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Coherent systems of genus 0. (English) Zbl 1072.14039

Coherent systems are pairs \((E,V)\) where \(E\) is a vector bundle of rank \(n\) and degree \(d\) over an algebraic curve, say \(C\), of genus \(g\), and \(V\) is a dimension \(k\) vector subspace of \(H^0(E)\). Interest in coherent systems stems from the fact that they admit geometrically and topologically rich moduli spaces which, moreover, relate in a useful way to higher rank Brill-Noether loci.
The moduli spaces, which parameterize equivalence classes of (semi)-stable coherent systems, depend on \(n,d,k\) and also on a parameter \(\alpha\) which appears in the definition of (semi)stability. These moduli spaces, denoted by \(G(\alpha; n,k,d)\), and their applications to Brill-Noether theory have previously been studied in [S. B. Bradlow, O. Garcia-Prada, V. Munoz, P. E. Newstead, Int. J. Math. 14, No. 7, 683–733 (2003; Zbl 1057.14041)]. Most of the specific results to date require \(C\) to have genus at least 2 and thus exclude the case in which \(C\) is the projective line. The paper under review addresses this omitted case.
The dominant special feature of the genus zero case is that all bundles decompose as sums of line bundles. Other than line bundles, there are thus no stable bundles over \(\mathbb P^1\), and neither is semistability possible unless the degree is divisible by the rank. The main general result in the paper under review is that whenever non-empty, \(G(\alpha; n,k,d)\) is smooth, irreducible and of the expected dimension (based on infinitesmal deformation theory). The authors also give explicit descriptions of the general elements in these moduli spaces and provide necessary and sufficient non-emptiness conditions for \(G(\alpha; n,k,d)\) in the cases that \(k=1\) or \(k=2\).
Notable differences between the \(g=0\) and the \(g>1\) cases can be seen in the dependence of \(G(\alpha; n,k,d)\) on \(\alpha\). For fixed \((n,d,k)\) with \(k<n\), if \(G(\alpha; n,k,d)\) is non-empty then \(\alpha\) lies in a bounded interval. If \(g>1\) then the allowed range for \(\alpha\) is from \(0\) to \(d/(n-k)\). If \(g=0\) the interval may be truncated at both ends, depending on the conjugacy class of \(d\) modulo \(n\) and modulo \((n-k)\). This is related to the fact that, as shown in this paper, there can be non-empty \(G(\alpha; n,k,d)\) even though there are no semistable bundles of the given rank and degree.

MSC:

14H60 Vector bundles on curves and their moduli
14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
32L10 Sheaves and cohomology of sections of holomorphic vector bundles, general results

Citations:

Zbl 1057.14041
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References:

[1] Atiyah M. F., Proc. London Math. Soc. 7 pp 414–
[2] DOI: 10.1112/S002461079900767X · Zbl 0954.32014 · doi:10.1112/S002461079900767X
[3] Bradlow S. B., J. Reine Angew. Math. 551 pp 123–
[4] Bradlow S. B., Int. J. Math. 14 pp 1–
[5] DOI: 10.1142/S0129167X94000024 · Zbl 0799.32022 · doi:10.1142/S0129167X94000024
[6] DOI: 10.1016/0022-4049(77)90010-X · Zbl 0374.55012 · doi:10.1016/0022-4049(77)90010-X
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