Mathematics – Algebraic Geometry
Scientific paper
2008-06-12
Mathematics
Algebraic Geometry
26 pages, 4 figures. Added the structure of the biggest component of the moduli space of sheaves of degree 0 on a cycle of pro
Scientific paper
10.1093/imrn/rnp094
We find some equivalences of the derived category of coherent sheaves on a Gorenstein genus one curve that preserve the (semi)-stability of pure dimensional sheaves. Using them we establish new identifications between certain Simpson moduli spaces of semistable sheaves on the curve. For rank zero, the moduli spaces are symmetric powers of the curve whilst for a fixed positive rank there are only a finite number of non-isomorphic spaces. We prove similar results for the relative semistable moduli spaces on an arbitrary genus one fibration with no conditions either on the base or on the total space. For a cycle $E_N$ of projective lines, we show that the unique degree 0 stable sheaves are the line bundles having degree 0 on every irreducible component and the sheaves $\mathcal{O}(-1)$ supported on one irreducible component. We also prove that the connected component of the moduli space that contains vector bundles of rank $r$ is isomorphic to the $r$-th symmetric product of the rational curve with one node.
Gómez Darío Sánchez
Lopez Martin Ana Cristina
Prieto Carlos Tejero
Ruiperez Daniel Hernandez
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