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On the reducibility of moduli spaces of stable 2-bundles over an elliptic surface. (English) Zbl 0744.14007

We consider the problem that the moduli spaces of stable 2-bundles \(E\) over a generic regular elliptic surface \(S\) with \(\hbox{det} (E)\simeq{\mathcal O}_ S\) are reducible for \(c_ 2(E)\) sufficiently large if \(S\) contains multiple fibres. Using the work of R. Friedman [Invent. Math. 96, No. 2, 283-332 (1989; Zbl 0671.14006)] we solve this problem in the cases when \(S\) has exactly one multiple fibre.

MSC:

14D20 Algebraic moduli problems, moduli of vector bundles
14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
14J27 Elliptic surfaces, elliptic or Calabi-Yau fibrations

Citations:

Zbl 0671.14006
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References:

[1] [D] Donaldson, S.K.: Polynomial invariants for smooth four-manifolds. Topology29, 257–315 (1990) · Zbl 0715.57007 · doi:10.1016/0040-9383(90)90001-Z
[2] [F] Friedman, R.: Rank two vector bundles over regular elliptic surfaces. Invent. Math.96, 283–332 (1989) · Zbl 0671.14006 · doi:10.1007/BF01393965
[3] [FM] Friedman, R., Morgan, J.W.: On the diffeomorphism types of certain algebraic surfaces I. J. Differ. Geom.27, 297–369 (1988) · Zbl 0669.57016
[4] [M] Mong, K.C.: Polynomial invariants for 4-manifolds of type (1,n) and a calculation forS 2{\(\times\)}S2. Q. J. Math., Oxf. (in press)
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