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Journal of the London Mathematical Society Advance Access originally published online on November 16, 2007
Journal of the London Mathematical Society 2008 77(1):15-32; doi:10.1112/jlms/jdm089
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© 2007 London Mathematical Society

Fourier–Mukai transforms for coherent systems on elliptic curves

Daniel Hernández Ruipérez and Carlos Tejero Prieto

Departamento de Matemáticas and
Instituto de Física Fundamental y Matemáticas
Universidad de Salamanca
Plaza de la Merced 1–4
37008 Salamanca
Spain
ruiperez{at}usal.es

We determine all the Fourier–Mukai transforms for coherent systems consisting of a vector bundle over an elliptic curve and a subspace of its global sections, showing that these transforms are indexed by positive integers. We prove that the natural stability condition for coherent systems, which depends on a parameter, is preserved by these transforms for small and large values of the parameter. By means of the Fourier–Mukai transforms we prove that certain moduli spaces of coherent systems corresponding to small and large values of the parameter are isomorphic. Using these results we draw some conclusions about the possible birational type of the moduli spaces. We prove that for a given degree d of the vector bundle and a given dimension of the subspace of its global sections there are at most d different possible birational types for the moduli spaces.

2000 Mathematics Subject Classification 14D20, 14H60, 14J60, 14H52.

This research has been partially supported by the research projects BFM2003-00097 of the Spanish MEC, SA114/04 of the ‘Junta de Castilla y León’ and by the European Scientific Exchange Programme ‘Geometry and topology of moduli spaces’ of the Royal Society of London and the Spanish CSIC under grant 15646.

Received March 10, 2006; revised March 14, 2007; published online November 16, 2007.


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