Stable sheaves on elliptic fibrations

We characterize the subscheme of the moduli space of torsion-free sheaves on an elliptic surface which are stable of relative degree zeero (with respect to a polarization of type aH+bf, H being the section and f the elliptic fibre) which is isomorphic, via the relative Fourier-Mukai transform, with the relative compactified Simpson Jacobian of the family of those curves D in the surface which are flat over the base of the elliptic fibration. This generalizes and completes earlier constructions due to Friedman, Morgan and Witten. We also study the relative moduli scheme of sheaves whose restriction to each fibre is torsion-free and semistable of rank n and degree zero for higher dimensional elliptic fibrations. The relative Fourier-Mukai transform induces an isomorphic between this relative moduli space and the relative n-th symmetric product of the fibration.Comment: AMS-LaTeX, 18 pages, XY-pic; new title, some modifications; final version as accepted in J. Geom. Phy

The role of CSCL pedagogical patterns as mediating artefacts for repurposing Open Educational Resources

Designing effective CSCL processes is a complex task that can be supported by existing good practices formulated as pedagogical patterns. From a cultural historical activity theory (CHAT) perspective previous research has shown that patterns served as Mediating Artefacts (MA) helping practitioners to make informed decisions and choices, being much closer to the practitioners' mindsets than complex learning design models, such as IMS-LD. However, a new challenge arises when the starting design element corresponds to Open Educational Resources (OER), i.e. free resources of high quality that are typically employed for individual learning. Recent research reported in this chapter has aimed to analyze the eventual contribution of CSCL patterns such as Collaborative Learning Flow Patterns (CLFP) in the repurposing process of existing OER for collaborative learning. Preliminary evidence coming from a set of workshops with educational technology experts shows that a small set of patterns drawn from a CSCL pattern language together with other MA, such as visual representations of Learning Designs, may be inspirational and effective in repurposing existing OER. Further research is under development that builds on the successful workshop format and involves practitioners in face-to-face and virtual workshops. This new set of experiences aims to analyze the effectiveness of the pedagogical patterns and other complementary MA in helping practitioners exploit the great potential of OER in the framework of the Open Learning Network (OLnet) project funded by The William and Flora Hewlett Foundation