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Hilbert stability of rank-two bundles on curves. (English) Zbl 0573.14005

Let k be an algebraically closed field, let g and d be integers with \(g\geq 2\) and d sufficiently large (the authors take \(d\geq 1000g(g-1))\), let W be a vector space of dimension \(n=d+2-2g\) over k, let G be the Grassmannian of all subspaces of W of codimension 2, and let E be the universal bundle of rank 2 on G. For any curve C of genus g and degree d on G, there is a natural map \(\phi_ 1: W=H^ 0(G,E)\to H^ 0(C,E_ C)\), where \(E_ C\) denotes the restriction of E to C. If m is an integer such that dim \(H^ 0(C,(\det E_ C)^ m)=P(m)=dm+1-g\), one can choose an isomorphism of \(\bigwedge^{P(m)}H^ 0(C,(\det E_ C)^ m)\) with k, and \(\phi_ 1\) then determines a linear map \(\phi^ m_ C: \bigwedge^{P(m)}S^ m(\bigwedge^ 2W)\to k\). We say that C is m- Hilbert stable (m-Hilbert semi-stable) if \(\phi^ m_ C\) is properly stable (semi-stable) under the action of SL(W) on the space of all such linear maps. The authors prove that, for given g and d with d sufficiently large, (i) there exists M such that, if \(m\geq M\) and C is smooth with \(\phi_ 1\) an isomorphism, then C is m-Hilbert stable (semi- stable) if and only if \(E_ C\) is a stable (semi-stable) bundle on C, and (ii) there exists M such that, if \(m\geq M\) and C is m-Hilbert semi- stable, then C is semi-stable as a curve and \(\phi_ 1\) is an isomorphism. In this second case, the bundle \(E_ C\) has one of a small number of possible forms described in terms of its restrictions to the components of C.
The concept of stability introduced here can be used to construct a compactification of the moduli space of bundles of rank 2 and odd degree d on an algebraic curve with a node. This compactification differs from that given by the Mumford-Seshadri theory; its construction sheds light on the way in which the moduli space in the smooth case can degenerate. [For more details, see a recent paper of the second named author in J. Diff. Geom. 19, 173-206 (1984).]
Reviewer: P.E.Newstead

MSC:

14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
14D20 Algebraic moduli problems, moduli of vector bundles
14H10 Families, moduli of curves (algebraic)
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