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Coherent sheaves on rational curves with simple double points and transversal intersections. (English) Zbl 1065.18009
The authors first review work of themselves and others [for example, Yu. A. Drozd and G.-M. Greuel, J. Algebra 246, 1–54 (2001; Zbl 1065.14041)] on the classification of indecomposable vector bundles on chains and cycles of projective lines (where adjacent curves in the configuration intersect transversally). Then they introduce (complicated!) combinatorial objects called strings and bands, and prove (Theorem 3.2) that there is a one-to-one correspondence between isomorphism classes of indecomposable objects in the derived category \(D^{-}(Coh_X)\) and equivalence classes of strings and bands, where \(X\) is a cycle of projective lines and \(Coh_X\) is the category of coherent sheaves on \(X\). Later this result is used to describe the various types of coherent sheaves on these varieties (including the case of a cycle of length one, which is the rational curve with one node).

MSC:
18E30 Derived categories, triangulated categories (MSC2010)
14H60 Vector bundles on curves and their moduli
16G20 Representations of quivers and partially ordered sets
34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations
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