Narrow Results

Let X be a curve of genus g. A coherent system on X consists of a pair (E,V), where E is an algebraic vector bundle over X of rank n and degree d and V is a subspace of dimension k of the space of sections of E. The stability of the coherent system depends on a parameter α. We study the variation of the moduli space of coherent systems when we move the parameter. As an application, we analyze the cases k=1,2,3 and n=2 explicitly. For small values of α, the moduli spaces of coherent systems are related to the Brill–Noether loci, the subschemes of the moduli spaces of stable bundles consisting of those bundles with at least a prescribed number of independent sections. The study of coherent systems is applied to find the dimension, prove the irreducibility, and in some cases calculate the Picard groups of the Brill–Noether loci with k≤3.

Let C be an algebraic curve of genus g ≥ 2. A coherent system on C consists of a pair (E,V), where E is an algebraic vector bundle over C of rank n and degree d and V is a subspace of dimension k of the space of sections of E. The stability of the coherent system depends on a parameter α. We study the geometry of the moduli space of coherent systems for different values of α when k ≤ n and the variation of the moduli spaces when we vary α. As a consequence, for sufficiently large α, we compute the Picard groups and the first and second homotopy groups of the moduli spaces of coherent systems in almost all cases, describe the moduli space for the case k = n - 1 explicitly, and give the Poincaré polynomials for the case k = n - 2. In an appendix, we describe the geometry of the "flips" which take place at critical values of α in the simplest case, and include a proof of the existence of universal families of coherent systems when GCD(n,d,k) = 1.

Let X be a general smooth projective algebraic curve of genus g ≥ 2 over ℂ. We prove that the moduli space G(α:n,d,k) of α-stable coherent systems of type (n,d,k) over X is empty if k > n and the Brill–Noether number β := β(n,d,n + 1) = β(1,d,n + 1) = g - (n + 1)(n - d + g) < 0. Moreover, if 0 ≤ β < g or β = g, n ∤g and for some α > 0, G(α : n,d,k) ≠ ∅ then G(α : n,d,k) ≠ ∅ for all α > 0 and G(α : n,d,k) = G(α′ : n,d,k) for all α,α′ > 0 and the generic element is generated. In particular, G(α : n,d,n + 1) ≠ ∅ if 0 ≤ β ≤ g and α > 0. Moreover, if β > 0 G(α : n,d,n + 1) is smooth and irreducible of dimension β(1,d,n + 1). We define a dual span of a generically generated coherent system. We assume d < g + n₁ ≤ g + n₂ and prove that for all α > 0, G(α : n₁,d, n₁ + n₂) ≠ ∅ if and only if G(α : n₂,d, n₁ + n₂) ≠ ∅. For g = 2, we describe G(α : 2,d,k) for k > n.

This paper contains results on stable bundles of rank 2 with space of sections of dimension 4 on a smooth irreducible projective algebraic curve C. There is a known lower bound on the degree for the existence of such bundles; the main result of the paper is a geometric criterion for this bound to be attained. For a general curve C of genus 10, we show that the bound cannot be attained, but that there exist Petri curves of this genus for which the bound is sharp. We interpret the main results for various curves and in terms of Clifford indices and coherent systems. The results can also be expressed in terms of Koszul cohomology and the methods provide a useful tool for the study of the geometry of the moduli space of curves.

When k < n, we study the coherent systems that come from a BGN extension in which the quotient bundle is strictly semistable. In this case we describe a stratification of the moduli space of coherent systems. We also describe the strata as complements of determinantal varieties and we prove that these are irreducible and smooth. These descriptions allow us to compute the Hodge polynomials of this moduli space in some cases. In particular, we give explicit computations for the cases in which (n, d, k) = (3, d, 1) and d is even, obtaining from them the usual Poincaré polynomials.

Over the past 20 years, a great deal of work has been done on the moduli spaces of coherent systems on algebraic curves. Until recently, however, there has been very little work on the fixed determinant case, except for the special case of rank 2 and canonical determinant. This situation has changed due to two papers of Osserman, who has obtained lower bounds for the dimensions of the fixed determinant moduli spaces in some cases. Our object in this paper is to show that some of Osserman's bounds are sharp.

Let C be a smooth irreducible projective curve of genus g and L a line bundle of degree d generated by a linear subspace V of H⁰(L) of dimension n + 1. We prove a conjecture of D. C. Butler on the semistability of the kernel of the evaluation map V ⊗ 𝒪_C → L and obtain new results on the stability of this kernel. The natural context for this problem is the theory of coherent systems on curves and our techniques involve wall crossing formulae in this theory.

The aim of this paper is to generalize the $m$-Segre invariant for vector bundles to coherent systems. Let $X$ be a non-singular irreducible complex projective curve of genus $g\ge 0$ and $G(\alpha ;n,d,k)$ be the moduli space of $\alpha$-stable coherent systems of type $(n,d,k)$ on $X$. For any pair of integers $(m,t)$ with $0<m<n$, $0\le t\le k$ we define the $(m,t)$-Segre invariant, and prove that it defines a lower semicontinuous function on the families of coherent systems. Thus, the $(m,t)$-Segre invariant induces a stratification of the moduli space $G(\alpha ;n,d,k)$ into locally closed subvarieties $G(\alpha ;n,d,k;m,t;s)$ according to the value $s$ of the function. We determine an above bound for the $(m,t)$-Segre invariant and compute a bound for the dimension of the different strata $G(\alpha ;n,d,k;m,t;s)$. Moreover, we give some conditions under which the different strata are nonempty. To prove the above results, we introduce the notion of coherent systems of subtype $(a)$.

Let $(C,\underline{w})$ be a polarized nodal reducible curve. In this paper, we consider coherent systems of type $(r,d,k)$ on $C$ with $k<r$. We prove that the moduli spaces of $\underline{w}$-stable coherent systems stabilize for large $\alpha$ and we generalize several results known for the irreducible case when we choose a good polarization. Then, we study in detail the components of moduli spaces containing coherent systems arising from locally free sheaves.

Three architectures of current interest—the pyramid, the linear array and the n-cube—are examined in relation to their ability to perform iconic and symbolic image processing. For each architecture an appropriate problem is identified and a matching implementation selected. An assessment of the characteristics of each implementation is given, together with an estimation of the performance on a number of fundamental operations. The relative merits of the designs are discussed but it is concluded that the success of a particular architecture depends crucially upon the type of application which is addressed, and that therefore no universally applicable ranking can be derived in the absence of a comprehensive benchmarking exercise.

In this paper, we consider the component importance analysis of coherent systems subject to common-cause failures. The purpose of component importance analysis is to obtain information regarding a component's contribution or importance to the system reliability. The results from the component importance analysis are key contributors to the system design, tuning, and maintenance activities. Various measures have been proposed for the component importance analysis, but little work has been done to compare their performance. In this research, we investigate and compare a set of nine existing importance measures and select the most informative and appropriate one for guiding the system maintenance activity. An important concern in the traditional fault tree reliability analysis, common-cause failures, is also considered in the component importance analysis. An example is designed and analyzed to show the selection process and to illustrate our efficient method for considering the effects of common-cause failures in the component importance analysis.