  We study the moduli space of stable sheaves of Euler characteristic 1
supported on curves of arithmetic genus 3 contained in a smooth quadric
surface. We show that this moduli space is rational. We compute its Betti
numbers by studying the variation of the moduli spaces of alpha-semi-stable
pairs. We classify the stable sheaves using locally free resolutions or
extensions. We give a global description: the moduli space is obtained from a
certain flag Hilbert scheme by performing two flips followed by a blow-down.
