Exact Sequences of Semistable Vector Bundles on Algebraic Curves

Let X be a smooth complex projective curve of genus g 1. Ifg 2, then assume further that X is either bielliptic or withgeneral moduli. Fix integers r, s, a, b with r > 1, s >1 and as br. Here we prove the existence of an exact sequence [formula] of semistable vector bundles on X with rk(H) = r, rk(Q) = s,deg(H) = a and deg(Q) = b. 1991 Mathematics Subject Classification14H60.     

Blowing-Ups Describing the Polarization Change of Moduli Schemes of Semistable Sheaves of General Rank

Let H and H′ be two ample line bundles over a smooth projective surface X, and M(H) (resp. M(H′)) the coarse moduli scheme of H-semistable (resp. H′-semistable) sheaves of fixed type (r, c ₁, c ₂). We construct a sequence of blowing-ups which describes how M(H) differs from M(H′) when r is arbitrary and the wall of fixed type separating H and H′ is not necessarily good. Means we here utilize are elementary transforms and the notion of a sheaf with flag.     

Rank 2 Ample Vector Bundles on Some Smooth Rational Surfaces

Several classification results for ample vector bundles of rank 2 on Hirzebruch surfaces, and on Del Pezzo surfaces, are obtained. In particular, we classify rank 2 ample vector bundles with _c2 less than seven on Hirzebruch surfaces, and with _c2 less than four on Del Pezzo surfaces.     

Restriction of Stable Rank Two Vector Bundles in Arbitrary Characteristic

Let X be a smooth variety defined over an algebraically closed field of arbitrary characteristic and 𝒪 _X (H) be a very ample line bundle on X. We show that for a semistable X-bundle E of rank two, there exists an integer m depending only on Δ(E) · H ^dim(X)?2 and H ^dim(X) such that the restriction of E to a general divisor in |mH| is again semistable. As corollaries, we obtain boundedness results, and weak versions of Bogomolov's Theorem and Kodaira's vanishing theorem for surfaces in arbitrary characteristic.     

二维向量丛的最大子线丛

本文中我们利用 Ａ．Ｂｅｒｔｒａｍ和 Ｂ． Ｆｅｉｂｅｒｇ证明的在 ｇ＝５的当 Ｓ（Ｅ）＜２时的一般代数曲线上二维特殊稳定向量丛的存在定理作为反例，说明进一步的Ｍａｒｕｙａｍａ猜想和Ａｒｒｏｎｄｏ－Ｓｏｌｓ猜想在ｇ＝５的一般代数曲线上均不能成立．     

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.     

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 .     

On the Motive of Moduli Spaces of Rank Two Vector Bundles over a Curve

We study the motive of the moduli spaces of rank two vector bundles on a curve. In the smooth case we obtain the Hodge numbers, intermediate Jacobians and number of points over a finite field as corollaries. In the singular case our computations yield the Poincaré–Hodge polynomial of Seshadri's smooth model.     

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.     

Test Elements for Free Solvable Groups of Rank 2

It is proved that a free solvable group of rank 2 of degree 3 contains test elements. Thereby we solve the Fine$ndash;Shpilrain problem posed in [9, Question 14.88].     

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.     

A Family of Finite Simple Groups Which Are 2-Recognizable by Their Elements Order

Abstract

Let G be a finite group and ω(G) the set of all orders of elements in G. Denote by h(ω(G)) the number of isomorphism classes of finite groups H satisfying ω(H) = ω(G), and put h(G) = h(ω(G)). A group G is called k-recognizable if h(G) = k < ∞ , otherwise G is called non-recognizable. In the present article we will show that the simple groups PSL(3, q), where q ≡ ±2(mod 5) and (6, (q ? 1)/2) = 2, are 2-recognizable. Therefore if q is a prime power and q ≡ 17, 33, 53 or 57 (mod 60), then the groups PSL(3, q) are 2-recognizable. Hence proving the existing of an infinite families of 2-recognizable simple groups.     

The Problem of the Classification of the Nilpotent Class 2 Torsion Free Groups up to Geometric Equivalence

In this article, we consider the problem of classification of the nilpotent class 2 finitely generated torsion free groups up to geometric equivalence. By a very easy technique it is proved that this problem is equivalent to the problem of classification of the complete in the Maltsev sense nilpotent torsion free finite rank groups up to isomorphism. This result leads to better understanding of the complexity of the problem of the classification of the quasi-varieties of the nilpotent class 2 groups. It is well known that the variety of the nilpotent class s groups is Noetherian for every s ∈ ?. So the problem of the classification of the quasi-varieties generated even by a single nilpotent class 2 finitely generated torsion free group is equivalent to the problem of classification of the complete in the Maltsev sense nilpotent torsion free finite rank groups up to isomorphism.     

New Component of the Moduli Space M(2;0,3) of Stable Vector Bundles on the Double Space P 3 of Index Two

We study the moduli scheme M(2;0,n) of rank-2 stable vector bundles with Chern classes c ₁=0, c ₂=n, on the Fano threefold X – the double space P ³ of index two. New component of this scheme is produced via the Serre construction using certain families of curves on X. In particular, we show that the Abel–Jacobi map :HJ(X) of any irreducible component H of the Hilbert scheme of X containing smooth elliptic quintics on X into the intermediate Jacobian J(X) of X factors by Stein through the quasi-finite (probably birational) map g:M of (an open part of) a component M of the scheme M(2;0,3) to a translate of the theta-divisor of J(X).