Fourier-Mukai transforms and semi-stable sheaves on nodal Weierstrass cubics

Abstract

We completely describe all semi-stable torsion free sheaves of degree zero on nodal cubic curves using the technique of Fourier-Mukai transforms. The Fourier-Mukai images of such sheaves are torsion sheaves of finite length, which we compute explicitly. We show how to obtain an SL(2,Z)-action (up to shifts) on the bounded derived category of coherent sheaves on any Weierstass cubic.