P. E. Newstead

Topological properties of some spaces of stable bundles

OUR OBJECT in this paper is to study the topology of certain spaces which arise in connection with the classification of holomorphic vector bundles. Mumford [5, 6] has introduced the concept of a stable bundle over a compact Riemann surface X and has proved that the set of stable bundles of fixed rank and degree over X has a natural structure of non-singular quasi-projective algebraic variety (see [I0]). It is not difficult to deduce that the set of such bundles of fixed determinant has a similar structure, and that the set of projective bundles arising from such bundles has a structure of quasi-projective variety (possibly with singularities). Narasimhan and Seshadri [7, 8] have shown that the topological type of these varieties depends only on the genus of X.

Coherent systems and Brill-Noether theory

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.

On the cohomology ring of the moduli space of rank 2 vector bundles on a curve

Abstract Let N g be the moduli space of stable holomorphic vector bundles of rank 2 and fixed determinant of odd degree over a smooth complex projective curve of genus g. This paper gives a complete and very simple description of the rational cohomology ring H ∗ ( N g ) . A structural formula is proved for H ∗ ( N g ) , which was originally conjectured by Mumford. It is shown that the first relation in genus g between the standard generators satisfies a recurrence relation, first found by Zagier, and that the invariant subring for the mapping class group is a complete intersection ring. A Grobner basis is found for the ideal of invariant relations; this leads to a natural monomial basis for H ∗ ( N g ) .

COHERENT SYSTEMS ON ELLIPTIC CURVES

In this paper we consider coherent systems (E,V) on an elliptic curve which are α-stable with respect to some value of a parameter α. We show that the corresponding moduli spaces, if non-empty, are smooth and irreducible of the expected dimension. Moreover we give precise conditions for non-emptiness of the moduli spaces. Finally we study the variation of the moduli spaces with α.

Clifford Indices for Vector Bundles on Curves

For smooth projective curves of genus g ≥ 4, the Clifford index is an important invariant which provides a bound for the dimension of the space of sections of a line bundle. This is the first step in distinguishing curves of the same genus. In this paper we generalise this to introduce Clifford indices for semistable vector bundles on curves. We study these invariants, giving some basic properties and carrying out some computations for small ranks and for general and some special curves. For curves whose classical Clifford index is two, we compute all values of our new Clifford indices.

COHERENT SYSTEMS OF GENUS 0

In this paper we begin the classification of coherent systems (E,V) on the projective line which are stable with respect to some value of a parameter α. In particular we show that the moduli spaces, if non-empty, are always smooth and irreducible of the expected dimension. We obtain necessary conditions for non-emptiness and, when dim V=1 or 2, we determine these conditions precisely. We also obtain partial results in some other cases.

Moduli Spaces of Coherent Systems of Small Slope on Algebraic Curves

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 0 < d ≤ 2n. We show that these spaces are irreducible whenever they are nonempty and obtain necessary and sufficient conditions for nonemptiness.

ON AN EXAMPLE OF MUKAI

In this note we use an example of Mukai to construct semistable bundles of rank 3 with 6 independent sections on a curve of genus 9 or 11 with Clifford index strictly less than the Clifford index of the curve. The example also allows us to show the non-emptiness of some Brill-Noether loci with negative expected dimension.

Vector Bundles of Rank 2 Computing Clifford Indices

Clifford indices of vector bundles on algebraic curves were introduced in a previous article of the authors. In this article we study bundles of rank 2 which compute these Clifford indices. This is of particular interest in the light of recently discovered counterexamples to a conjecture of Mercat.

Nonemptiness of Brill–Noether Loci in M(2, K)

Let C be a smooth projective complex curve of genus g ≥ 2. We investigate the Brill–Noether locus consisting of stable bundles of rank 2 and canonical determinant having at least k independent sections. Using the Hecke correspondence, we construct a fundamental class, which determines the nonemptiness of this locus at least when C is a Petri curve. We prove that in many expected cases the Brill–Noether locus is nonempty. For some values of k, the result is best possible.