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Vector bundles and torsion free sheaves on degenerations of elliptic curves. (English) Zbl 1111.14027

Catanese, Fabrizio (ed.) et al., Global aspects of complex geometry. Berlin: Springer (ISBN 3-540-35479-4/hbk). 83-128 (2006).
This is a survey paper which gives an account of the classification of vector bundles and torsion free sheaves on projective curves of arithmetic genus one which have certain types of singularities. The authors present together some results contained in previous joint works of some of them. In the second part some classical results about vector bundles on projective smooth curves are recalled. In particular Atiyah’s classical results about indecomposable vector bundles on elliptic curves are presented as an application of the Fourier-Mukai transform. The approach is similar to the one contained in A. Polishchuk’s book “Abelian varieties, theta functions, and the Fourier transform” [Cambridge Tracts in Mathematics. 153 (2003; Zbl 1018.14016)] and to the one described by G. Hein and D. Ploog [Beitr. Algebra Geom. 46, No. 2, 423–434 (2005; Zbl 1093.14047)]. For the case of higher genus previous work of Y. A. Drozd and G.-M. Greuel [J. Algebra 246, No. 1, 1–54 (2001; Zbl 1065.14041)] is reported.
The third part is devoted to the case of singular projective curves of arithmetic genus one. The approach considered in this section is to study sheaves on a singular curve in terms of sheaves on the normalization. A particular case is that of the classification of indecomposable vector bundles and torsion sheaves on cycles and chains of projective lines is given in terms of Bondarenko’s algorithm of bands and strips, which is described out of work of the authors. This gives the classification of indescomposable vector bundles on a elliptic nodal curve; thanks to the fact that on such a curve, any coherent simple sheaf is stable, stable sheaves are also described. Furthermore, a description of stable bundles on a elliptic cuspidal curve is given.
In the last part the paper reports the application of the Fourier-Mukai transform on an elliptic integral singular curve to obtain more information on the moduli spaces of semiestable and stable sheaves on such curves. Using the fact that the Fourier-Mukai transform preserves stability and semistability for nonzero degree (a result proven by the first time by C. Bartocci, U. Bruzzo, J. M. Muñoz Porras and the reviewer [Math. Nachr. 238, 23–36 (2002; Zbl 1033.14007)]), isomorphisms between different moduli spaces can be established. Moreover, one can describe the moduli space of semistable sheaves of degree zero as the moduli of skyscraper sheaves. Using the classification of skyscraper sheaves supported on the unique singular point, the authors describe semistable sheaves of degree zero on both an elliptic nodal curve and an elliptic cuspidal curve, showing the differences between both cases. The differences come from the differences between the categories of modules of finite length over the completions of the local ring at the singular point in the nodal and in the cuspidal curve; these categories has been studied by Y. A. Drozd [Funct. Anal. Appl. 6, 286–288 (1972); translation from Funkts. Anal. Prilozh. 6, No. 4, 41–43 (1972; Zbl 0289.13009)]. It should be mentioned that the classification of stable or semistable vector bundles on more general singular elliptic curves (like other types of Kodaira fibres) is a difficult problem. In this more general case the classification depends on the polarization. The description of semistable torsion free sheaves of polarized rank one has been made by A. C. López [J. Geom. Phys. 56, No. 3, 375–385 (2006; Zbl 1085.14034) and J. Reine Angew. Math. 582, 1–39 (2005; Zbl 1078.14033)].
For the entire collection see [Zbl 1099.14001].

MSC:

14H60 Vector bundles on curves and their moduli
14H52 Elliptic curves
14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
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