteristic of bundles on Z _k and prove existence of families of bundles with prescribed numerical invariants. Our numerical calculations are performed using a Macaulay 2 algorithm, which is available for download at http://www.maths.ed.ac.uk/~s0571100/Instanton/.     

Generalized Adjunction of k‐Very Ample Vector Bundles on Algebraic Surfaces II

We study the k‐very ampleness of the adjoint bundle K_S + det E associated to a (k — 1)‐very ample vector bundle E with degree greater than or equal to 4k + 5 on an algebraic surface S. We classify polarized surfaces (S, E) which the k‐very ampleness of K_S + det E fails.     

A Universal Construction for Moduli Spaces of Decorated Vector Bundles over Curves

Let $X$ be a smooth projective curve over the field of complex numbers, and fix a homogeneous representation $\rho\colon \mathop{\rm GL}(r)\rightarrow \mathop{\rm GL}(V)$. Then one can associate to every vector bundle $E$ of rank $r$ over $X$ a vector bundle $E_\rho$ with fibre $V$. We would like to study triples $(E,L,\phi)$ where $E$ is a vector bundle of rank $r$ over $X$, $L$ is a line bundle over $X$, and $\phi\colon E_\rho\rightarrow L$ is a nontrivial homomorphism. This setup comprises well known objects such as framed vector bundles, Higgs bundles, and conic bundles. In this paper, we will formulate a general (parameter dependent) semistability concept for such triples, which generalizes the classical Hilbert--Mumford criterion, and we establish the existence of moduli spaces for the semistable objects. In the examples which have been studied so far, our semistability concept reproduces the known ones. Therefore, our results give in particular a unified construction for many moduli spaces considered in the literature.     

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.     

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.     

代数曲线上的Hadamard定理

本文在紧Riemann曲面上引入了拟距离函数和圆环域的概念,并给出了这种圆环域上的Hadamard定理.     

Vector Bundles on Certain Non-algebraic Surfaces

It is well-known that every holomorphic vector bundle is filtrable on a projective algebraic